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  1. <?xml version="1.0" encoding="iso-8859-1"?>
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  3.  <title>The String Coffee Table</title>
  4.  <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/" />
  5.  <link rel="self" href="http://golem.ph.utexas.edu/string/atom10.xml" />
  6.  <link rel="replies" type="application/atom+xml" href="http://golem.ph.utexas.edu/string/comments.atom"/>
  7.  <updated>2006-12-05T15:03:56Z</updated>
  8.  <subtitle>A Group Blog on Physics</subtitle>
  9.  <id>tag:golem.ph.utexas.edu,2004:StringCoffeeTable/2</id>
  10.  <generator uri="http://www.movabletype.org/" version="3.36">Movable Type</generator>
  11.  <icon>http://golem.ph.utexas.edu/string/images/favicon.ico</icon>
  12.  <rights>Copyright (c) 2006, String Coffee Table Collective</rights>
  13.  <entry>
  14.    <title type="html">Postdoctoral Position at the interface of Algebra, Conformal Field Theory and String Theory</title>
  15.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/001064.html" />
  16.    <updated>2006-12-05T15:03:56Z</updated>
  17.    <published>2006-12-05T15:02:25+00:00</published>
  18.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.1064</id>
  19.    <summary type="text">Postdoc position in algebra, CFT and strings.</summary>
  20.    <author>
  21.      <name>urs</name>
  22.      <uri>http://www.math.uni-hamburg.de/home/schreiber</uri>
  23.      <email>[email protected]</email>
  24.    </author>
  25.    
  26.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/001064.html">
  27.      <div xmlns="http://www.w3.org/1999/xhtml">
  28. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  29. <p>The Mathematics Department of the <a href="http://www.uni-hamburg.de/index_e.html">University of Hamburg</a> has a postdoctoral position
  30. available in the area of  <strong>Algebra, Conformal Field Theory and String Theory</strong> which is part of the Collaborative Research Centre 676 <a href="http://golem.ph.utexas.edu/string/archives/000783.html">&#8220;Particles, Strings and the Early Universe:
  31. the Structure of Matter and Space-Time&#8221;</a> funded by the German Science Foundation
  32. (DFG).</p>
  33.  
  34. <p>The position starts in the <strong>fall of 2007</strong> and is for a period of 2 years with the possibility of an extension for an additional year. The candidate is expected to do research at the
  35. interface of Algebra, Conformal Field Theory and String Theory. </p>
  36.  
  37. <p>Applicants must have a PhD in Theoretical Physics or Mathematics.</p>
  38.  
  39. <p>See the <a href="http://www.math.uni-hamburg.de/home/schreiber/Stelle.pdf">full announcement</a>.</p>
  40.  
  41. <p><em>Further links:</em></p>
  42.  
  43. <p><a href="http://www.math.uni-hamburg.de/home/schweigert/">Prof. Ch. Schweigert&#8217;s homepage</a></p>
  44.  
  45. <p><em>Some entries discussing the group&#8217;s work:</em></p>
  46.  
  47. <p><a href="http://golem.ph.utexas.edu/string/archives/000813.html">The FRS Theorem on <abbr title="Rational Conformal Field Theory">RCFT</abbr></a></p>
  48.  
  49. <p><a href="http://golem.ph.utexas.edu/string/archives/000747.html">FRS Reviews</a></p>
  50.  
  51. <p><a href="http://golem.ph.utexas.edu/string/archives/000708.html">Unoriented Strings and Gerbe Holonomy</a></p>
  52.  
  53. <p><a href="http://golem.ph.utexas.edu/string/archives/000783.html">some projected research</a></p>
  54.  
  55.      </div>
  56.    </content>
  57.  </entry>
  58.  <entry>
  59.    <title type="html">More polymer oscillators</title>
  60.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/001011.html" />
  61.    <updated>2006-11-01T10:03:08Z</updated>
  62.    <published>2006-10-31T15:00:40+00:00</published>
  63.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.1011</id>
  64.    <summary type="text"> The same day as the "Lessons from the LQG string" appeared on hep-th, there was another paper by Corichi,...</summary>
  65.    <author>
  66.      <name>robert</name>
  67.      <uri>http://atdotde.de/</uri>
  68.      <email>[email protected]</email>
  69.    </author>
  70.    
  71.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/001011.html">
  72.      <div xmlns="http://www.w3.org/1999/xhtml">
  73. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  74.  
  75. <p>The same day as the <a href="http://arxiv.org/abs/hep-th/0610193">&#8220;Lessons from the <abbr title="Loop Quantum Gravity">LQG</abbr> string&#8221;</a> appeared on hep-th, there was <a href="http://arxiv.org/abs/gr-qc/0610072">another paper by Corichi, Vukasniac and Zapata</a> crosslisted from gr-qc discussing the loopy oscillator and coming to conclusions which at first sight comes just to the opposite conclusions. Their abstract starts out with &#8220;<em>In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics.</em>&#8221; while I derived that at high frequencies the absorption spectrum of the polymer oscillator is quite distinct from the usual Fock/Schr&#x000F6;dinger version. How could this be?</p>
  76.  
  77. <p>Luckily, their paper is written in a very clear manner and free from the notational ballast which makes many <abbr>LQG</abbr> papers hard to read. With only a brief read one can find the resolution: The two papers are doing different things. That&#8217;s not too surprising. Let me spell this out in a bit more detail. In one sentence: I tried to take the polymer oscillator literally and work out the conclusions from what I am given while they apply some limiting/regularisation/renormalisation procedure to the system to finally end up with the usual Fock description.</p>
  78.  
  79. <p>As a warning I should say that what I am going to present here is probably some kind of caricature of their paper. I have had some email exchange with the authors from Mexico and they have been very helpful and responded to many questions and I am extremely thankful. What I write here is the result of my learning process but might not be the way they would summarise their paper. So: All the errors in this presentation are mine! </p>
  80.  
  81. <p>The first difference is that the two papers start from different polymer Hilbert spaces: In my case, I have a basis labelled by points in phase space and the Weyl operators by translations in the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> directions. Because of the singular scalar product these actions are not continuous and neither <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> nor <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> exist as operators, only <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>e</mi> <mi>iaX</mi></msup></mrow><annotation encoding='application/x-tex'>e^{iaX}</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>e</mi> <mi>ibP</mi></msup></mrow><annotation encoding='application/x-tex'>e^{ibP}</annotation></semantics></math> (for real <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>a,b</annotation></semantics></math>). This has the advantage that the classical time evolution translates directly to a unitary operator in that Hilbert space: It just rotates the phase space by an angle proportional to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math>.</p>
  82.  
  83. <p>They start with a Hilbert space where a basis is labelled by points on the real line and there is an <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> operator which even has normalisable eigenfunctions. However, there is still no <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> operator but what would be <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>e</mi> <mi>ibP</mi></msup></mrow><annotation encoding='application/x-tex'>e^{ibP}</annotation></semantics></math> acts by translations by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>. This Hilbert space does not come with a nice time evolution of the oscillator but we will see below what they do instead.
  84. However, this difference is I think only technical and does not really matter in the following.</p>
  85.  
  86. <p>Then they go through some mathematically involved (projective) limiting procedure and play the &#8220;go to the dual space&#8221;-game several times. The result is that they pick a squence of subspaces of countable dimension, namely at stage <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> they consider only the span of vectors over points of the form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>m</mi><mo stretchy="false">/</mo><msup><mn>2</mn> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>m/2^n</annotation></semantics></math> for integer <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>m</mi></mrow><annotation encoding='application/x-tex'>m</annotation></semantics></math>. These are in one to one correspondance to characteristic functions of the interval <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">/</mo><msup><mn>2</mn> <mi>n</mi></msup><mo>,</mo><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mn>2</mn> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(m/2^n,(m+1)/2^n)</annotation></semantics></math> in the usual Hilbert space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>L^2(R)</annotation></semantics></math>. This mapping however is not in isometry: In <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>L^2(R)</annotation></semantics></math> these characteristic functions have a norm given by their length, i.e. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>n</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>2^{&#x2d;n}</annotation></semantics></math> while they have norm 1 in the polymer space. Now comes the trick: You redefine (&#8220;renormalise&#8221;) the norm on the polymer side by copying the norm form the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>L^2(R)</annotation></semantics></math> side. When doing this you should remind yourself that the norm/scalar product is where the choice of state showed up in the GNS construction. So, by redefining the norm you effectively revise your choice of state. And the two descriptions (Fock and polymer) only differed by the choice of state&#8230;</p>
  87.  
  88. <p>At each finite stage of this regularisation procedure, you have broken most translation operators <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>e</mi> <mi>ibP</mi></msup></mrow><annotation encoding='application/x-tex'>e^{ibP}</annotation></semantics></math>, only the ones for which <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mrow><msup><mn>2</mn> <mi>n</mi></msup></mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>{2^n}b</annotation></semantics></math> is an integer survive. But you can use those to come up with finite difference versions <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>P</mi> <mi>fd</mi></msub></mrow><annotation encoding='application/x-tex'>P_{fd}</annotation></semantics></math> for what would be the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> operator and use it to define a regularised oscillator Hamiltonian <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>H</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msubsup><mi>P</mi> <mi>fd</mi> <mn>2</mn></msubsup><mo>+</mo><msup><mi>X</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>H=\frac{1}{2}(P_{fd}^2+X^2)</annotation></semantics></math>. </p>
  89.  
  90. <p>Finially you take the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi><mo>&#x02192;</mo><mn>&#x0221E;</mn></mrow><annotation encoding='application/x-tex'>n\to\infty</annotation></semantics></math> limit everywhere. To nobodies&#8217; surprise you end up with the usual Hamiltonian in the usual Fock space. Strictly speaking, you have only defined the operators <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>e</mi> <mi>ibP</mi></msup></mrow><annotation encoding='application/x-tex'>e^{ibP}</annotation></semantics></math> for those rational <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math> which have a denominator which is a power of two. But as you are taking limits anyway, you can use these and continuity to define them for all <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>. Of course, as Drs Stone and von Neumann have told you long ago, there is no other continious choice of representation of Weyl operators than the standart one.</p>
  91.  
  92. <p>So what do we learn? As Giuseppe put it to me (of course with his better manners in more polite words): Both papers agree that the polymer representation sucks. In my paper, I show that how much it sucks and in their paper they show how you can redefine it away and proceed to the usual Fock space.</p>
  93.  
  94. <p>But I should warn you, dear reader: The original motivation for considering polymer representations at all (not so much for the oscillator but for gauge theories and gravity) was that it gives an easy (trivial) implementation of diffeomorphism symmetries. This is a central part of all this &#8220;background independance&#8221; stuff. </p>
  95.  
  96. <p>But the procedure these people suggest is to introduce a regulator (the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><msup><mn>2</mn> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>1/2^n</annotation></semantics></math> equal partitioning), do the calculation and then remove the regulator. This regulator is nothing but a background! And it breaks many of the nice symmetries you wanted to maintain. </p>
  97.  
  98. <p>So there are two obvious questions: 1) In systems more involved than the harmonic oscillator (which is just the free theory in 0+1 dimensions), is it possible to renormalise scalar products and operators in a way that the limit exists? This question is like the continuum limit for a lattice regularisation: In nice theories (like <abbr title="Quantum ChromoDynamics">QCD</abbr>) it exists, in other cases there is no good continuum limit like for example QED, because the theory is not asymptotically free. And in the case of gravity I would be worried that the well known non-renormalisability (in the usual treatment) shows up when you try to remove the regulator and find the whole thing exploding. </p>
  99.  
  100. <p>But let&#8217;s assume for a second this problem does not occur or you have found a way to solve it. Then there is still question 2),  the anomalies: The regularisation has broken many essential symmetries. Thus it is non-trivial that these reappear in the continuum limit. And we know: In general they don&#8217;t. The polymer state didn&#8217;t have this problem as it preserved the symmetries. But now they are explicitly broken. So you are thrown back to the situation of the conventional treatment (with a <abbr title="UltraViolet">UV</abbr> cut-off say). If you don&#8217;t believe this, you are welcome to upgrade the content of the paper to the case of the bosonic string and show how Diff(<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math>) reappears in the continuum limit.</p>
  101.  
  102.  
  103.      </div>
  104.    </content>
  105.  </entry>
  106.  <entry>
  107.    <title type="html">Lessons from the LQG string</title>
  108.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000990.html" />
  109.    <updated>2006-10-18T06:33:20Z</updated>
  110.    <published>2006-10-17T15:15:13+00:00</published>
  111.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.990</id>
  112.    <summary type="text">It&#8217;s now two years, that Giuseppe and I have put out out our paper comparing the usual quantisation of the...</summary>
  113.    <author>
  114.      <name>robert</name>
  115.      <uri>http://atdotde.de/</uri>
  116.      <email>[email protected]</email>
  117.    </author>
  118.    <category term="LQG" />
  119.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000990.html">
  120.      <div xmlns="http://www.w3.org/1999/xhtml">
  121. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div><p>It&#8217;s now two years, that Giuseppe and I have put out out our <a href="http://arxiv.org/abs/hep-th/0409182">paper</a> comparing the usual quantisation of the bosonic string to Thiemann&#8217;s loop inspired version. A bit to my surprise, that paper was of interest to a number of people and the months afterwards I was lucky to tour half of Europe to give seminars about it (in that respect it was my most successful paper ever; the only talk I have given more often is my popular science talk <a href="http://www.aei.mpg.de/~helling/startrek/">&#8220;Phaser, Wurmloch, Warpantriebe&#8221;</a> about physics with a Star Trek spin prepared for the Max Planck society public outreach).</p>
  122.  
  123. <p>That paper had quite a resonance in the blogosphere as well, but its results have not always been presented in a way we intended them. This might also be because the paper was in large parts quite technical and some of the main messages were burried in mathematical arguments.</p>
  124.  
  125. <p>So I thought it might be a good idea to put out <a href="http://arxiv.org/abs/hep-th/0610193">a &#8220;mainly prose&#8221; version</a> of the argument which leaves out the technicalities to bring home the main messages. This I did and you should be able to find it on hep-th as you read this.</p>
  126.  
  127. <p>Remember the philosophy of this investigation: The loopy people always insist that diffeomorphism invariance is so central to gravity that it is important to build it into a theory of quantum gravity right from the beginning and all the problems one has with perturbatively quantising <span class="caps">GR</span> are due to ignoring this important symmetry or at least not building it into the formalism but expanding around some background.</p>
  128.  
  129. <p>As <span class="caps">GR</span> is a complicated interacting theory it is easy to get lost in the technical difficulties and one should consider simpler examples to test such claims.</p>
  130.  
  131. <p>The world sheet theory of the bosonic string is such an example as it is extremely simple being a free theory but still has an infinite dimensional symmetry of diffeomorphisms of the lightcone coordinates. It is thus the ideal testbed for approaches to diffeomorphism invariant theories where one can compute everything and check if it makes sense.</p>
  132.  
  133. <p>The first part of today&#8217;s paper explains all this and shows that the difference in the treatments can be summarised by saying that the usual Fock space quantisation of the string uses a Hilbert space built upon a <em>covariant</em> state whereas the loopy approach insists on <em>invariance</em> of that state which is a much stronger requirement.</p>
  134.  
  135. <p>My point is that covariance is the property which is physically required (and in fact states in the classical field theory are covariant but not invariant) and thus statements like the <a href="http://arxiv.org/abs/gr-qc/0504147"><span class="caps">LOST</span> theorem</a> have too strict assumtions.</p>
  136.  
  137. <p>If you insists on invariance you end up with a Hilbert space representation which is not continuous as this is what <span class="caps">LOST</span> like theorems tell you. The question now is if this discontinuity makes your theory useless as a quantum theory. Well, everybody is free to set up the rules of the game they call &#8220;quantisation&#8221; and in the end only theories which do not disagree with experiments are good theories. But as we are all well aware, there are not too many experiments performed today which study properties of quantum gravity or bosonic string and thus this test is not available for the time being.</p>
  138.  
  139. <p>A weaker test would be to apply your rules of quantisation to other systems which are available for experimentation and see what they give there. Thus the second part (as in the original paper with Giuseppe) deals with a loop inspired quantisation of the harmonic oscillator. The old paper was criticised for providing a solid argument that it is observationally possible to distinguish the loopy oscillator from the Fock oscillator.</p>
  140.  
  141. <p>The second part of the new paper I think provides such an argument: It couples the oscillator to an electromagnetic radiation field and computes the absorption spectrum. Remember that usually the absorption for a transition between states <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">&#x0007C;</mo><mi>m</mi><mo stretchy="false">&#x027E9;</mo></mrow><annotation encoding='application/x-tex'>|m\rangle</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">&#x0007C;</mo><mi>m</mi><mo>&#x02032;</mo><mo stretchy="false">&#x027E9;</mo></mrow><annotation encoding='application/x-tex'>|m&apos;\rangle</annotation></semantics></math> goes like</p>
  142.  
  143. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>&#x003A9;</mi><mo>&#x02212;</mo><msub><mi>&#x003C9;</mi> <mi>m</mi></msub><mo>+</mo><msub><mi>&#x003C9;</mi> <mrow><mi>m</mi><mo>&#x02032;</mo></mrow></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow></mfrac><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>\frac{1}{(\Omega&#x2d;\omega_m+\omega_{m&apos;})^2}\, .</annotation></semantics></math></p>
  144.  
  145. <p>Here, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003A9;</mi></mrow><annotation encoding='application/x-tex'>\Omega</annotation></semantics></math> is the frequency of the radiation.  Now, the loopy result is proportional to</p>
  146.  
  147. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mfrac><mn>1</mn><mrow><mi>sin</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>&#x003A9;</mi><mo>&#x02212;</mo><mi>m</mi><mo>+</mo><mi>m</mi><mo>&#x02032;</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>N</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding='application/x-tex'>\frac{1}{\sin((\Omega&#x2d;m+m&apos;)/N)^2}</annotation></semantics></math></p>
  148.  
  149. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> is a large natural number characterising the states. Thus if <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003A9;</mi><mo>&#x0226A;</mo><mi>N</mi></mrow><annotation encoding='application/x-tex'>\Omega\ll N</annotation></semantics></math> the two expressions agree but for large <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003A9;</mi></mrow><annotation encoding='application/x-tex'>\Omega</annotation></semantics></math> they don&#8217;t (don&#8217;t worry about an overall constant).</p>
  150.  
  151. <p>Thus if I am only allowed to measure within a finite frequency band for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003A9;</mi></mrow><annotation encoding='application/x-tex'>\Omega</annotation></semantics></math> the states can be made similar by choosing <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> large enough. But once that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> is chosen the experimenter can reveal the difference by studying the behaviour at large frequencies.</p>
  152.  
  153. <p>So are they the same or not? Well, that&#8217;s a long story for which you have to read the paper.</p>
  154.  
  155. <p>After you&#8217;ve done that, you can come back here and comment.</p>
  156.  
  157.      </div>
  158.    </content>
  159.  </entry>
  160.  <entry>
  161.    <title type="html">The Master constraint program in LQG</title>
  162.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000915.html" />
  163.    <updated>2006-09-02T05:19:10Z</updated>
  164.    <published>2006-09-02T05:20:32+00:00</published>
  165.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.915</id>
  166.    <summary type="text">I&#8217;m a little reluctant to post much on the master constraint program because I haven&#8217;t read much on it. But...</summary>
  167.    <author>
  168.      <name>aaron</name>
  169.      <uri>http://zippy.ph.utexas.edu/~abergman/</uri>
  170.      <email>[email protected]</email>
  171.    </author>
  172.    <category term="LQG" />
  173.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000915.html">
  174.      <div xmlns="http://www.w3.org/1999/xhtml">
  175. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div><p>I&#8217;m a little reluctant to post much on the master constraint program because I haven&#8217;t read much on it. But I thought I&#8217;d post this if others want to comment on the subject.</p>
  176.  
  177. <p>My initial question is how does the master constraint program work in classical mechanics? In particular, say we are given some symplectic manifold and some set of constraints. The master constraint is<math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>M</mi><mo>=</mo><msup><mi>K</mi> <mi>IJ</mi></msup><msub><mi>C</mi> <mi>I</mi></msub><msub><mi>C</mi> <mi>J</mi></msub><mo>.</mo></mrow><annotation encoding='application/x-tex'>M = K^{IJ} C_I C_J .</annotation></semantics></math>Using this, how does one obtain the constrained phase space?</p>
  178.  
  179. <p>Or is this the wrong question to ask?</p>
  180.  
  181.      </div>
  182.    </content>
  183.  </entry>
  184.  <entry>
  185.    <title type="html">The Harmonic Oscillator in LQG</title>
  186.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000914.html" />
  187.    <updated>2006-09-02T05:17:39Z</updated>
  188.    <published>2006-09-02T03:11:15+00:00</published>
  189.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.914</id>
  190.    <summary type="text">I&#8217;ve been trying to understand Thomas Thiemann&#8217;s riposte to the papers of Nicolai, Peeters and Zamaklar, Nicolai and Peeters and...</summary>
  191.    <author>
  192.      <name>aaron</name>
  193.      <uri>http://zippy.ph.utexas.edu/~abergman/</uri>
  194.      <email>[email protected]</email>
  195.    </author>
  196.    <category term="LQG" />
  197.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000914.html">
  198.      <div xmlns="http://www.w3.org/1999/xhtml">
  199. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div><p>I&#8217;ve been trying to understand Thomas Thiemann&#8217;s <a href = "http://arxiv.org/abs/hep-th/0608210">riposte</a> to the papers of <a href = "http://www.arxiv.org/abs/hep-th/0501114">Nicolai, Peeters and Zamaklar</a>, <a href = "http://www.arxiv.org/abs/hep-th/0601129">Nicolai and Peeters</a> and <a href = "http://arxiv.org/abs/hep-th/0409182">Helling and Policastro</a>. I&#8217;m fairly busy right now with a paper of my own and moving, so I&#8217;ll concentrate on the part where he describes how <span class="caps"><abbr title="Loop Quantum Gravity">LQG</abbr></span>-quantization replicates the usual quantization of the harmonic oscillator. Maybe later, I can get to trying to understand the master constraint program.</p>
  200.  
  201. <p>I&#8217;ve already posted some comments at <a href = "http://christinedantas.blogspot.com/2006/08/lqg-inside.html">Christine Dantas&#8217;s blog</a>, but I thought I might also try to post them here. I really would love to see some sort of discussion on these points. One of the things I think any scientist should be able to do is to get in front of a chalkboard and be able to communicate a pretty good idea about what&#8217;s going on with their work. Think of this as a long distance chalkboard, and I&#8217;m the skeptical visitor.</p>
  202.  
  203. <p>Anyways, I will try in this post to summarize my understanding of the construction in Thiemann. I hope people will correct me if I get it wrong. (And I hope I get the algebra correct&#8230;.)</p>
  204. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div><p>In <abbr>LQG</abbr>-quantization, we start with a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebra, in this case the algebra given by the exponentiated position and momentum operators. This is often used because <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>q</mi></mrow><annotation encoding='application/x-tex'>q</annotation></semantics></math> are unbounded operators. In particular, we have</p>
  205.  
  206. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>U</mi><mo>=</mo><msup><mi>e</mi> <mi>iap</mi></msup><mspace width="1em"/><mi>V</mi><mo>=</mo><msup><mi>e</mi> <mi>ibq</mi></msup></mrow><annotation encoding='application/x-tex'>U = e^{iap} \quad V = e^{ibq}</annotation></semantics></math></p>
  207.  
  208. <p>and</p>
  209.  
  210. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>UV</mi><mo>=</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>a</mi><mi>b</mi></mrow></msup><mi>VU</mi></mrow><annotation encoding='application/x-tex'>UV = e^{i a b} VU</annotation></semantics></math></p>
  211.  
  212. <p>The Stone-von Neumann theorem tells us that ordinarily all unitary representations of this algebra are unitarily equivalent. In <span class="caps"><abbr>LQG</abbr>,</span> however, we have weakly continuous representations, so we can&#8217;t appeal to this any more. Unfortunately, the usual harmonic oscillator Hamiltonian</p>
  213.  
  214. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>H</mi><mo>=</mo><msup><mi>p</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>q</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>H = p^2 + q^2</annotation></semantics></math></p>
  215.  
  216. <p>isn&#8217;t an element in this <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebra. However, we can define a one parameter family of operators:</p>
  217.  
  218. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>H</mi> <mi>&#x003F5;</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msup><mi>sin</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>&#x003F5;</mi><mi>p</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>sin</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>&#x003F5;</mi><mi>q</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>&#x003F5;</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>H_\epsilon = (sin^2(\epsilon p) + sin^2(\epsilon q)) / \epsilon^2</annotation></semantics></math></p>
  219.  
  220. <p>which are elements in the algebra. Classically, of course, this expression does converge to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> as <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003F5;</mi><mo>&#x02192;</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\epsilon \to 0</annotation></semantics></math>. It would be nice now to determine the spectrum of this operator, but apparently for the usual <abbr>LQG</abbr> representation, we cannot. Thus, let us define the raising and lowing operators</p>
  221.  
  222. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msubsup><mi>a</mi> <mi>&#x003F5;</mi> <mo>&#x02020;</mo></msubsup><mo>=</mo><mo stretchy="false">(</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>q</mi><mi>&#x003F5;</mi><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>p</mi><mi>&#x003F5;</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>&#x003F5;</mi></mrow><annotation encoding='application/x-tex'>a_\epsilon^\dagger = (sin(q\epsilon) + i sin(p\epsilon))/\epsilon</annotation></semantics></math></p>
  223.  
  224. <p>and similarly for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>a</mi> <mi>&#x003F5;</mi></msub></mrow><annotation encoding='application/x-tex'>a_\epsilon</annotation></semantics></math>. It is not true <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>a</mi> <mi>&#x003F5;</mi> <mo>&#x02020;</mo></msubsup><msub><mi>a</mi> <mi>&#x003F5;</mi></msub><mo>+</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>a^\dagger_\epsilon a_\epsilon + 1/2</annotation></semantics></math> is <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>H</mi> <mi>&#x003F5;</mi></msub></mrow><annotation encoding='application/x-tex'>H_\epsilon</annotation></semantics></math>, but it is true to order <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x003F5;</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>\epsilon^2</annotation></semantics></math> if I haven&#8217;t screwed up my algebra. Now, define the operators<math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>b</mi> <mrow><mi>n</mi><mo>,</mo><mi>&#x003F5;</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>a</mi> <mi>&#x003F5;</mi></msub><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><msubsup><mi>a</mi> <mi>&#x003F5;</mi> <mo>&#x02020;</mo></msubsup><msub><mi>a</mi> <mi>&#x003F5;</mi></msub><mo stretchy="false">(</mo><msubsup><mi>a</mi> <mi>&#x003F5;</mi> <mo>&#x02020;</mo></msubsup><msup><mo stretchy="false">)</mo> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>b_{n,\epsilon} = \frac{1}{n!}(a_\epsilon)^n a^\dagger_\epsilon a_\epsilon (a^\dagger_\epsilon)^n</annotation></semantics></math>For the usual harmonic oscillator, we have that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>b</mi> <mi>n</mi></msub><mo stretchy="false">&#x0007C;</mo><mn>0</mn><mo stretchy="false">&#x027E9;</mo><mo>=</mo><mi>n</mi><mo stretchy="false">&#x0007C;</mo><mn>0</mn><mo stretchy="false">&#x027E9;</mo></mrow><annotation encoding='application/x-tex'>b_n|0\rangle = n|0\rangle</annotation></semantics></math> and so the vev of the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>b</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>b_n</annotation></semantics></math> is <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>. By weak continuity, we can ensure that, for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi><mo>&lt;</mo><mi>N</mi></mrow><annotation encoding='application/x-tex'>n \lt N</annotation></semantics></math> and for any <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B4;</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\delta \gt 0</annotation></semantics></math>, there exists an <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003F5;</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\epsilon_0</annotation></semantics></math> such that for all <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003F5;</mi><mo>&lt;</mo><msub><mi>&#x003F5;</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\epsilon \lt \epsilon_0</annotation></semantics></math>:<math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">&#x0007C;</mo><mo stretchy="false">&#x027E8;</mo><mn>0</mn><mo stretchy="false">&#x0007C;</mo><msub><mi>b</mi> <mrow><mi>n</mi><mo>,</mo><mi>&#x003F5;</mi></mrow></msub><mo stretchy="false">&#x0007C;</mo><mn>0</mn><mo stretchy="false">&#x027E9;</mo><mo>&#x02212;</mo><mi>n</mi><mo stretchy="false">&#x0007C;</mo><mo>&lt;</mo><mi>&#x003B4;</mi></mrow><annotation encoding='application/x-tex'>|\langle 0|b_{n,\epsilon}|0\rangle &#x2d; n| \lt \delta</annotation></semantics></math></p>
  225.  
  226. <p>We&#8217;re still nowhere near the <abbr>LQG</abbr> harmonic oscillator, though. To summarize what we&#8217;ve done so far, we have defined a one parameter family of operators that exist in the algebra generated by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> that, such that if we interpret them as in the algebra with the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>q</mi></mrow><annotation encoding='application/x-tex'>q</annotation></semantics></math>, they would go to the usual Hamiltonian and raising and lowering operators. There, the vevs of the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>b</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>b_n</annotation></semantics></math> give exactly the spectrum of the Hamiltonian. We would have ensired that the vevs of a finite number of the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>b</mi> <mrow><mi>n</mi><mo>,</mo><mi>&#x003F5;</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>b_{n,\epsilon}</annotation></semantics></math> are close to the integers, but ut&#8217;s not at all clear to me what these vevs have to do with the spectrum of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>H</mi> <mi>&#x003F5;</mi></msub></mrow><annotation encoding='application/x-tex'>H_\epsilon</annotation></semantics></math>.</p>
  227.  
  228. <p>Regardless, we don&#8217;t necessarily have the harmonic oscillator vacuum floating around. It is a theorem (apparently) that the space of traceclass operators on our (GNS-)Hilbert space is dense in the space of states on the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebra. Thus, we can find a traceclass operator, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C1;</mi></mrow><annotation encoding='application/x-tex'>\rho</annotation></semantics></math>, such <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>Tr</mi><mo stretchy="false">(</mo><mi>&#x003C1;</mi><msub><mi>b</mi> <mrow><mi>n</mi><mo>,</mo><mi>&#x003F5;</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>Tr(\rho b_{n,\epsilon})</annotation></semantics></math> is as close to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">&#x027E8;</mo><msub><mi>b</mi> <mrow><mi>n</mi><mo>,</mo><mi>&#x003F5;</mi></mrow></msub><mo stretchy="false">&#x027E9;</mo></mrow><annotation encoding='application/x-tex'>\langle b_{n,\epsilon} \rangle</annotation></semantics></math> in the Harmonic oscillator vaccum as we desire. Since that&#8217;s as close to the integers as we specify, we&#8217;ve produced a state (actually, an infinite number of them) such that, for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi><mo>&lt;</mo><mi>N</mi></mrow><annotation encoding='application/x-tex'>n\lt N</annotation></semantics></math>,<math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">&#x0007C;</mo><mi>Tr</mi><mo stretchy="false">(</mo><mi>&#x003C1;</mi><msub><mi>b</mi> <mrow><mi>n</mi><mo>,</mo><mi>&#x003F5;</mi></mrow></msub><mo stretchy="false">)</mo><mo>&#x02212;</mo><mi>n</mi><mo stretchy="false">&#x0007C;</mo><mo>&lt;</mo><mi>&#x003B4;</mi></mrow><annotation encoding='application/x-tex'>|Tr(\rho b_{n,\epsilon}) &#x2d; n| \lt \delta</annotation></semantics></math></p>
  229.  
  230. <p>So, what have we proven? We&#8217;ve produced an infinite number of mixed states for the GNS-representation provided by <abbr>LQG</abbr>-quantization. In each of these mixed states, the vevs of the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>b</mi> <mrow><mi>n</mi><mo>,</mo><mi>&#x003F5;</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>b_{n,\epsilon}</annotation></semantics></math> are within a specified closeness to the integers.</p>
  231.  
  232. <p>Let&#8217;s grant for the moment that the vevs of these <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>b</mi> <mrow><mi>n</mi><mo>,</mo><mi>&#x003F5;</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>b_{n,\epsilon}</annotation></semantics></math> operators are really related to what we measure for the Harmonic oscillator energy states. Given a particular choice of mixed state, there is a prediction that at some accuracy, these values will differ from the standard prediction. Presumably, something similar will hold for any comparison of <abbr>LQG</abbr>-quantization and standard quantization.</p>
  233.  
  234. <p>What we have not furnished at this point is a first principles method for determining the <i>correct</i> mixed state. I don&#8217;t see how we have any predictivity without this.</p>
  235.  
  236. <p>What&#8217;s more, there are plenty of other traceclass operators. For example, take the density matrix for the harmonic oscillator:<math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x003C1;</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mo stretchy="false">&#x0007C;</mo><mn>0</mn><mo stretchy="false">&#x027E9;</mo><mo stretchy="false">&#x027E8;</mo><mn>0</mn><mo stretchy="false">&#x0007C;</mo><mo>+</mo><mo stretchy="false">&#x0007C;</mo><mn>1</mn><mo stretchy="false">&#x027E9;</mo><mo stretchy="false">&#x027E8;</mo><mn>1</mn><mo stretchy="false">&#x0007C;</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\rho = \frac{1}{2}(|0\rangle \langle 0| + |1\rangle \langle 1|)</annotation></semantics></math>Then,<math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>Tr</mi><mo stretchy="false">(</mo><mi>&#x003C1;</mi><msub><mi>b</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>Tr(\rho b_n) = \frac{1}{2}(n + (n+1)^2)</annotation></semantics></math>I can just as well produce a density matrix such that the vevs of the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>b</mi> <mrow><mi>n</mi><mo>,</mo><mi>&#x003F5;</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>b_{n,\epsilon}</annotation></semantics></math> give these numbers to the specified accuracy. Can I distinguish from first principles why I should choose one of the above density matrices that give us &#8216;good&#8217; answers rather than one of the ones that give these values?</p>
  237.  
  238. <p>Given this wide variety of &#8216;energy levels&#8217; for the Harmonic oscillator, it seems to me that <span class="caps"><abbr>LQG</abbr></span> has not predicted (or retrodicted in this case) anything. Worse, no matter how accurate traditional quantization appears to be, we can always find a state that reproduces our measurement to the needed accuracy, so <span class="caps"><abbr>LQG</abbr></span> does not predict when we should see deviations from standard quantization.</p>
  239.  
  240. <p>This isn&#8217;t completely awful in situations like the Harmonic oscillator where traditional quantum mechanics tells us the answer to look for. But what happens in quantum gravity, then, when we don&#8217;t have a traditional quantization to guide us? Does this surfeit of mixed states cease to exist? How do we get any predictions at all?</p>
  241.      </div>
  242.    </content>
  243.  </entry>
  244.  <entry>
  245.    <title type="html"><![CDATA[<i>Not Even Wrong</i>]]></title>
  246.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000898.html" />
  247.    <updated>2006-08-19T03:33:07Z</updated>
  248.    <published>2006-08-19T02:54:31+00:00</published>
  249.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.898</id>
  250.    <summary type="text">The physics blog-wars have hit the world of publishing with Peter Woit's Not Even Wrong and Lee Smolin's The Trouble...</summary>
  251.    <author>
  252.      <name>aaron</name>
  253.      <uri>http://zippy.ph.utexas.edu/~abergman/</uri>
  254.      <email>[email protected]</email>
  255.    </author>
  256.    <category term="strings" />
  257.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000898.html">
  258.      <div xmlns="http://www.w3.org/1999/xhtml">
  259. <p>The physics blog-wars have hit the world of publishing with <a href = "http://www.math.columbia.edu/~woit/wordpress">Peter Woit</a>&#8217;s <i>Not Even Wrong</i> and <a href = "http://www.thetroublewithphysics.com/">Lee Smolin</a>&#8217;s <i>The Trouble with Physics</i>. Having given up blogging long ago, I still seem to have spent an inordinate amount of time in these internet trenches. Since Dr. Woit was kind enough to send me a review copy of his book, once more into the breach I suppose. Here is my contribution to the chorus of reviews that will surely be appearing. Like the book, it is aimed at the general public rather than towards physicists.</p>
  260.  
  261. <p><a href = "http://zippy.ph.utexas.edu/~abergman/Review.pdf">Review of <i>Not Even Wrong</i></a></p>
  262.  
  263. <p>Any comments and corrections are greatly appreciated. The first paragraph follows after the jump.</p>
  264.  
  265.  
  266. <p>String theory, the enormously ambitious and speculative endeavor that has, for the past thirty years, attempted to unify our understanding of quantum mechanics and gravity has failed to live up to its initial promise. Its relative domination of the field of fundamental theoretical physics has long led to criticism within the scientific community. In the last few years, however, a number of popular books and television shows have made the case for string theory to the public. There is certainly a place, then, for these criticisms to also be presented to the public. More so, the sociology of modern theoretical physics could provide a fascinating context in which to present a reasonably disinterested discussion of the pros and cons of both string theory as a research program and the way in which modern theoretical physics is pursued. Dr. Woit has instead chosen to write a tendentious account providing little guidance as to why, even in the face of such criticism, so many have chosen to work on string theory. After reading this book and some of the unfortunate innuendo it contains, one might conclude not that string theorists are honest researchers doing the best they can to understand the nature of the universe, but rather are misguided devotees of a failed cult mired in self-delusion.
  267. </p><p>
  268. &#8230;</p>
  269.      </div>
  270.    </content>
  271.  </entry>
  272.  <entry>
  273.    <title type="html">The n-Category Caf&amp;#x000E9;</title>
  274.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000894.html" />
  275.    <updated>2006-08-17T11:31:22Z</updated>
  276.    <published>2006-08-17T12:05:30+00:00</published>
  277.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.894</id>
  278.    <summary type="text">New group blog: "The n-Category Cafe".</summary>
  279.    <author>
  280.      <name>urs</name>
  281.      <uri>http://www.math.uni-hamburg.de/home/schreiber</uri>
  282.      <email>[email protected]</email>
  283.    </author>
  284.    <category term="blog" />
  285.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000894.html">
  286.      <div xmlns="http://www.w3.org/1999/xhtml">
  287. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  288. <p>A new group blog has been created. </p>
  289.  
  290. <blockquote>
  291. <p>
  292. <a href="http://golem.ph.utexas.edu/category/">The <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-Category Caf&#x000E9;</a>.
  293. </p>
  294. </blockquote>
  295.  
  296. <p>It&#8217;s hosted by <a href="http://math.ucr.edu/home/baez/README.html">John Baez</a>, <a href="http://en.wikipedia.org/wiki/David_Corfield">David Corfield</a> and myself.   </p>
  297.  
  298. <p>The caf&#x000E9; is supposed to be the right place for the sort of discussion of mathematical physics that you know from John&#8217;s <a href="http://math.ucr.edu/home/baez/TWF.html">This Week&#8217;s Finds in Mathematical Physics</a>, maybe from some of the <a href="http://golem.ph.utexas.edu/string/archives.html">stuff</a> that I have been posting here, hopefully close to the constructive style that is <a href="http://www.dcorfield.pwp.blueyonder.co.uk/2006/08/klein-2-geometry-iv.html">practised</a> on <a href="http://www.dcorfield.pwp.blueyonder.co.uk/blog.html">David&#8217;s blog</a>.</p>
  299.  
  300. <p>We are very glad to be able to <a href="http://golem.ph.utexas.edu/category/2006/08/inaugural_post.html">use</a> <a href="http://golem.ph.utexas.edu/~distler/">Jacques Distler</a>&#8217;s <a href="http://golem.ph.utexas.edu/~distler/blog/archives/000826.html">sophisticated</a> blog technology.  </p>
  301.  
  302. <p>I will probably move much of my activity from the coffee table to the caf&#x000E9;.</p>
  303.  
  304.      </div>
  305.    </content>
  306.  </entry>
  307.  <entry>
  308.    <title type="html">Synthetic Transitions</title>
  309.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000883.html" />
  310.    <updated>2007-01-28T21:57:55Z</updated>
  311.    <published>2006-07-28T19:58:31+00:00</published>
  312.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.883</id>
  313.    <summary type="text">On deriving 2-connection transition laws using synthetic differential geometry.</summary>
  314.    <author>
  315.      <name>urs</name>
  316.      <uri>http://www.math.uni-hamburg.de/home/schreiber</uri>
  317.      <email>[email protected]</email>
  318.    </author>
  319.    <category term="mathematical physics" />
  320.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000883.html">
  321.      <div xmlns="http://www.w3.org/1999/xhtml">
  322. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  323. <p>On the occasion of the availability of the new edition of Anders Kock&#8217;s <a href="http://home.imf.au.dk/kock/galley.pdf">book</a> on synthetic differential geometry (<a href="http://golem.ph.utexas.edu/string/archives/000655.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>) I want to go through an exercise which I wanted to type long time ago already.</p>
  324.  
  325. <p>I&#8217;ll redo the derivation of the transition laws for 2-connections (<a href="http://golem.ph.utexas.edu/string/archives/000689.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>) using synthetic language. This greatly simplifies the derivation, to the extent that the equations in terms of differential forms become almost identical to the diagrammatic equations that we derive them from.</p>
  326. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  327. <p>The main two facts of synthetic differential geometry that I&#8217;ll need are the following.</p>
  328.  
  329. <p>1) A function from <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-simplices to some group, which sends degenerate <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-simplices to the identity element <em>is</em> a (group-valued) <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-form. For the additive group <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0211D;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}</annotation></semantics></math> this <em>is</em> an ordiary <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-form. More generally, this function is to be thought of as the infinitesimal exponential of a Lie algebra-valued <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>-form.</p>
  330.  
  331. <p>2) Given a group-valued 1-form </p>
  332.  
  333. <div class="numberedEq"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>&#x02192;</mo><mi>y</mi><mo stretchy="false">)</mo><mo>&#x021A6;</mo><mi>exp</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  334.  (x \to y) \mapsto \exp(A_{x,y})
  335.  \,,
  336. </annotation></semantics></math></div>
  337.  
  338. <p>its gauge covariant curvature is the 2-form</p>
  339.  
  340. <div class="numberedEq"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>&#x02192;</mo><mi>y</mi><mo>&#x02192;</mo><mi>z</mi><mo stretchy="false">)</mo><mo>&#x021A6;</mo></mtd> <mtd><mi>exp</mi><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  341. \begin{aligned}
  342.  (x \to y \to z)
  343.   \mapsto &amp;
  344.   \exp F(x,y,z)
  345.  \\
  346.  &amp;=
  347.  \exp(A(x,y))\exp(A(y,z))\exp(A(z,x))
  348. \end{aligned}
  349.  \,.
  350. </annotation></semantics></math></div>
  351.  
  352. <p>So fix some space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> together with a good covering by open sets <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math>. On each <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math> we have a transport 2-functor which sends infinitesimal 2-simplices </p>
  353.  
  354. <div class="numberedEq"><span>(3)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><mi>x</mi></mtd> <mtd><mo>&#x02192;</mo></mtd> <mtd><mi>y</mi></mtd> <mtd><mo>&#x02192;</mo></mtd> <mtd><mi>z</mi></mtd> <mtd><mo>&#x02192;</mo></mtd> <mtd><mi>x</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">&#x02193;</mo></mtd> <mtd/> <mtd/> <mtd/> <mtd/> <mtd/> <mtd><mo stretchy="false">&#x02193;</mo></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd columnspan="5"><mover><mo>&#x02192;</mo><mspace height=".0ex" depth=".0ex" width="11.0em"/></mover></mtd> <mtd><mi>x</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  355. \array{
  356.  x &amp; \to &amp; y &amp; \to &amp; z &amp; \to &amp; x
  357.  \\
  358.  \downarrow &amp;&amp;&amp;&amp;&amp;&amp; \downarrow
  359.  \\
  360.  x &amp; \cellopts{\colspan{5} }\overset{\space{0}{0}{110}}{\to} &amp; x
  361. }
  362. </annotation></semantics></math></div>
  363.  
  364. <p>to elements of the 2-group <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>H</mi><mo>&#x02192;</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>H \to G</annotation></semantics></math>:</p>
  365.  
  366. <div class="numberedEq"><span>(4)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable columnalign="right center center center center center left"><mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><mi>A</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><mi>A</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><mi mathvariant="normal">Id</mi><mo stretchy="false">&#x02193;</mo></mtd> <mtd/> <mtd/> <mtd><mi>exp</mi><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd/> <mtd/> <mtd><mo stretchy="false">&#x02193;</mo><mi mathvariant="normal">Id</mi></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd columnspan="5"><munder><mo>&#x02192;</mo><mrow><mspace height=".0ex" depth=".0ex" width="11.0em"/><mi mathvariant="normal">Id</mi><mspace height=".0ex" depth=".0ex" width="11.0em"/></mrow></munder></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  367. \array{\arrayopts{\colalign{right center center center center center left}}
  368.  \bullet &amp; \overset{\exp A(x,y)}{\to} &amp; \bullet &amp; \overset{\exp A(y,z)}{\to} &amp;
  369.  \bullet &amp; \overset{\exp A(z,x)}{\to} &amp; \bullet
  370.  \\
  371.  \mathrm{Id}\downarrow &amp;&amp;&amp;\exp B(x,y,z)&amp;&amp;&amp; \downarrow\mathrm{Id}
  372.  \\
  373.  \bullet &amp;\cellopts{\colspan{5}} \underset{\space{0}{0}{110}\mathrm{Id}\space{0}{0}{110}}{\to}  &amp; \bullet
  374. }
  375. \,.
  376. </annotation></semantics></math></div>
  377.  
  378. <p>From the source-target relation in 2-groups we read off that the 1-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and 2-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> satisfy the fake flatness condition</p>
  379.  
  380. <div class="numberedEq"><span>(5)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>F</mi> <mi>A</mi></msub><mo>+</mo><mi>B</mi><mo>=</mo><mn>0</mn><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  381.  F_A + B = 0
  382.  \,.
  383. </annotation></semantics></math></div>
  384.  
  385. <p>On double intersections, two such 2-functors are related by a pseudonatural transformation. This is a 1-form</p>
  386.  
  387. <div class="numberedEq"><span>(6)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>&#x02192;</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"/><mo>&#x021A6;</mo><mspace width="thickmathspace"/><mrow><mtable columnalign="right center left"><mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo></mtd> <mtd><mi>exp</mi><msub><mi>a</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">&#x02193;</mo><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><munder><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  388.  (x \to y)
  389.  \;
  390.  \mapsto
  391.  \;
  392.  \array{\arrayopts{\colalign{right center left}}
  393.    \bullet &amp;\overset{\exp A_i(x,y)}{\to}&amp; \bullet
  394.    \\
  395.    g_{ij}(x) \downarrow &amp;\exp a_{ij}(x,y)&amp;  \downarrow g_{ij}(y)
  396.   \\
  397.    \bullet &amp;\underset{\exp A_j(x,y)}{\to}&amp; \bullet
  398.  }
  399.  \,.
  400. </annotation></semantics></math></div>
  401.  
  402. <p>Its values are composed horizontally using whiskering in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mo>&#x02192;</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(H\to G)</annotation></semantics></math>. We may hence think of this as a 1-form taking values in the semidirect product group <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi><mo>&#x022C9;</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>G \ltimes H</annotation></semantics></math>. I&#8217;ll denote the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>-part of its synthetic curvature by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>exp</mi><msub><mi>F</mi> <mrow><msub><mi>a</mi> <mi>ij</mi></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\exp F_{a_{ij},A_i}</annotation></semantics></math> below.</p>
  403.  
  404. <p>The mere existence of the 2-cell on the right above means that</p>
  405.  
  406. <div class="numberedEq"><span>(7)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>exp</mi><msub><mi>a</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>exp</mi><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>exp</mi><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  407.  t(\exp a_{ij}(x,y)) \exp A_i(x,y) g_{ij}(x) = g_{ij}(y) \exp A_j(x,y)
  408.  \,,
  409. </annotation></semantics></math></div>
  410.  
  411. <p>which is equivalent to the transition law for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>A</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>A_i</annotation></semantics></math>.</p>
  412.  
  413. <p>In order to qualify as a pseudonatural transformation, this 1-form must satisfy the equation</p>
  414.  
  415. <div class="numberedEq"><span>(8)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd/> <mtd><mrow><mtable columnalign="right center center center center center left"><mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><mi mathvariant="normal">Id</mi><mo stretchy="false">&#x02193;</mo></mtd> <mtd/> <mtd/> <mtd><mi>exp</mi><msub><mi>B</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd/> <mtd/> <mtd><mo stretchy="false">&#x02193;</mo><mi mathvariant="normal">Id</mi></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd columnspan="5"><mover><mo>&#x02192;</mo><mrow><mspace height=".1ex" depth=".0ex" width="12.5em"/><mi mathvariant="normal">Id</mi><mspace height=".1ex" depth=".0ex" width="12.5em"/></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo></mtd> <mtd/> <mtd/> <mtd><mi mathvariant="normal">Id</mi></mtd> <mtd/> <mtd/> <mtd><mo stretchy="false">&#x02193;</mo><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd columnspan="5"><munder><mo>&#x02192;</mo><mrow><mspace height=".1ex" depth=".0ex" width="12.5em"/><mi mathvariant="normal">Id</mi><mspace height=".1ex" depth=".0ex" width="12.5em"/></mrow></munder></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr></mtable></mrow></mtd></mtr> <mtr><mtd><mphantom><mi>M</mi></mphantom></mtd></mtr> <mtr><mtd><mo>=</mo></mtd> <mtd><mrow><mtable columnalign="right center right center right center left"><mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo></mtd> <mtd><mi>exp</mi><msub><mi>a</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo></mtd> <mtd><mi>exp</mi><msub><mi>a</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo></mtd> <mtd><mi>exp</mi><msub><mi>a</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">&#x02193;</mo><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><mi mathvariant="normal">Id</mi><mo stretchy="false">&#x02193;</mo></mtd> <mtd/> <mtd/> <mtd><mi>exp</mi><msub><mi>B</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd/> <mtd/> <mtd><mo stretchy="false">&#x02193;</mo><mi mathvariant="normal">Id</mi></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd columnspan="5"><munder><mo>&#x02192;</mo><mrow><mspace height=".1ex" depth=".0ex" width="15.0em"/><mi mathvariant="normal">Id</mi><mspace height=".1ex" depth=".0ex" width="15.0em"/></mrow></munder></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  416. \begin{aligned}
  417. &amp;
  418. \array{\arrayopts{\colalign{right center center center center center left}}
  419.  \bullet &amp; \overset{\exp A_i(x,y)}{\to} &amp; \bullet &amp; \overset{\exp A_i(y,z)}{\to} &amp;
  420.  \bullet &amp; \overset{\exp A_i(z,x)}{\to} &amp;\bullet
  421.  \\
  422.  \mathrm{Id}\downarrow &amp;&amp;&amp;\exp B_i(x,y,z)&amp;&amp;&amp;\downarrow\mathrm{Id}
  423.  \\
  424.  \bullet &amp;\cellopts{\colspan{5}}  \overset{\space{1}{0}{125}\mathrm{Id}\space{1}{0}{125}}{\to} &amp;\bullet
  425.  \\
  426.  g_{ij}(x) \downarrow &amp;&amp;&amp;\mathrm{Id} &amp;&amp;&amp;\downarrow g_{ij}(x)
  427.  \\
  428.  \bullet &amp;\cellopts{\colspan{5}}  \underset{\space{1}{0}{125}\mathrm{Id}\space{1}{0}{125}}{\to} &amp;\bullet
  429. }  
  430. \\
  431. \phantom{M}\\
  432. =&amp;
  433. \array{\arrayopts{\colalign{right center right center right center left}}
  434.  \bullet &amp;\overset{A_i(x,y)}{\to}&amp;
  435.  \bullet &amp;\overset{A_i(y,z)}{\to}&amp;
  436.  \bullet &amp;\overset{A_i(z,x)}{\to}&amp;\bullet
  437.  \\
  438.  g_{ij}(x)\downarrow
  439.  &amp;\exp a_{ij}(x,y)&amp;
  440.  g_{ij}(y)\downarrow
  441.  &amp;\exp a_{ij}(y,z)&amp;
  442.  g_{ij}(z)\downarrow
  443.  &amp;\exp a_{ij}(z,x)&amp;
  444.  \downarrow g_{ij}(x)
  445.  \\
  446.  \bullet &amp; \overset{\exp A_j(x,y)}{\to} &amp; \bullet &amp; \overset{\exp A_j(y,z)}{\to} &amp;
  447.  \bullet &amp; \overset{\exp A_j(z,x)}{\to} &amp;\bullet
  448.  \\
  449.  \mathrm{Id}\downarrow &amp;&amp;&amp;\exp B_j(x,y,z)&amp;&amp;&amp;\downarrow\mathrm{Id}
  450.  \\
  451.  \bullet &amp; \cellopts{\colspan{5}}  \underset{\space{1}{0}{150}\mathrm{Id}\space{1}{0}{150}}{\to} &amp;\bullet
  452. }  
  453. \end{aligned}
  454. </annotation></semantics></math></div>
  455.  
  456. <p>But using the two SDG facts stated above, together with the rules for composition in the 2-group <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mo>&#x02192;</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(H\to G)</annotation></semantics></math>, this immediately says that</p>
  457.  
  458. <div class="numberedEq"><span>(9)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>B</mi> <mi>i</mi></msub><mo>=</mo><mi>&#x003B1;</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>B</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>F</mi> <mrow><msub><mi>a</mi> <mi>ij</mi></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub></mrow></msub><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  459.  B_i = \alpha(g_ij)(B_j) + F_{a_{ij},A_{i}}
  460.  \,.
  461. </annotation></semantics></math></div>
  462.  
  463. <p>Next, on triple intersections the pseudonatural transformations <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>exp</mi><msub><mi>a</mi> <mi>ij</mi></msub></mrow><annotation encoding='application/x-tex'>\exp a_{ij}</annotation></semantics></math> are related by an isomodification. This says that there is a 0-form</p>
  464.  
  465. <div class="numberedEq"><span>(10)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>x</mi><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mo>&#x021A6;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mrow><mtable columnalign="right center left"><mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mi mathvariant="normal">Id</mi></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><mrow><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x02193;</mo></mrow></mtd> <mtd/> <mtd rowspan="3" rowalign="top"><mrow><mo>&#x02193;</mo><mphantom><mspace height=".0ex" depth="4.0ex" width=".0em"/></mphantom><msub><mi>g</mi> <mi>ik</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><msub><mi>f</mi> <mi>ijk</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mrow><msub><mi>g</mi> <mi>jk</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x02193;</mo></mrow></mtd> <mtd/></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><munder><mo>&#x02192;</mo><mi mathvariant="normal">Id</mi></munder></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  466.  x
  467.  \;\;\;
  468.  \mapsto
  469.  \;\;\;
  470. \array{\arrayopts{\colalign{right center left}}
  471.  \bullet &amp; \overset{\mathrm{Id}}{\to} &amp;\bullet
  472.  \\
  473.  \left. g_{ij}(x)\right\downarrow&amp;&amp;
  474. \cellopts{\rowspan{3}\rowalign{top}} \left\downarrow \phantom{\space{0}{40}{0}}g_{ik}(x)\right.
  475.  \\
  476.  \bullet &amp; f_{ijk}(x)
  477.  \\
  478.  \left. g_{jk}(x)\right\downarrow &amp;
  479.  \\
  480.  \bullet &amp; \underset{\mathrm{Id}}{\to} &amp;\bullet
  481. }
  482. </annotation></semantics></math></div>
  483.  
  484. <p>which satisfies</p>
  485.  
  486. <div class="numberedEq"><span>(11)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd/> <mtd><mrow><mtable columnalign="right center left"><mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo></mtd> <mtd><mi>exp</mi><msub><mi>a</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">&#x0007C;</mo><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><msub><mi>g</mi> <mi>jk</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo></mtd> <mtd><mi>exp</mi><msub><mi>a</mi> <mi>jk</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">&#x02193;</mo><msub><mi>g</mi> <mi>jk</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><munder><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr></mtable></mrow></mtd></mtr> <mtr><mtd><mphantom><mi>M</mi></mphantom></mtd></mtr> <mtr><mtd><mo>=</mo></mtd> <mtd><mrow><mtable columnalign="right center center center center center left"><mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mi mathvariant="normal">Id</mi></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mi mathvariant="normal">Id</mi></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><mrow><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x02193;</mo></mrow></mtd> <mtd/> <mtd rowspan="3" rowalign="top"><mrow><mo>&#x0007C;</mo><mspace height=".0ex" depth="4.0ex" width=".0em"/></mrow></mtd> <mtd/> <mtd rowspan="3" rowalign="top"><mrow><mo>&#x0007C;</mo><mspace height=".0ex" depth="4.0ex" width=".0em"/></mrow></mtd> <mtd/> <mtd><mrow><mo>&#x0007C;</mo><msub><mi>g</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><msub><mi>f</mi> <mi>ijk</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>exp</mi><msub><mi>a</mi> <mi>ik</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><msubsup><mi>f</mi> <mi>ijk</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr> <mtr><mtd><mrow><msub><mi>g</mi> <mi>jk</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x02193;</mo></mrow></mtd> <mtd/> <mtd/> <mtd/> <mtd><mrow><mo>&#x02193;</mo><msub><mi>g</mi> <mi>jk</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mtd></mtr> <mtr><mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mi mathvariant="normal">Id</mi></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>exp</mi><msub><mi>A</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>&#x02022;</mo></mtd> <mtd><mover><mo>&#x02192;</mo><mi mathvariant="normal">Id</mi></mover></mtd> <mtd><mo>&#x02022;</mo></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  487. \begin{aligned}
  488. &amp;\array{\arrayopts{\colalign{right center left}}
  489.  \bullet &amp; \overset{\exp A_i(x,y)}{\to} &amp;\bullet
  490.  \\
  491.  g_{ij}(x)\downarrow &amp;\exp a_{ij}(x,y)&amp;| g_{ij}(y)
  492.  \\
  493.  \bullet &amp; \overset{\exp A_j(x,y)}{\to} &amp;\bullet
  494.  \\
  495.  g_{jk}(x)\downarrow &amp;\exp a_{jk}(x,y)&amp;\downarrow g_{jk}(y)
  496.  \\
  497.  \bullet &amp; \underset{\exp A_k(x,y)}{\to} &amp;\bullet
  498. }
  499. \\
  500. \phantom{M}\\
  501. =
  502. &amp;
  503. \array{\arrayopts{\colalign{right center center center center center left}}
  504.  \bullet &amp; \overset{\mathrm{Id}}{\to} &amp; \bullet &amp; \overset{\exp A_i(x,y)}{\to} &amp; \bullet
  505.  &amp; \overset{\mathrm{Id}}{\to} &amp;\bullet
  506.  \\
  507.  \left. g_{ij}(x)\right\downarrow &amp;&amp;\cellopts{\rowspan{3}\rowalign{top}}\left\vert \space{0}{40}{0}\right.&amp;&amp;\cellopts{\rowspan{3}\rowalign{top}}\left\vert \space{0}{40}{0}\right.&amp;&amp;\left\vert g_{ij}(y)\right.
  508.  \\
  509.  \bullet &amp; f_{ijk}(x) &amp; \exp a_{ik}(x,y) &amp;  f_{ijk}^{&#x2d;1}(y) &amp;\bullet
  510.  \\
  511.  \left. g_{jk}(x)\right\downarrow &amp;&amp;&amp;&amp;\left\downarrow g_{jk}(y)\right.
  512.  \\
  513.  \bullet &amp; \overset{\mathrm{Id}}{\to} &amp; \bullet &amp; \overset{\exp A_k(x,y)}{\to} &amp; \bullet
  514.  &amp; \overset{\mathrm{Id}}{\to} &amp;\bullet
  515. }
  516. \end{aligned}
  517. </annotation></semantics></math></div>
  518.  
  519. <p>Here we just need to collect exponents to get the expected transition law.</p>
  520.  
  521. <p>Finally, there is the tetrahedron law on quadruple intersections. This only involves 0-forms and SDG does not tell us anything here that we did no know before.</p>
  522.  
  523. <p>In conclusion, SDG here mainly helps handling the otherwise somewhat subtle curvature <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>F</mi> <mrow><msub><mi>a</mi> <mi>ij</mi></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>F_{a_{ij},A_{i}}</annotation></semantics></math> in the transition on double intersections, and makes reading off the law on triple intersections a little more systematic.</p>
  524.      </div>
  525.    </content>
  526.  </entry>
  527.  <entry>
  528.    <title type="html">Quillen&apos;s Superconnections -- Functorially</title>
  529.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000882.html" />
  530.    <updated>2006-10-23T11:35:27Z</updated>
  531.    <published>2006-07-26T20:11:05+00:00</published>
  532.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.882</id>
  533.    <summary type="text">On how to interpret the superconnections appearing on brane/anti-brane configurations in terms of functorial transport.</summary>
  534.    <author>
  535.      <name>urs</name>
  536.      <uri>http://www.math.uni-hamburg.de/home/schreiber</uri>
  537.      <email>[email protected]</email>
  538.    </author>
  539.    <category term="mathematical physics" />
  540.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000882.html">
  541.      <div xmlns="http://www.w3.org/1999/xhtml">
  542. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  543. <p>As explained for instance in </p>
  544.  
  545. <p>Richard J. Szabo
  546. <br/><em>Superconnections, Anomalies and Non-BPS Brane Charges</em>
  547. <br/><a href="http://arxiv.org/abs/hep-th/0108043">hep-th/0108043</a></p>
  548.  
  549. <p>a special case of Quillen&#8217;s concept of <em>superconnections</em> can be used to elegantly subsume both the gauge connection as well as the tachyon field on non-BPS D-branes into a single entity. </p>
  550.  
  551. <p>Assuming that this is not just a coincidence, one might ask what it <em>really means</em>. What notion of functorial parallel transport (<a href="http://golem.ph.utexas.edu/string/archives/000753.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>) is encoded in these superconnections? </p>
  552.  
  553. <p>I&#8217;ll give an interpretation below. With hindsight, it is absolutely obvious. But I haven&#8217;t seen it discussed before, and - trivial as it may be - it deserves to be stated.</p>
  554. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  555. <p><strong>Superconnections.</strong></p>
  556.  
  557. <p>Let <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi><mo>&#x02192;</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> be some <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x02124;</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_2</annotation></semantics></math>-graded vector bundle. Let <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003A9;</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mi>&#x003A9;</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x02297;</mo><mi>&#x00393;</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\Omega(X,E) = \Omega(X)\otimes \Gamma(E)</annotation></semantics></math> be the space of differential forms taking values in sections of this bundle. This is an <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003A9;</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\Omega(X)</annotation></semantics></math>-module in the obvious way. It inherits a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x02124;</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_2</annotation></semantics></math>-grading from the combined <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x02124;</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_2</annotation></semantics></math>-grading of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003A9;</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\Omega(X)</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x00393;</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\Gamma(E)</annotation></semantics></math>.</p>
  558.  
  559. <p>A <strong>superconnection</strong> on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is defined to be any odd graded endomorphism <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>A</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{A}</annotation></semantics></math> of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003A9;</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\Omega(X,E)</annotation></semantics></math> which satisfies the Leibnitz rule</p>
  560.  
  561. <div class="numberedEq"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>A</mi></mstyle><mo>,</mo><mi>&#x003C9;</mi><mo stretchy="false">]</mo><mo>=</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>&#x003C9;</mi></mrow><annotation encoding='application/x-tex'>
  562.  [\mathbf{A},\omega] = \mathbf{d}\omega
  563. </annotation></semantics></math></div>
  564.  
  565. <p>
  566. for all <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C9;</mi><mo>&#x02208;</mo><mi>&#x003A9;</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\omega \in \Omega(X)</annotation></semantics></math>.</p>
  567.  
  568. <p>Such superconnections arise in the form of ordinary connections plus an odd element <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003A9;</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi mathvariant="normal">End</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\Omega(X,\mathrm{End}(E))</annotation></semantics></math>:</p>
  569.  
  570. <div class="numberedEq"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mstyle mathvariant="bold"><mi>A</mi></mstyle><mo>=</mo><mo>&#x02207;</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>A</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  571.  \mathbf{A} = \nabla + A
  572.  \,.
  573. </annotation></semantics></math></div>
  574.  
  575. <p>Hence, in particular, they may contain a 0-form contribution, taking values in odd-graded endomorphisms of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>.</p>
  576.  
  577. <p><br/><strong>Superconnections on D-branes.</strong></p>
  578.  
  579. <p>For the applications to D-branes, all we need of superconnections is this additional 0-form degree of freedom. </p>
  580.  
  581. <p>The graded bundle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi><mo>=</mo><msup><mi>E</mi> <mo>+</mo></msup><mo>&#x02295;</mo><msup><mi>E</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>E = E^+ \oplus E^&#x2d;</annotation></semantics></math> is interpreted as the Chan-Paton bundle of some D-branes plus that of some anti D-branes. The tachyon field arises from strings stretching beween branes and anti-branes, and is hence a 0-form taking values in odd endomorphisms of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>. Combined with the ordinary connections on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>E</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>E^+</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>E</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>E^&#x2d;</annotation></semantics></math> we get a superconnection.</p>
  582.  
  583. <p>While this way of looking at tachyon fields is very fruitful, it is noteworthy that we have severely restricted the full freedom of superconnections. This might indicate that the &#8220;super&#8221;-point of view is not precisely the most natural one describing this situation. I shall now argue for what I feel is a more natural way of looking at the situation.</p>
  584.  
  585. <p><br/><strong>Functorial reformulation.</strong></p>
  586.  
  587. <p>Consider first just a stack of D-branes, without any anti-D-branes. They carry a gerbe module (a twisted vector bundle) and parallel transport of open strings ending on the brane and coupled to a possibly non-vanishing Kalb-Ramond fields is described by a 2-functor which takes endpoints of strings to fibers of an algebra bundle of compact operators, which takes open strings to bimodules for the algebras associated to the endpoints, and which takes pieces of worldsheet to homomorphisms between the bimodule of the incoming and the outgoing string.</p>
  588.  
  589. <p>This in particular encodes a parallel transport </p>
  590.  
  591. <div class="numberedEq"><span>(3)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi mathvariant="normal">tra</mi><mo>:</mo><msub><mi>P</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><mi mathvariant="normal">Trans</mi><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>+</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  592. \mathrm{tra} : P_2(X) \to \mathrm{Trans}(E^+)
  593. </annotation></semantics></math></div>
  594.  
  595. <p>in the (twisted) bundle on the D-brane which takes paths</p>
  596.  
  597. <div class="numberedEq"><span>(4)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>x</mi><mover><mo>&#x02192;</mo><mrow><msub><mi>&#x003B3;</mi> <mn>1</mn></msub></mrow></mover><mi>y</mi><mover><mo>&#x02192;</mo><mrow><msub><mi>&#x003B3;</mi> <mn>2</mn></msub></mrow></mover><mi>z</mi></mrow><annotation encoding='application/x-tex'>
  598.  x \overset{\gamma_1}{\to} y \overset{\gamma_2}{\to} z
  599. </annotation></semantics></math></div>
  600.  
  601. <p>in the base manifold (the D-brane&#8217;s worldvoume) to morphisms of fibers</p>
  602.  
  603. <div class="numberedEq"><span>(5)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msubsup><mi>E</mi> <mi>x</mi> <mo>+</mo></msubsup><mover><mo>&#x02192;</mo><mrow><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msub><mi>&#x003B3;</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><msubsup><mi>E</mi> <mi>y</mi> <mo>+</mo></msubsup><mover><mo>&#x02192;</mo><mrow><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msub><mi>&#x003B3;</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover><msubsup><mi>E</mi> <mi>z</mi> <mo>+</mo></msubsup><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  604.  E_x^+
  605.  \overset{\mathrm{tra}(\gamma_1)}{\to}
  606.  E_y^+
  607.  \overset{\mathrm{tra}(\gamma_2)}{\to}
  608.  E_z^+
  609.  \,.
  610. </annotation></semantics></math></div>
  611.  
  612. <p>If we conveniently take the point of view of synthetic differential geometry (<a href="http://golem.ph.utexas.edu/string/archives/000655.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>) and assume <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>z</mi></mrow><annotation encoding='application/x-tex'>z</annotation></semantics></math> to be infinitesimal neighbours, then this assignment <em>is</em> an endomorphism-valued connection 1-form.</p>
  613.  
  614. <p>The above paths live in some sort of (2-)path (2-)groupoid <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>P</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>P_2(X)</annotation></semantics></math>, roughly encoding open strings ending on the stack of D-branes.</p>
  615.  
  616. <p>Now, quite obviously, if we want to incorporate in addition a stack of anti D-branes with a bundle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>E</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>E^&#x2d;</annotation></semantics></math> over them, we need to enrich <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>P</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>P_2(X)</annotation></semantics></math> by additional morphisms encoding the strings stretching between <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>E</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>E^+</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>E</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>E^&#x2d;</annotation></semantics></math>. </p>
  617.  
  618. <p>Let&#8217;s take two copies of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>P</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>P_2(X)</annotation></semantics></math>, and throw in precisely one 1-morphism going between every ordered pair of two copies of the same object in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>P</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>P_2(X)</annotation></semantics></math>. These will encode paths with no spatial extension, but whose endpoints lie on different copies of stacks of branes. Call the 2-category freely generated this way <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{P}_2(X)</annotation></semantics></math>.</p>
  619.  
  620. <p>Proceeding analogously for the transport category <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">Trans</mi><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>+</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{Trans}(E^+)</annotation></semantics></math>, we obtain
  621. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>Trans</mi></mstyle><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>+</mo></msup><mo>,</mo><msup><mi>E</mi> <mo>&#x02212;</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{Trans}(E^+,E^&#x2d;)</annotation></semantics></math>, which is generated from <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">Trans</mi><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>+</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{Trans}(E^+)</annotation></semantics></math>, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">Trans</mi><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>&#x02212;</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{Trans}(E^&#x2d;)</annotation></semantics></math> and all morphisms between fibers over the same point.</p>
  622.  
  623. <p><br/><strong>The connection data encoded by the functor.</strong></p>
  624.  
  625. <p>Now a connection of the brane/anti-brane stack is a functor</p>
  626.  
  627. <div class="numberedEq"><span>(6)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi mathvariant="normal">tra</mi><mo>:</mo><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><mstyle mathvariant="bold"><mi>Trans</mi></mstyle><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>+</mo></msup><mo>,</mo><msup><mi>E</mi> <mo>&#x02212;</mo></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  628.  \mathrm{tra} : \mathbf{P}_2(x) \to \mathbf{Trans}(E^+,E^&#x2d;)
  629.  \,.
  630. </annotation></semantics></math></div>
  631.  
  632. <p>A 1-morphism diagram in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{P}_2(X)</annotation></semantics></math> would look for instance like this</p>
  633.  
  634. <div class="numberedEq"><span>(7)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>x</mi> <mo>+</mo></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>1</mn> <mo>+</mo></msubsup></mrow></mover></mtd> <mtd><msup><mi>y</mi> <mo>+</mo></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>2</mn> <mo>+</mo></msubsup></mrow></mover></mtd> <mtd><msup><mi>z</mi> <mo>+</mo></msup></mtd></mtr> <mtr><mtd><msub><mi>t</mi> <mi>x</mi></msub><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><msub><mi>t</mi> <mi>y</mi></msub><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><msub><mi>t</mi> <mi>z</mi></msub><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd></mtr> <mtr><mtd><msup><mi>x</mi> <mo>&#x02212;</mo></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>1</mn> <mo>&#x02212;</mo></msubsup></mrow></mover></mtd> <mtd><msup><mi>y</mi> <mo>&#x02212;</mo></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>2</mn> <mo>&#x02212;</mo></msubsup></mrow></mover></mtd> <mtd><msup><mi>z</mi> <mo>&#x02212;</mo></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  635. \array{
  636.  x^+ &amp;\overset{\gamma_1^+}{\to}&amp; y^+ &amp;\overset{\gamma_2^+}{\to}&amp; z^+
  637.  \\
  638.  t_x \downarrow\;\;\; &amp;&amp; t_y \downarrow\;\;\; &amp;&amp; t_z \downarrow \;\;\;
  639.  \\
  640.  x^&#x2d; &amp;\overset{\gamma_1^&#x2d;}{\to}&amp; y^&#x2d; &amp;\overset{\gamma_2^&#x2d;}{\to}&amp; z^&#x2d;
  641. }
  642. \,,
  643. </annotation></semantics></math></div>
  644.  
  645. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x003B3;</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>\gamma^+</annotation></semantics></math> is a path with endpoints on the stack of branes, and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x003B3;</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>\gamma^&#x2d;</annotation></semantics></math> is the same path (as a path in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>) but with the endpoints taken to sit on the stack of anti-branes. The morphism <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>t</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>t_x</annotation></semantics></math> are constant paths (as paths in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>) with one endpoint on the branes, the other on the anti-branes.</p>
  646.  
  647. <p>Hitting this with our functor produces a diagram</p>
  648.  
  649. <div class="numberedEq"><span>(8)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>+</mo></msubsup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msup><mi mathvariant="normal">tra</mi> <mo>+</mo></msup><mo stretchy="false">(</mo><msub><mi>&#x003B3;</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msubsup><mi>E</mi> <mi>y</mi> <mo>+</mo></msubsup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msup><mi mathvariant="normal">tra</mi> <mo>+</mo></msup><mo stretchy="false">(</mo><msub><mi>&#x003B3;</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msubsup><mi>E</mi> <mi>z</mi> <mo>+</mo></msubsup></mtd></mtr> <mtr><mtd><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mi>y</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mi>z</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd></mtr> <mtr><mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x02212;</mo></msubsup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msup><mi mathvariant="normal">tra</mi> <mo>&#x02212;</mo></msup><mo stretchy="false">(</mo><msub><mi>&#x003B3;</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msubsup><mi>E</mi> <mi>y</mi> <mo>&#x02212;</mo></msubsup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msup><mi mathvariant="normal">tra</mi> <mo>&#x02212;</mo></msup><mo stretchy="false">(</mo><msub><mi>&#x003B3;</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msubsup><mi>E</mi> <mi>z</mi> <mo>&#x02212;</mo></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  650. \array{
  651.  E_x^+ &amp;\overset{\mathrm{tra}^+(\gamma_1)}{\to}&amp; E_y^+ &amp;\overset{\mathrm{tra}^+(\gamma_2)}{\to}&amp; E_z^+
  652.  \\
  653.  \mathrm{tra}(t_x) \downarrow\;\;\;\;\;\; &amp;&amp; \mathrm{tra}(t_y) \downarrow\;\;\;\;\;\; &amp;&amp; \mathrm{tra}(t_z) \downarrow \;\;\;\;\;\;
  654.  \\
  655.  E_x^&#x2d; &amp;\overset{\mathrm{tra}^&#x2d;(\gamma_1)}{\to}&amp; E_y^&#x2d; &amp;\overset{\mathrm{tra}^&#x2d;(\gamma_2)}{\to}&amp; E_z^&#x2d;
  656. }
  657. </annotation></semantics></math></div>
  658.  
  659. <p>in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>Trans</mi></mstyle><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>+</mo></msup><mo>,</mo><msup><mi>E</mi> <mo>&#x02212;</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{Trans}(E^+,E^&#x2d;)</annotation></semantics></math>.</p>
  660.  
  661. <p>We read off what data the new functor encodes: it contains an ordinary connection <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi mathvariant="normal">tra</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>\mathrm{tra}^+</annotation></semantics></math> on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>E</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>E^+</annotation></semantics></math>, as well as an ordinary connection <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi mathvariant="normal">tra</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>\mathrm{tra}^&#x2d;</annotation></semantics></math> on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>E</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>E^&#x2d;</annotation></semantics></math>. Both are given by ordinary 1-forms.</p>
  662.  
  663. <p>In addition, there is now an assignment of morphisms <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo>:</mo><msubsup><mi>E</mi> <mi>x</mi> <mo>+</mo></msubsup><mo>&#x02192;</mo><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x02212;</mo></msubsup></mrow><annotation encoding='application/x-tex'>\mathrm{tra}(t_x) : E^+_x \to E^&#x2d;_x</annotation></semantics></math> for every point <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>. Since <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>x</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>x^+</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>x</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>x^&#x2d;</annotation></semantics></math> belong to the same point in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, this is a morphism-valued <em>0-form</em>. </p>
  664.  
  665. <p>It&#8217;s precisely the tachyon field 0-form that we expect to see.</p>
  666.  
  667. <p>And of course there is a similar 0-form with morphisms going the other way, not depicted in the above diagram.</p>
  668.  
  669. <p>In conclusion, the transport functor on the path category which allows paths to end on points colored by two different colors encodes precisely the information contained in the superconnections which appear on brane/anti-brane pairs. In fact, as I vaguely indicated, the construction seamlessly generlizes to KR-twisted bundles and the NS-NS surface transport associated with that.</p>
  670.  
  671. <p><br/><strong>Curvatures functorially.</strong></p>
  672.  
  673. <p>One can also nicely deduce the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-form curvatures from that, by transporting around infinitesimal loops.</p>
  674.  
  675. <p>First, there is a 0-form curvature obtained by transport along</p>
  676.  
  677. <div class="numberedEq"><span>(9)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>x</mi> <mo>+</mo></msup></mtd></mtr> <mtr><mtd><msub><mi>t</mi> <mi>x</mi></msub><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd></mtr> <mtr><mtd><msup><mi>x</mi> <mo>&#x02212;</mo></msup></mtd></mtr> <mtr><mtd><msub><mover><mi>t</mi><mo stretchy="false">&#x000AF;</mo></mover> <mi>x</mi></msub><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd></mtr> <mtr><mtd><msup><mi>x</mi> <mo>+</mo></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  678. \array{
  679.  x^+
  680.  \\
  681.  t_x \downarrow \;\;
  682.  \\
  683.  x^&#x2d;
  684.  \\
  685.  \bar t_x \downarrow\;\;
  686.  \\
  687.  x^+
  688. }
  689. \,.
  690. </annotation></semantics></math></div>
  691.  
  692. <p>Next, there is a 1-form curvature obtained by transport around</p>
  693.  
  694. <div class="numberedEq"><span>(10)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>x</mi> <mo>+</mo></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msup><mi>&#x003B3;</mi> <mo>+</mo></msup></mrow></mover></mtd> <mtd><msup><mi>y</mi> <mo>+</mo></msup></mtd></mtr> <mtr><mtd><msub><mover><mi>t</mi><mo stretchy="false">&#x000AF;</mo></mover> <mi>x</mi></msub><mo stretchy="false">&#x02191;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mo stretchy="false">&#x02193;</mo><msub><mi>t</mi> <mi>y</mi></msub></mtd></mtr> <mtr><mtd><msup><mi>x</mi> <mo>&#x02212;</mo></msup></mtd> <mtd><mover><mo>&#x02190;</mo><mrow><msup><mi>&#x003B3;</mi> <mo>&#x02212;</mo></msup></mrow></mover></mtd> <mtd><msup><mi>y</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  695. \array{
  696. x^+ &amp;\overset{\gamma^+}{\rightarrow}&amp; y^+
  697.  \\
  698.  \bar t_x \uparrow \;\; &amp;&amp; \;\; \downarrow t_y
  699.  \\
  700. x^&#x2d; &amp;\overset{\gamma^&#x2d;}{\leftarrow}&amp; y^n
  701. }
  702. \,,
  703. </annotation></semantics></math></div>
  704.  
  705. <p>for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> first order infinitesimal neighbours.</p>
  706.  
  707. <p>Finally, there are two ordinary curvatures obtained, as usual, by transporting around infinitesimal loops</p>
  708.  
  709. <div class="numberedEq"><span>(11)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>x</mi> <mo>&#x000B1;</mo></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>1</mn> <mo>&#x000B1;</mo></msubsup></mrow></mover></mtd> <mtd><msup><mi>y</mi> <mo>&#x000B1;</mo></msup></mtd></mtr> <mtr><mtd><msubsup><mover><mi>&#x003B3;</mi><mo stretchy="false">&#x000AF;</mo></mover> <mn>4</mn> <mo>&#x000B1;</mo></msubsup><mo stretchy="false">&#x02191;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mo stretchy="false">&#x02193;</mo><msubsup><mi>&#x003B3;</mi> <mn>2</mn> <mo>&#x000B1;</mo></msubsup></mtd></mtr> <mtr><mtd><msup><mi>v</mi> <mo>&#x000B1;</mo></msup></mtd> <mtd><mover><mo>&#x02190;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>3</mn> <mo>&#x000B1;</mo></msubsup></mrow></mover></mtd> <mtd><msup><mi>z</mi> <mo>&#x000B1;</mo></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  710. \array{
  711. x^\pm &amp;\overset{\gamma_1^\pm}{\rightarrow}&amp; y^\pm
  712.  \\
  713.  \bar \gamma^\pm_4 \uparrow \;\;\; &amp;&amp; \;\;\; \downarrow \gamma_2^\pm
  714.  \\
  715. v^\pm &amp;\overset{\gamma_3^\pm}{\leftarrow}&amp; z^\pm
  716. }
  717. \,,
  718. </annotation></semantics></math></div>
  719.  
  720. <p>The 0-curvature is just the square of the tachyon field. The 1-form curvature is a gauge covariant derivative of the tachyon field, with respect to the two gauge connections on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>E</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>E^+</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>E</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>E^&#x2d;</annotation></semantics></math>.</p>
  721.  
  722. <p>I believe it is easy to see that this does reproduce the three curvature equations (2.12), (2.13) and (2.14) in Richard Szabo&#8217;s text.</p>
  723.  
  724. <p><br/><strong>Gauge tranformations functorially.</strong></p>
  725.  
  726. <p>Gauge transformations are discussed similarly. For us, a gauge transformation is a natural isomorphism</p>
  727.  
  728. <div class="numberedEq"><span>(12)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi mathvariant="normal">tra</mi><mover><mo>&#x02192;</mo><mi>g</mi></mover><mi mathvariant="normal">tra</mi><mo>&#x02032;</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  729.  \mathrm{tra} \overset{g}{\to} \mathrm{tra}&apos;
  730.  \,.
  731. </annotation></semantics></math></div>
  732.  
  733. <p>If we assume, as usual, the base space to be fixed,
  734. then this is an assignment of an isomorphism</p>
  735.  
  736. <div class="numberedEq"><span>(13)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>+</mo></msup><mo stretchy="false">)</mo><mo>:</mo><msubsup><mi>E</mi> <mi>x</mi> <mo>+</mo></msubsup><mo>&#x02192;</mo><msubsup><mi>E</mi> <mi>x</mi> <mo>+</mo></msubsup></mrow><annotation encoding='application/x-tex'>
  737.  g(x^+) : E^+_x \to E^+_x
  738. </annotation></semantics></math></div>
  739.  
  740. <p>to every point <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>x</mi> <mo>+</mo></msup></mrow><annotation encoding='application/x-tex'>x^+</annotation></semantics></math> on the stack of D-branes, and an isoomorphism</p>
  741.  
  742. <div class="numberedEq"><span>(14)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>&#x02212;</mo></msup><mo stretchy="false">)</mo><mo>:</mo><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x02212;</mo></msubsup><mo>&#x02192;</mo><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x02212;</mo></msubsup></mrow><annotation encoding='application/x-tex'>
  743.  g(x^&#x2d;) : E^&#x2d;_x \to E^&#x2d;_x
  744. </annotation></semantics></math></div>
  745.  
  746. <p>to every point on the stack of anti D-branes,
  747. such that all diagrams of the form</p>
  748.  
  749. <div class="numberedEq"><span>(15)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x000B1;</mo></msubsup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msup><mi>&#x003B3;</mi> <mo>&#x000B1;</mo></msup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x000B1;</mo></msubsup></mtd></mtr> <mtr><mtd><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>&#x000B1;</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mo stretchy="false">&#x02193;</mo><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>&#x000B1;</mo></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x000B1;</mo></msubsup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msup><mi>&#x003B3;</mi> <mo>&#x000B1;</mo></msup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x000B1;</mo></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  750. \array{
  751.  E_x^\pm &amp;\overset{\mathrm{tra}(\gamma^\pm)}{\to}&amp; E_x^\pm
  752.  \\
  753.  g(x^\pm) \downarrow \;\;\; &amp;&amp; \;\;\; \downarrow g(x^\pm)
  754.  \\
  755.  E_x^\pm &amp;\overset{\mathrm{tra}(\gamma^\pm)}{\to}&amp; E_x^\pm
  756. }
  757. </annotation></semantics></math></div>
  758.  
  759. <p>and</p>
  760.  
  761. <div class="numberedEq"><span>(16)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>+</mo></msubsup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>+</mo></msup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>+</mo></msubsup></mtd></mtr> <mtr><mtd><msub><mi>t</mi> <mi>x</mi></msub><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mo stretchy="false">&#x02193;</mo><mi>t</mi><msub><mo>&#x02032;</mo> <mi>x</mi></msub></mtd></mtr> <mtr><mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x02212;</mo></msubsup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>&#x02212;</mo></msup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x02212;</mo></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  762. \array{
  763.  E_x^+ &amp;\overset{g(x^+)}{\to}&amp; E_x^+
  764.  \\
  765.  t_x\downarrow \;\; &amp;&amp; \;\; \downarrow t&apos;_x
  766.  \\
  767.  E_x^&#x2d; &amp;\overset{g(x^&#x2d;)}{\to}&amp; E_x^&#x2d;
  768. }
  769. </annotation></semantics></math></div>
  770.  
  771. <p>commute.</p>
  772.  
  773. <p>Clearly, the first reproduces the ordinary gauge transformations of the two ordinary connections, while the second gives the correct transformation of the tachyon field (compare Szabo&#8217;s equation (2.22)).</p>
  774.  
  775. <p><br/><strong>Example.</strong></p>
  776.  
  777. <p>An especially important form of tachyon fields are those discussed for instance on p. 29 and p. 38 of Szabo&#8217;s text. Here the bundle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi><mo>=</mo><msup><mi>E</mi> <mo>+</mo></msup><mo>&#x02295;</mo><msup><mi>E</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>E = E^+ \oplus E^&#x2d;</annotation></semantics></math> contains in particular a spinor bundle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>S</mi><mo>=</mo><msup><mi>S</mi> <mo>+</mo></msup><mo>&#x02295;</mo><msup><mi>S</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>S = S^+ \oplus S^&#x2d;</annotation></semantics></math> with the usual <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x02124;</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_2</annotation></semantics></math> spinor grading.</p>
  778.  
  779. <p>We may hence consider tachyon fields which on a local coordinate patch <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{x\}</annotation></semantics></math> look like</p>
  780.  
  781. <div class="numberedEq"><span>(17)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>:</mo></mtd> <mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>+</mo></msubsup></mtd> <mtd><mo>&#x02192;</mo></mtd> <mtd><msubsup><mi>E</mi> <mi>x</mi> <mo>&#x02212;</mo></msubsup></mtd></mtr> <mtr><mtd/> <mtd/> <mtd><mi>&#x03D5;</mi></mtd> <mtd><mo>&#x021A6;</mo></mtd> <mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>x</mi> <mi>i</mi></msup><msub><mi>&#x003B3;</mi> <mi>i</mi></msub><mi>&#x03D5;</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  782. \array{
  783.  \mathrm{tra}(t_x) &amp;:&amp; E^+_x &amp;\to&amp; E_x^&#x2d;
  784.  \\
  785.  &amp;&amp; \phi &amp;\mapsto&amp; f(x) x^i \gamma_i \phi
  786. }
  787. \,,
  788. </annotation></semantics></math></div>
  789.  
  790. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003B3;</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\gamma_i</annotation></semantics></math> are some generators of a representation of the relevant Clifford algebra on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>. If the tachyon field going the other way looks similar, the corresponding 0-form curvature</p>
  791.  
  792. <div class="numberedEq"><span>(18)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>F</mi> <mi>T</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mi>x</mi></msub><mo>&#x02218;</mo><msub><mover><mi>t</mi><mo stretchy="false">&#x000AF;</mo></mover> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  793.  F_T(x) = \mathrm{tra}(t_x \circ \bar t_x)
  794. </annotation></semantics></math></div>
  795.  
  796. <p>looks like</p>
  797.  
  798. <div class="numberedEq"><span>(19)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>F</mi> <mi>T</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">&#x0007C;</mo><mi>x</mi><msup><mo stretchy="false">&#x0007C;</mo> <mn>2</mn></msup><mo>&#x022C5;</mo><msub><mi mathvariant="normal">Id</mi> <mrow><msubsup><mi>E</mi> <mi>x</mi> <mo>+</mo></msubsup></mrow></msub><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  799.  F_T(x) = f(x)^2 |x|^2 \cdot \mathrm{Id}_{E_x^+}
  800.  \,.
  801. </annotation></semantics></math></div>
  802.  
  803. <p>This indicates that all the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> D-branes in the rank-<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> bundle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> decy into a single one, following the tachyon profile <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>f</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>f^2</annotation></semantics></math>.</p>
  804.  
  805. <p>Notice that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{tra}(t_x)</annotation></semantics></math> here is like a Dirac operator expressed in a plane-wave basis, i.e. Fourier-transformed/T-dualized. Compare this with the general relationship between tachyon fields and operators appearing in spectral triples and Fredholm modules (<a href="http://golem.ph.utexas.edu/string/archives/000880.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).</p>
  806.  
  807. <p><br/><strong>Complexes of D-branes and derived categories.</strong></p>
  808.  
  809. <p>While I am just giving a rather obvious reformulation of superconnections in terms of certain functors, I might add that the above seamlessly generalizes to the setup usually considered in detail for instance for topological strings, where we don&#8217;t just have branes/anti-branes, but an entire <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02124;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math>-grading of branes. </p>
  810.  
  811. <p>This <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02124;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math>-grading plays a major role in deriving that D-branes are described by derived categories (<a href="http://golem.ph.utexas.edu/string/archives/000538.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>), since it is responsible for the fact that there is not just one tachyon field</p>
  812.  
  813. <div class="numberedEq"><span>(20)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>E</mi> <mo>+</mo></msup><mover><mo>&#x02192;</mo><mi>T</mi></mover><msup><mi>E</mi> <mo>&#x02212;</mo></msup></mrow><annotation encoding='application/x-tex'>
  814.   E^+ \overset{T}{\to} E^&#x2d;
  815. </annotation></semantics></math></div>
  816.  
  817. <p>but an entire complex</p>
  818.  
  819. <div class="numberedEq"><span>(21)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x022EF;</mi><mo>&#x02192;</mo><msup><mi>E</mi> <mn>0</mn></msup><mover><mo>&#x02192;</mo><mrow><msup><mi>T</mi> <mn>1</mn></msup></mrow></mover><msup><mi>E</mi> <mn>1</mn></msup><mover><mo>&#x02192;</mo><mrow><msup><mi>T</mi> <mn>2</mn></msup></mrow></mover><msup><mi>E</mi> <mn>1</mn></msup><mo>&#x02192;</mo><mi>&#x022EF;</mi></mrow><annotation encoding='application/x-tex'>
  820.  \cdots \to  E^0 \overset{T^1}{\to} E^1 \overset{T^2}{\to} E^1 \to \cdots
  821. </annotation></semantics></math></div>
  822.  
  823. <p>of them.</p>
  824.  
  825. <p>There is nothing more natural than generalizing the above setup to this situation. Now we take <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{P}_2(X)</annotation></semantics></math> to contain diagrams of the form</p>
  826.  
  827. <div class="numberedEq"><span>(22)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><mi>&#x022EE;</mi></mtd> <mtd/> <mtd><mi>&#x022EE;</mi></mtd> <mtd/> <mtd><mi>&#x022EE;</mi></mtd></mtr> <mtr><mtd><msup><mi>x</mi> <mn>0</mn></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>1</mn> <mn>0</mn></msubsup></mrow></mover></mtd> <mtd><msup><mi>y</mi> <mn>0</mn></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>2</mn> <mn>0</mn></msubsup></mrow></mover></mtd> <mtd><msup><mi>z</mi> <mn>0</mn></msup></mtd></mtr> <mtr><mtd><msubsup><mi>t</mi> <mi>x</mi> <mn>1</mn></msubsup><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><msubsup><mi>t</mi> <mi>y</mi> <mn>1</mn></msubsup><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><msubsup><mi>t</mi> <mi>z</mi> <mn>1</mn></msubsup><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd></mtr> <mtr><mtd><msup><mi>x</mi> <mn>1</mn></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>1</mn> <mn>1</mn></msubsup></mrow></mover></mtd> <mtd><msup><mi>y</mi> <mn>1</mn></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>2</mn> <mn>1</mn></msubsup></mrow></mover></mtd> <mtd><msup><mi>z</mi> <mn>1</mn></msup></mtd></mtr> <mtr><mtd><msubsup><mi>t</mi> <mi>x</mi> <mn>2</mn></msubsup><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><msubsup><mi>t</mi> <mi>y</mi> <mn>2</mn></msubsup><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd> <mtd/> <mtd><msubsup><mi>t</mi> <mi>z</mi> <mn>2</mn></msubsup><mo stretchy="false">&#x02193;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mtd></mtr> <mtr><mtd><msup><mi>x</mi> <mn>2</mn></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>1</mn> <mn>2</mn></msubsup></mrow></mover></mtd> <mtd><msup><mi>y</mi> <mn>2</mn></msup></mtd> <mtd><mover><mo>&#x02192;</mo><mrow><msubsup><mi>&#x003B3;</mi> <mn>2</mn> <mn>2</mn></msubsup></mrow></mover></mtd> <mtd><msup><mi>z</mi> <mn>1</mn></msup></mtd></mtr> <mtr><mtd><mi>&#x022EE;</mi></mtd> <mtd/> <mtd><mi>&#x022EE;</mi></mtd> <mtd/> <mtd><mi>&#x022EE;</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  828. \array{
  829.  \vdots &amp;&amp; \vdots &amp;&amp; \vdots
  830.  \\
  831.  x^0 &amp;\overset{\gamma_1^0}{\to}&amp; y^0 &amp;\overset{\gamma_2^0}{\to}&amp; z^0
  832.  \\
  833.  t^1_x \downarrow\;\;\; &amp;&amp; t^1_y \downarrow\;\;\; &amp;&amp; t^1_z \downarrow \;\;\;
  834.  \\
  835.  x^1 &amp;\overset{\gamma_1^1}{\to}&amp; y^1 &amp;\overset{\gamma_2^1}{\to}&amp; z^1
  836.  \\
  837.  t^2_x \downarrow\;\;\; &amp;&amp; t^2_y \downarrow\;\;\; &amp;&amp; t^2_z \downarrow \;\;\;
  838.  \\
  839.  x^2 &amp;\overset{\gamma_1^2}{\to}&amp; y^2 &amp;\overset{\gamma_2^2}{\to}&amp; z^1
  840.  \\
  841.  \vdots &amp;&amp; \vdots &amp;&amp; \vdots
  842. }
  843. \,.
  844. </annotation></semantics></math></div>
  845.  
  846. <p>These encode strings stretching between stacks of D-branes of arbitrary ghost charge - or whatever you call the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02124;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math>-grading here.</p>
  847.  
  848. <p>All of the above discussion generalizes straightforwardly to this setup in the obvious way.</p>
  849.  
  850. <p>While everything is encoded in the single functor</p>
  851.  
  852. <div class="numberedEq"><span>(23)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi mathvariant="normal">tra</mi><mo>:</mo><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><mstyle mathvariant="bold"><mi>Trans</mi></mstyle><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>&#x02022;</mo></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  853.  \mathrm{tra} : \mathbf{P}_2(X) \to \mathbf{Trans}(E^\bullet)
  854.  \,,
  855. </annotation></semantics></math></div>
  856.  
  857. <p>we may again manifestly write this in terms of its components <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi mathvariant="normal">tra</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathrm{tra}^n</annotation></semantics></math> encoding ordinary connections on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>E</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>E^n</annotation></semantics></math>, together with almost-morphisms</p>
  858.  
  859. <div class="numberedEq"><span>(24)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi mathvariant="normal">tra</mi> <mrow><mi>n</mi><mo>&#x02212;</mo><mn>1</mn></mrow></msup><mover><mo>&#x02192;</mo><mrow><msup><mi>T</mi> <mi>n</mi></msup></mrow></mover><msup><mi mathvariant="normal">tra</mi> <mi>n</mi></msup><mover><mo>&#x02192;</mo><mrow><msup><mi>T</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mover><msup><mi mathvariant="normal">tra</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>
  860.  \mathrm{tra}^{n&#x2d;1} \overset{T^n}{\to} \mathrm{tra}^n \overset{T^{n+1}}{\to} \mathrm{tra}^{n+1}
  861. </annotation></semantics></math></div>
  862.  
  863. <p>endoced by the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">tra</mi><mo stretchy="false">(</mo><msubsup><mi>t</mi> <mi>x</mi> <mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{tra}(t^n_x)</annotation></semantics></math>. Notice that the failure of these to be natural transformations in measured precisely by the 1-form curvature of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">tra</mi></mrow><annotation encoding='application/x-tex'>\mathrm{tra}</annotation></semantics></math>, i.e. by the gauge-covariant derivative of the tachyon.</p>
  864.  
  865. <p>Because, as I vaguely indicated, all the above 1-functorial discussion secretly sits inside a 2-functorial description of surface transport, there is the possibility that we realize that what looks like the failure of a natural transformation of 1-functors is actually a pseudonatural transformation of 2-functors. But I won&#8217;t go into that at the moment.</p>
  866.  
  867. <p>In any case, we may consider the situation where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>T</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x02218;</mo><msup><mi>T</mi> <mi>n</mi></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>T^{n+1}\circ T^n = 0</annotation></semantics></math>, in wich case we get a complex of vector bundles with connection, with morphisms being tachyon fields - as known from the derived category description of D-branes. </p>
  868.  
  869. <p>Without any additional effort, we actually have the ability here to describe a complex of <em>twisted</em> bundles with connection, even a complex of twisted bundles together with their associated Kalb-Ramond gerbes with connection. But I won&#8217;t go into that at the moment.</p>
  870.  
  871. <p><br/><strong>Quivers</strong></p>
  872.  
  873. <p>Assume all our D-branes are pointlike in some sense (&#8220;fractional&#8221;, maybe). The vector bundles over them are then just vector spaces, and all ordinary connections <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi mathvariant="normal">tra</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathrm{tra}^n</annotation></semantics></math> in our superconnection <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">tra</mi></mrow><annotation encoding='application/x-tex'>\mathrm{tra}</annotation></semantics></math> disappear. We are left only with the tachyon component of the superconnections. </p>
  874.  
  875. <p>In this case, any diagram in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{P}_2(X)</annotation></semantics></math> consists only of point-like D-branes and strings stretching between these. This is usually called a <em>quiver diagram</em> describing the D-brane configuration (<a href="http://golem.ph.utexas.edu/string/archives/000794.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).</p>
  876.  
  877. <p>Applying our superconnection transport functor <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">tra</mi><mo>:</mo><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><mstyle mathvariant="bold"><mi>Trans</mi></mstyle><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>&#x02022;</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{tra} : \mathbf{P}_2(X) \to \mathbf{Trans}(E^\bullet)</annotation></semantics></math> on this quiver yields nothing but a quiver representation (<a href="http://golem.ph.utexas.edu/string/archives/000536.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).</p>
  878.  
  879. <p>That&#8217;s not supposed to be deep. But it&#8217;s true.</p>
  880.      </div>
  881.    </content>
  882.  </entry>
  883.  <entry>
  884.    <title type="html">K-Theory for Dummies, II</title>
  885.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000880.html" />
  886.    <updated>2006-07-27T10:06:22Z</updated>
  887.    <published>2006-07-26T10:42:39+00:00</published>
  888.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.880</id>
  889.    <summary type="text">Some remarks on K-theory and D-branes.</summary>
  890.    <author>
  891.      <name>urs</name>
  892.      <uri>http://www.math.uni-hamburg.de/home/schreiber</uri>
  893.      <email>[email protected]</email>
  894.    </author>
  895.    <category term="mathematical physics" />
  896.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000880.html">
  897.      <div xmlns="http://www.w3.org/1999/xhtml">
  898. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  899. <p>Before finishing the <a href="http://golem.ph.utexas.edu/string/archives/000879.html">last entry</a> I should review some basic facts about K-theory and D-branes, beyond of what I had in my previous notes (<a href="http://golem.ph.utexas.edu/string/archives/000627.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).</p>
  900.  
  901. <p>Apart from the Brodzki-Mathai-Rosenberg-Szabo paper (<a href="http://golem.ph.utexas.edu/string/archives/000879.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>) I&#8217;ll mainly follow</p>
  902.  
  903. <p>T. Asakawa, S. Sugimoto, S. Terashima
  904. <br/><em>D-branes, Matrix Theory and K-homology</em>
  905. <br/><a href="http://arxiv.org/abs/hep-th/0108085">hep-th/0108085</a></p>
  906.  
  907. <p>which is based in part on </p>
  908.  
  909. <p>Richard J. Szabo
  910. <br/><em>Superconnections, Anomalies and Non-BPS Brane Charges</em>
  911. <br/><a href="http://arxiv.org/abs/hep-th/0108043">hep-th/0108043</a></p>
  912.  
  913. <p>and</p>
  914.  
  915. <p>Jeffrey A. Harvey, Gregory Moore
  916. <br/><em>Noncommutative Tachyons and K-Theory</em>
  917. <br/><a href="http://arxiv.org/abs/hep-th/0009030">hep-th/0009030</a>.</p>
  918. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  919. <p>Since Witten&#8217;s original claim that D-branes are described by K-theory</p>
  920.  
  921. <p>Edward Witten
  922. <br/><em>Overview Of K-Theory Applied To Strings</em>
  923. <br/><a href="http://arxiv.org/abs/hep-th/0007175">hep-th/0007175</a></p>
  924.  
  925. <p>several refinements of the precise relationship have been discussed. Usually, the decategorification and Grothedieck group completion performed in forming K-theory from the category of vector bundles is identified with the physical process of partial mutual annihilation of
  926. space-filling D9 brane and anti-brane pairs, thereby realizing all lower-dimensional branes as decay products of D9-brane configurations.</p>
  927.  
  928. <p>T. Asakawa, S. Sugimoto and S. Terashima in their paper promote an alternative point of view, which, as they aim to demonstrate, makes more direct contact with the conception of K-theory in terms of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebras. Namely, they use the fact that one can go the other way around, and realize all higher-dimensional branes as composites of non-BPS D(-1) branes, using a certain flavor of what is called <em>Matrix Theory</em>.</p>
  929.  
  930. <p>From this point of view the world is described by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi><mo>&#x02192;</mo><mn>&#x0221E;</mn></mrow><annotation encoding='application/x-tex'>N\to \infty</annotation></semantics></math> objects called <em>non-BPS D-instantons</em>, whose geometric configuration is encoded by ten operators </p>
  931.  
  932. <div class="numberedEq"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>&#x003A6;</mi> <mi>&#x003BC;</mi></msup><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  933.  \Phi^\mu
  934.  \,,
  935. </annotation></semantics></math></div>
  936.  
  937. <p>called the <em>scalar fields</em>, as well as an operator</p>
  938.  
  939. <div class="numberedEq"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>T</mi><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  940.  T
  941.  \,,
  942. </annotation></semantics></math></div>
  943.  
  944. <p>called the <em>tachyon field</em>, all represented on some seperable Hilbert space </p>
  945.  
  946. <div class="numberedEq"><span>(3)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x0210B;</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  947.  \mathcal{H}
  948.  \,.
  949. </annotation></semantics></math></div>
  950.  
  951. <p>The mathematically inclined reader not familiar with this might (or might not) get an impression of what is going on here by looking at</p>
  952.  
  953. <p>Alain Connes, Michael R. Douglas, Albert Schwarz
  954. <br/><em>Noncommutative Geometry and Matrix Theory</em>
  955. <br/><a href="http://arxiv.org/abs/hep-th/9711162">hep-th/9711162</a>.</p>
  956.  
  957. <p>The dynamics of these funny objects is encoded by a functional which contains terms like</p>
  958.  
  959. <div class="numberedEq"><span>(4)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi mathvariant="normal">tr</mi><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><msup><mi>T</mi> <mn>2</mn></msup></mrow></msup><mrow><mo>[</mo><msup><mi>&#x003A6;</mi> <mi>&#x003BC;</mi></msup><mo>,</mo><msup><mi>&#x003A6;</mi> <mi>&#x003BD;</mi></msup><mo>]</mo></mrow><mrow><mo>[</mo><msub><mi>&#x003A6;</mi> <mi>&#x003BC;</mi></msub><mo>,</mo><msub><mi>&#x003A6;</mi> <mi>&#x003BD;</mi></msub><mo>]</mo></mrow><mo>+</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><msup><mi>T</mi> <mn>2</mn></msup></mrow></msup><mrow><mo>[</mo><msup><mi>&#x003A6;</mi> <mi>&#x003BC;</mi></msup><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mrow><mo>[</mo><msub><mi>&#x003A6;</mi> <mi>&#x003BC;</mi></msub><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  960.  \mathrm{tr}\left(
  961.   e^{&#x2d;T^2}\left[\Phi^\mu, \Phi^\nu\right]\left[\Phi_\mu, \Phi_\nu\right]
  962.  +
  963.   e^{&#x2d;T^2}\left[\Phi^\mu, T\right]\left[\Phi_\mu, T\right]
  964.  \right)
  965.  \,.
  966. </annotation></semantics></math></div>
  967.  
  968. <p>This is something one derives from boundary string field theory.</p>
  969.  
  970. <p>One expects to be able to make full sense of this only if at least </p>
  971.  
  972. <div class="numberedEq"><span>(5)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><msup><mi>T</mi> <mn>2</mn></msup></mrow></msup></mrow><annotation encoding='application/x-tex'>
  973.  e^{&#x2d;T^2}
  974. </annotation></semantics></math></div>
  975.  
  976. <p>is trace class, hence <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> is an unbounded operator, and</p>
  977.  
  978. <div class="numberedEq"><span>(6)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">[</mo><msup><mi>&#x003A6;</mi> <mi>&#x003BC;</mi></msup><mo>,</mo><msup><mi>&#x003A6;</mi> <mi>&#x003BD;</mi></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"/><mo>,</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mo stretchy="false">[</mo><mi>T</mi><mo>,</mo><msup><mi>&#x003A6;</mi> <mi>&#x003BC;</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>
  979.  [\Phi^\mu,\Phi^\nu] \,,\;\;\;\;   [T,\Phi^\mu]  
  980. </annotation></semantics></math></div>
  981.  
  982. <p>are in</p>
  983.  
  984. <div class="numberedEq"><span>(7)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>&#x0210B;</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  985.  B(\mathcal{H})
  986.  \,,
  987. </annotation></semantics></math></div>
  988.  
  989. <p>the space of bounded operators on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0210B;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{H}</annotation></semantics></math>.</p>
  990.  
  991. <p>Anyone who has run across a spectral triple before should now have a d&#x000E9;j&#x000E0; vu. If we like Fredholm modules better than spectral triples, we may instead consider the normalized tachyon operator</p>
  992.  
  993. <div class="numberedEq"><span>(8)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>T</mi> <mi>b</mi></msub><mo>=</mo><mfrac><mi>T</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>T</mi> <mn>2</mn></msup><msup><mo stretchy="false">)</mo> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  994.  T_b = \frac{T}{(1+T^2)^{1/2}}
  995. \,.
  996. </annotation></semantics></math></div>
  997.  
  998. <p>This is now bounded and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><msup><mi>T</mi> <mn>2</mn></msup></mrow></msup></mrow><annotation encoding='application/x-tex'>e^{&#x2d;T^2}</annotation></semantics></math> being trace class implies that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>T</mi> <mi>b</mi> <mn>2</mn></msubsup><mo>&#x02212;</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>T_b^2 &#x2d; 1</annotation></semantics></math> is compact. Similarly the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">[</mo><mi>T</mi><mo>,</mo><msup><mi>&#x003A6;</mi> <mi>&#x003BC;</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>[T,\Phi^\mu]</annotation></semantics></math> are now required to be compact operators - and we have obtained a Fredholm module from our spectral triple describing D-instanton dynamics.</p>
  999.  
  1000. <p>This story is thought to extend to an entire dictionary, which should look roughly like this.</p>
  1001.  
  1002. <div class="numberedEq"><span>(9)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mtext>math</mtext></mstyle></mtd> <mtd><mstyle mathvariant="bold"><mtext>physics</mtext></mstyle></mtd></mtr> <mtr><mtd><mtext>Dirac operator</mtext><mi>D</mi></mtd> <mtd><mtext>Tachyon field</mtext><mi>T</mi></mtd></mtr> <mtr><mtd><mtext>Fredholm operator</mtext><mi>F</mi></mtd> <mtd><mtext>normalized Tachyon field</mtext><mfrac><mi>T</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>T</mi> <mn>2</mn></msup><msup><mo stretchy="false">)</mo> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mtd></mtr> <mtr><mtd><mtext>Hilbert space</mtext><mi>&#x0210B;</mi></mtd> <mtd><mtext>space of</mtext><mi>N</mi><mo>=</mo><mn>&#x0221E;</mn><mtext>D-instanton Chan-Paton labels</mtext></mtd></mtr> <mtr><mtd><msup><mi>C</mi> <mo>*</mo></msup><mtext>-algebra</mtext><mi>A</mi></mtd> <mtd><mtext>spacetime encoded in scalar fields</mtext><mi>&#x003A6;</mi></mtd></mtr> <mtr><mtd><mtext>spectral triple</mtext><mo stretchy="false">(</mo><mi>&#x0210B;</mi><mo>,</mo><mi>D</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mtext>D-brane configuration</mtext></mtd></mtr> <mtr><mtd><mtext>fredholm module</mtext><mo stretchy="false">(</mo><mi>&#x0210B;</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mtext>D-brane configuration</mtext></mtd></mtr> <mtr><mtd><mtext>unitary equivalence of Fredholm modules</mtext></mtd> <mtd><mtext>gauge equivalence of D-brane configurations</mtext></mtd></mtr> <mtr><mtd><mtext>operator homotopy of Fredholm modules</mtext></mtd> <mtd><mtext>deformation of tachyon configuration</mtext></mtd></mtr> <mtr><mtd><mtext>spectral action functional of</mtext><mi>D</mi></mtd> <mtd><mtext>dynamics of tachyon condensation induced by</mtext><mi>T</mi></mtd></mtr> <mtr><mtd><mtext>representation</mtext><mo>*</mo><mtext>-homomorphism</mtext><mi>&#x03D5;</mi><mo>:</mo><mi>A</mi><mo>&#x02192;</mo><mi mathvariant="normal">B</mi><mo stretchy="false">(</mo><mi>&#x0210B;</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>embedding</mi><mi>of</mi><mi>D</mi><mo>&#x02212;</mo><mi>brane</mi><mi>in</mi><mi>spacetime</mi></mtd></mtr> <mtr><mtd><mi mathvariant="normal">im</mi><mi>&#x03D5;</mi></mtd> <mtd><mtext>the (algebra describing the) world-volume of the embedded D-brane</mtext></mtd></mtr> <mtr><mtd><mi mathvariant="normal">ker</mi><mi>F</mi></mtd> <mtd><mtext>D-branes left over after tachyon condensation</mtext></mtd></mtr> <mtr><mtd><mi mathvariant="normal">coker</mi><mi>F</mi></mtd> <mtd><mtext>anti D-branes left over after tachyon condensation</mtext></mtd></mtr> <mtr><mtd><mi mathvariant="normal">ind</mi><mi>F</mi><mo>=</mo><mi mathvariant="normal">dim</mi><mi mathvariant="normal">ker</mi><mi>F</mi><mo>&#x02212;</mo><mi mathvariant="normal">dim</mi><mi mathvariant="normal">coker</mi><mi>F</mi></mtd> <mtd><mtext>net number of D-branes left after tachyon condensation</mtext></mtd></mtr> <mtr><mtd><mtext>K-homology</mtext><msup><mi>K</mi> <mo>&#x02022;</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>D</mi><mo>&#x02212;</mo><mi>branes</mi><mi>modulo</mi><mi>gauge</mi><mi>equivalence</mi></mtd></mtr> <mtr><mtd><mtext>K-cohomology (-theory)</mtext><msub><mi>K</mi> <mo>&#x02022;</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>D</mi><mo>&#x02212;</mo><mi>brane</mi><mi>RR</mi><mi>charges</mi><mi>modulo</mi><mi>gauge</mi><mi>equivalence</mi></mtd></mtr> <mtr><mtd><mtext>index pairing</mtext><msup><mi>K</mi> <mo>&#x02022;</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>&#x000D7;</mo><msub><mi>K</mi> <mo>&#x02022;</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><mi>&#x02124;</mi></mtd> <mtd><mtext>net number of D-branes after tachyon condensation for given RR-charge</mtext></mtd></mtr> <mtr><mtd><mtext>degenerate Fredholm module</mtext><msup><mi>F</mi> <mn>2</mn></msup><mo>&#x02212;</mo><mn>1</mn><mo>=</mo><mn>0</mn></mtd> <mtd><mtext>D-instantons completely disappearing after tachyon condensation</mtext></mtd></mtr> <mtr><mtd><mtext>KK-theory</mtext></mtd> <mtd><mo>?</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  1003. \array{
  1004.  \mathbf{\text{math}} &amp; \mathbf{\text{physics}}
  1005.  \\
  1006.  \text{Dirac operator} D &amp; \text{Tachyon field} T
  1007.  \\
  1008.  \text{Fredholm operator} F &amp; \text{normalized Tachyon field} \frac{T}{(1 + T^2)^{1/2}}
  1009.  \\
  1010.  \text{Hilbert space} \mathcal{H} &amp; \text{space of}N=\infty \text{D&#x2d;instanton Chan&#x2d;Paton labels}
  1011.  \\
  1012.  C^*\text{&#x2d;algebra} A &amp; \text{spacetime encoded in scalar fields} \Phi  
  1013.  \\
  1014.  \text{spectral triple} (\mathcal{H},D,A) &amp; \text{D&#x2d;brane configuration}
  1015.  \\
  1016.  \text{fredholm module}(\mathcal{H},F,A) &amp; \text{D&#x2d;brane configuration}
  1017.  \\
  1018.  \text{unitary equivalence of Fredholm modules}
  1019.  &amp;
  1020.  \text{gauge equivalence of D&#x2d;brane configurations}
  1021.  \\
  1022.  \text{operator homotopy of Fredholm modules} &amp; \text{deformation of tachyon configuration}
  1023.  \\
  1024.  \text{spectral action functional of} D &amp; \text{dynamics of tachyon condensation induced by}T
  1025.  \\
  1026.  \text{representation} *\text{&#x2d;homomorphism} \phi : A \to \mathrm{B}(\mathcal{H}) &amp; embedding of D&#x2d;brane in spacetime
  1027.  \\
  1028.  \mathrm{im} \phi &amp; \text{the (algebra describing the) world&#x2d;volume of the embedded D&#x2d;brane}
  1029.  \\
  1030.  \mathrm{ker}F &amp; \text{D&#x2d;branes left over after tachyon condensation}
  1031.  \\
  1032.  \mathrm{coker}F &amp; \text{anti D&#x2d;branes left over after tachyon condensation}
  1033.  \\
  1034.  \mathrm{ind}F = \mathrm{dim}\mathrm{ker}F &#x2d; \mathrm{dim}\mathrm{coker}F
  1035.  &amp;
  1036.  \text{net number of D&#x2d;branes left after tachyon condensation}
  1037.  \\
  1038.  \text{K&#x2d;homology} K^\bullet(A) &amp; D&#x2d;branes modulo gauge equivalence
  1039. \\
  1040.  \text{K&#x2d;cohomology (&#x2d;theory)} K_\bullet(A) &amp; D&#x2d;brane RR charges modulo gauge equivalence
  1041.  \\
  1042.  \text{index pairing} K^\bullet(A)\times K_\bullet(A) \to \mathbb{Z}
  1043.    &amp; \text{net number of D&#x2d;branes after tachyon condensation for given RR&#x2d;charge}
  1044.  \\
  1045.  \text{degenerate Fredholm module} F^2 &#x2d; 1 = 0 &amp; \text{D&#x2d;instantons completely disappearing after tachyon condensation}
  1046.  \\
  1047.  \text{KK&#x2d;theory} &amp; ?
  1048. }
  1049. </annotation></semantics></math></div>
  1050.  
  1051. <p>Asakawa, Sugimoto and Terashima are mainly interested in identifying the very last entry on the right. They argue in </p>
  1052.  
  1053. <p>T. Asakawa, S. Sugimoto, S. Terashima
  1054. <br/><em>D-branes and KK-theory in Type I String Theory</em>
  1055. <br/><a href="http://arxiv.org/abs/hep-th/0202165">hep-th/0202165</a></p>
  1056.  
  1057. <p>that the KK-theory <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>KK</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>KK(A,B)</annotation></semantics></math> describes D-brane configurations on product spaces <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi><mo>&#x02297;</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A\otimes B</annotation></semantics></math> (with <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> the corresponding <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebras.) I am not sure that I precisely follow this, but it is certainly compelling to associate KK-theory to pairs of D-brane configurations. Given that D-branes &#8220;are modules&#8221; (<a href="http://golem.ph.utexas.edu/string/archives/000795.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>), KK-theory seems to describe the associated bimodules.</p>
  1058.  
  1059. <p>But before I get into this KK issue I should try to explain some of the entries of the above dictionary.</p>
  1060.  
  1061. <p><strong>Tachyon fields as Dirac operators</strong></p>
  1062.  
  1063. <p>One of the crucial keys for unlocking the above dictionary is the fact that the tachyon field operators that we are talking about indeed are Dirac operators in situations where they describe &#8220;geometric&#8221; D-branes.</p>
  1064.  
  1065. <p>Seiji Terashima
  1066. <br/><em>A Construction of Commutative D-branes from Lower Dimensional Non-BPS D-branes</em>
  1067. <br/><a href="http://arxiv.org/abs/hep-th/0101087">hep-th/0101087</a>.</p>
  1068.  
  1069. <p><strong>D-branes as spectral triples or Fredholm modules</strong></p>
  1070.  
  1071. <p>Recall the target space definition of a D-brane (<a href="http://golem.ph.utexas.edu/string/archives/000879.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>), as a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi mathvariant="normal">spin</mi> <mi>&#x02102;</mi></msup></mrow><annotation encoding='application/x-tex'>\mathrm{spin}^\mathbb{C}</annotation></semantics></math> manifold <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> with a Chan-Paton K-class <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> over it embedded into spacetime by means of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x03D5;</mi><mo>:</mo><mi>W</mi><mo>&#x02192;</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\phi : W \to X</annotation></semantics></math>.</p>
  1072.  
  1073. <p>This data is equivalently encoded in a spectral triple <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>&#x0210B;</mi><mo>,</mo><mi>&#x003C1;</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><mi>B</mi><mo stretchy="false">(</mo><mi>&#x0210B;</mi><mo stretchy="false">)</mo><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{H},\rho : C(X) \to B(\mathcal{H}),D)</annotation></semantics></math> or, alternatively, a Fredholm module (hence a class in K-homology) by setting</p>
  1074.  
  1075. <div class="numberedEq"><span>(10)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x0210B;</mi><mo>=</mo><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><msub><mi>S</mi> <mi>W</mi></msub><mo>&#x02297;</mo><mi>E</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1076.  \mathcal{H} = L^2(W, S_W \otimes E)
  1077.  \,,
  1078. </annotation></semantics></math></div>
  1079.  
  1080. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>S</mi> <mi>W</mi></msub></mrow><annotation encoding='application/x-tex'>S_W</annotation></semantics></math> is the spinor bundle on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math>, by letting the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>*</annotation></semantics></math>-homomorphism</p>
  1081.  
  1082. <div class="numberedEq"><span>(11)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x003C1;</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><mi>B</mi><mo stretchy="false">(</mo><mi>&#x0210B;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1083.  \rho : C(X) \to B(\mathcal{H})
  1084. </annotation></semantics></math></div>
  1085.  
  1086. <p>be given by precomposition with <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x03D5;</mi></mrow><annotation encoding='application/x-tex'>\phi</annotation></semantics></math> and taking <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> to be the Dirac operator on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math>.</p>
  1087.  
  1088. <p>Notice that in particular the embedding of the D-brane is indeed encoded in an algebra homomorphism, as stated in the above table (which is of course nothing but a result of applying the contravariant functor from topological spaces to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebras).</p>
  1089.  
  1090. <p>We don&#8217;t want to distinguish D-branes up to gauge transformations, up continuous deformations of the tachyon potential <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02243;</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>\simeq D</annotation></semantics></math> and up to addition of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>-brane configurations which disappear after tachyon condensation. Dividing out the space of Fredholm modules by these three equivalence relations, following the above dictionary, yields the K-homology group, and our D-brane defines a class of that. Which K-homology group precisely (which degree, real or complex, depends on whether we are in ty IIA, IIB or I, and on details which I have not mentioned yet.)</p>
  1091.  
  1092. <p><strong>&#8220;Bi-branes&#8221; as bimodules</strong></p>
  1093.  
  1094. <p>Let&#8217;s follow Asakawa et al. again and think of the Hilbert space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0210B;</mi></mrow><annotation encoding='application/x-tex'>\mathcal{H}</annotation></semantics></math> appearing in our spectral triples as the space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0210B;</mi><mo>&#x02243;</mo><msup><mi>l</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><mo lspace="0em" rspace="thinmathspace">C</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{H} \simeq l^2(N,\C)</annotation></semantics></math> of square summable series with amplitudes valued in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02102;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{C}</annotation></semantics></math>, one for each of the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> Chan-Paton labels. Since <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02102;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{C}</annotation></semantics></math> is the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebra of a point, we might imagine that it makes sense to consider Chan-Paton amplitudes indexed by a more general algebra <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, encoding not just a point but some other space. This would lead to a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>-Hilbert space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>l</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>l^2(N,B)</annotation></semantics></math>, where linearity with respect to the complex numbers is replaced by linearity with respect to some <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebra <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>.</p>
  1095.  
  1096. <p>As I said, Askawa et al. argue that this does describe D-branes on product manifolds. While I am wondering if it should not rather describe configurations with open strings stretched between a space encoded by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and one encoded by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, I am willing to accept that it seems not incredibly unnatural to extend the above table and generalize Fredholm modules to bimodules:</p>
  1097.  
  1098. <p><strong>Kasparov&#8217;s KK-Theory</strong></p>
  1099.  
  1100. <p>So we start with two <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebra <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>. We want to construct something like an <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>-<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>-bimodule with a Fredholm operator represented on it. The fact that left and right bimodule action commute can be rephrased as saying that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> acts <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>-linearly, if we like. In particular, if <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi><mo>=</mo><mo lspace="0em" rspace="thinmathspace">C</mo></mrow><annotation encoding='application/x-tex'>B=\C</annotation></semantics></math>, this would just say that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> acts by ordinary linear maps on some vector space.</p>
  1101.  
  1102. <p>So generalizing this, one says that a <strong>Hilbert <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>-module</strong> is a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>-module with a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>-valued (instead of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>-valued) inner product. An (odd) Kasparaov <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>-<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> bimodule is then a Hilbert <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>-module on which <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> acts by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>-linear self-adjoint operators, together with a self-adjoint operator <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>, playing the role of a Fredholm operator. </p>
  1103.  
  1104. <p>The definition (inlcuding the details which I skipped) is such that for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi><mo>=</mo><mo lspace="0em" rspace="thinmathspace">C</mo></mrow><annotation encoding='application/x-tex'>B = \C</annotation></semantics></math> this reduces to an ordinary Fredholm module.</p>
  1105.  
  1106. <p>The duality between K-theory and K-homology in this language amounts to switching between Kasparav <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>-<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo lspace="0em" rspace="thinmathspace">C</mo></mrow><annotation encoding='application/x-tex'>\C</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo lspace="0em" rspace="thinmathspace">C</mo></mrow><annotation encoding='application/x-tex'>\C</annotation></semantics></math>-<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> bimodules.</p>
  1107.  
  1108. <p>The important point is that on Kasparov bimodules we do have a product induced from the ordinary tensor product of modules:</p>
  1109.  
  1110. <div class="numberedEq"><span>(12)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>KK</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>&#x000D7;</mo><msub><mi>KK</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><msub><mi>KK</mi> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1111.  KK_i(A,B) \times KK_j(B,C) \to KK_{i+j}(A,C)
  1112.  \,.
  1113. </annotation></semantics></math></div>
  1114.  
  1115. <p>While I am not sure yet about the <em>physical</em> interpretation of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>KK</mi> <mo>&#x02022;</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>KK_\bullet(A,B)</annotation></semantics></math> for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> both different from the ground field, KK-theory certainly provides us with a very natural way to obtain K-cohomology as the dual to the Fredholm-module K-cohomology description of D-branes described above.</p>
  1116.  
  1117. <p>We simply observe that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>KK</mi><mo stretchy="false">(</mo><mi>&#x02102;</mi><mo>,</mo><mi>&#x02102;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>KK(\mathbb{C},\mathbb{C})</annotation></semantics></math> is the group of equivalence classes of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02102;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{C}</annotation></semantics></math>-modules equipped with a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02102;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{C}</annotation></semantics></math>-action. But this are nothing but vector spaces, hence the K-theory of a point, hence isomorphic to the natural numbers <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02124;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math>.</p>
  1118.  
  1119. <p>Therefore the index pairing of K-theory with K-homology (<a href="http://golem.ph.utexas.edu/string/archives/000627.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>) now essentially just becomes the product of a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02102;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{C}</annotation></semantics></math>-<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> bimodule (a K-theory class) with a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>-<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02102;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{C}</annotation></semantics></math>-bimodule (a K-homology class).</p>
  1120.  
  1121. <p>The real power of the KK-formuation, though, as emphasized by Brodzki, Mathai, Rosenberg and Szabo, is that, since we are in a braided context, we may perform the KK-product only over certain factors of the left and right algebras. This allows to encode Poincar&#x000E9; duality. Maybe I&#8217;ll talk about that another time.</p>
  1122.      </div>
  1123.    </content>
  1124.  </entry>
  1125.  <entry>
  1126.    <title type="html"><![CDATA[Brodzki, Mathai, Rosenberg &amp; Szabo on D-Branes, RR-Fields and Duality]]></title>
  1127.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000879.html" />
  1128.    <updated>2006-07-26T13:14:44Z</updated>
  1129.    <published>2006-07-20T22:36:01+00:00</published>
  1130.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.879</id>
  1131.    <summary type="text">Mathai et al give a detailed analysis of the nature of D-branes, RR-charges and T-duality using and extending the topological/algebraic machinery known from "topological T-duality". </summary>
  1132.    <author>
  1133.      <name>urs</name>
  1134.      <uri>http://www.math.uni-hamburg.de/home/schreiber</uri>
  1135.      <email>[email protected]</email>
  1136.    </author>
  1137.    <category term="mathematical physics" />
  1138.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000879.html">
  1139.      <div xmlns="http://www.w3.org/1999/xhtml">
  1140. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1141. <p>I have begun reading </p>
  1142.  
  1143. <p>Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard J. Szabo
  1144. <br/><em>D-Branes, RR-Fields and Duality on Noncommutative Manifolds</em>
  1145. <br/><a href="http://arxiv.org/abs/hep-th/0607020">hep-th/0607020</a> .</p>
  1146.  
  1147. <p>This is a detailed study of the concepts appearing in the title, using and extending the topological and algebraic machinery known from &#8220;topological T-duality&#8221; (<a href="http://golem.ph.utexas.edu/string/archives/000827.html">I</a>, <a href="http://golem.ph.utexas.edu/string/archives/000828.html">II</a>, <a href="http://golem.ph.utexas.edu/string/archives/000829.html">III</a>, <a href="http://golem.ph.utexas.edu/string/archives/000838.html">IV</a>, <a href="http://golem.ph.utexas.edu/~distler/blog/archives/000837.html">V</a>). The motivation is to formulate everything in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebraic language, in order to get, both, a powerful language for the ordinary situation as well as a generalization to noncommutative spacetimes.</p>
  1148.  
  1149. <p><br/><strong>Warning:</strong> I am still editing this entry.</p>
  1150. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1151. <p>The main result presented is a generalization of the formula for D-brane charge known from</p>
  1152.  
  1153. <p>Ruben Minasian, Gregory Moore
  1154. <br/><em>K-theory and Ramond-Ramond charge</em>
  1155. <br/><a href="http://arxiv.org/abs/hep-th/9710230">hep-th/9710230</a>,</p>
  1156.  
  1157. <p>which is roughly the Chern class of the Chan-Paton bundle times the square root of the Todd class.</p>
  1158.  
  1159. <p>The authors of the present paper manage to find an analogue of the Todd class for general <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi mathvariant="normal">spin</mi> <mi>c</mi></msup></mrow><annotation encoding='application/x-tex'>\mathrm{spin}^c</annotation></semantics></math> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebras, such that in terms of this the D-brane charge in the corresponding noncommutative geometry is still given by a formula of the familiar form.</p>
  1160.  
  1161. <p>Before starting, I would like to record some of the definitions.</p>
  1162.  
  1163. <p>Let <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a manifold - spacetime - assumed to be spin, of dimension 10 and equipped with a Riemannian metric.</p>
  1164.  
  1165. <p>First, an elementary definition, for the sake of completeness. The goal is to very precisely state what we are talking about.</p>
  1166.  
  1167. <p><strong>Definition 1)</strong> <em>An (non-twisted)</em> <strong>D-brane</strong> <em>in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is an embedded <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi mathvariant="normal">spin</mi> <mi>c</mi></msup></mrow><annotation encoding='application/x-tex'>\mathrm{spin}^c</annotation></semantics></math>-manifold</em></p>
  1168.  
  1169. <div class="numberedEq"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x03D5;</mi><mo>:</mo><mi>W</mi><mo>&#x02192;</mo><mi>X</mi><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1170.  \phi : W \to X
  1171.  \,,
  1172. </annotation></semantics></math></div>
  1173.  
  1174. <p><em>together with a K-theory class</em></p>
  1175.  
  1176. <div class="numberedEq"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>E</mi><mo>&#x02208;</mo><msup><mi>K</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1177.  E \in K^0(W)
  1178.  \,,
  1179. </annotation></semantics></math></div>
  1180.  
  1181. <p><em>specifying the</em> <strong>Chan-Paton bundle</strong> <em>on the D-brane according to</em></p>
  1182.  
  1183. <p>Edward Witten
  1184. <br/><em>Overview Of K-Theory Applied To Strings</em>
  1185. <br/><a href="http://arxiv.org/abs/hep-th/0007175">hep-th/0007175</a>.</p>
  1186.  
  1187. <p>The point of this being &#8220;non-twisted&#8221; is that, more generally, there is a Kalb-Ramond field on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, manifested in terms of an abeluian gerbe with connection and curving. In the presence of non-trivial such gerbes </p>
  1188.  
  1189. <p>Daniel S. Freed, Edward Witten
  1190. <br/><em>Anomalies in String Theory with D-Branes</em>
  1191. <br/><a href="http://arxiv.org/abs/hep-th/9907189">hep-th/9907189</a></p>
  1192.  
  1193. <p>tells us that we have a twisted Chan-Paton bundle, with the twist given by the class of the gerbe pulled back to the D-brane and equal to the third Stieffel-Whitney-class of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> (<a href="http://golem.ph.utexas.edu/string/archives/000795.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).</p>
  1194.  
  1195. <p>So we really need a more refined definition of D-brane. (And I should maybe remark that we are talking here about what one could call &#8220;geometric D-branes&#8221;, those coming from submanifolds, as opposed to the more general ones arising as arbitrary <abbr title="Conformal Field Theory">CFT</abbr> boundary conditions. No <abbr>CFT</abbr> is considered explicitly in the present case.)</p>
  1196.  
  1197. <p><strong>Definition 2)</strong> <em>A (possibly twisted)</em> <strong>D-brane</strong> <em>in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> in the presence of a Kalb-Ramond background field, i.e. in the presence of an abelian gerbe with connection and curving <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo>&#x02192;</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\mathbf{G} \to X</annotation></semantics></math> (<a href="http://golem.ph.utexas.edu/string/archives/000868.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>) on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, with characteristic class</em></p>
  1198.  
  1199. <div class="numberedEq"><span>(3)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>H</mi><mo>&#x02208;</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1200.  H \in H^3(X,\mathbb{Z})
  1201. </annotation></semantics></math></div>
  1202.  
  1203. <p><em>is an embedded oriented submanifold</em></p>
  1204.  
  1205. <div class="numberedEq"><span>(4)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x03D5;</mi><mo>:</mo><mi>W</mi><mo>&#x02192;</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>
  1206.  \phi : W \to X
  1207. </annotation></semantics></math></div>
  1208.  
  1209. <p><em>together with a Chan-Paton K-theory class</em></p>
  1210.  
  1211. <div class="numberedEq"><span>(5)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>E</mi><mo>&#x02208;</mo><msup><mi>K</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1212.  E \in K^0(W)
  1213. </annotation></semantics></math></div>
  1214.  
  1215. <p><em>such that the Freed-Witten anomaly cancellation condition</em></p>
  1216.  
  1217. <div class="numberedEq"><span>(6)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>&#x03D5;</mi> <mo>*</mo></msup><mi>H</mi><mo>=</mo><msub><mi>W</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1218.  \phi^* H = W_3(W)
  1219. </annotation></semantics></math></div>
  1220.  
  1221. <p><em>holds, where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>W</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>W_3</annotation></semantics></math> denotes the third Stieffel/Whitney class</em>.</p>
  1222.  
  1223. <p>There are various ways how to encode the gerbe <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo>&#x02192;</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\mathbf{G} \to X</annotation></semantics></math> in terms of ordinary differential geometric structures. As before in the context of topological T-duality, one here finds it convenient to think of the gerbe in terms of its (possibly infinite-rank) gerbe modules (<a href="http://golem.ph.utexas.edu/string/archives/000795.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>), hence <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>PU</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&#x02243;</mo><mi>BU</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>&#x02243;</mo><mi>&#x02102;</mi><msup><mi>P</mi> <mn>&#x0221E;</mn></msup></mrow><annotation encoding='application/x-tex'>PU(H)\simeq BU(1) \simeq \mathbb{C}P^\infty</annotation></semantics></math>-bundles, for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> some seperable Hilbert space, or rather the associated vector bundles of compact operators on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>. I believe we can think of these as coming from the endomorphism bundle of our gerbe module. </p>
  1224.  
  1225. <p>This latter point of view will be the one best suited for the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebraic setup, since we may associate to such algebra bundles naturally the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebra of sections of the bundle.</p>
  1226.  
  1227. <p>(I am <a href="">still</a> wondering if from this algebra one can get a globally well defined version of the Seiberg-Witten Moyal star, which is locally given by something like <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B8;</mi><mo>&#x0221D;</mo><mo stretchy="false">(</mo><mi>g</mi><mo>&#x000B1;</mo><mi>B</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\theta \propto (g \pm B)^{&#x2d;1}</annotation></semantics></math>, for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> the local gerbe connection 2-form. But nobody seems to know.)</p>
  1228.  
  1229. <p>In order to say something about the <strong>charges</strong> of these D-branes, we need a couple of pairings involving K-cohomology and -homology (<a href="http://golem.ph.utexas.edu/string/archives/000627.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).</p>
  1230.  
  1231. <p>A D-brane as defined above, can be taken to define a class in the K-<em>hom</em>ology of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p>
  1232.  
  1233. <div class="numberedEq"><span>(7)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>K</mi> <mo>&#x02022;</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1234.  K_\bullet(X)
  1235.  \,,
  1236. </annotation></semantics></math></div>
  1237.  
  1238. <p>or, more precisely, in the presence of a Kalb-Ramond-gerbe, a class in the <em>twisted</em> K-homology</p>
  1239.  
  1240. <div class="numberedEq"><span>(8)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>K</mi> <mo>&#x02022;</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1241.  K_\bullet(X,H)
  1242.  \,.
  1243. </annotation></semantics></math></div>
  1244.  
  1245. <p>Next, we want to define the <strong>RR-charge</strong> of a given D-brane.  In words, this is computed by</p>
  1246.  
  1247. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> pushing the Chan-Paton bundle from the worldvolume forward (the &#8220;wrong way&#8221; (<a href="http://golem.ph.utexas.edu/string/archives/000830.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>)) intto spacetime <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p>
  1248.  
  1249. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> computing its Chern class there</p>
  1250.  
  1251. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> cup-multiplying this Chern class with half of the Atiyah-Hirzebruch class of spacetime</p>
  1252.  
  1253. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> pulling the result back to the world-volume <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math></p>
  1254.  
  1255. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> and evaluating it on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math>.</p>
  1256.  
  1257. <p>If we denote the ordinary Chern class of a bundle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> with <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">ch</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{ch}(N)</annotation></semantics></math>, and the <strong>modified Chern class</strong> by</p>
  1258.  
  1259. <div class="numberedEq"><span>(9)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi mathvariant="normal">Ch</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi mathvariant="normal">ch</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>&#x0222A;</mo><msqrt><mrow><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow></msqrt><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1260.  \mathrm{Ch}(N) := \mathrm{ch}(N) \cup \sqrt{\hat A(H)}
  1261.  \,,
  1262. </annotation></semantics></math></div>
  1263.  
  1264. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover></mrow><annotation encoding='application/x-tex'>\hat A</annotation></semantics></math> is the Atiyah-Hirzebruch class, then the above reads in formulas</p>
  1265.  
  1266. <p><strong>Definition 3)</strong>
  1267. The RR-charge of a D-brane with world volume <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x03D5;</mi><mo>:</mo><mi>W</mi><mo>&#x02192;</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\phi : W \to X</annotation></semantics></math> and Chan-Paton bundle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>
  1268. is</p>
  1269.  
  1270. <div class="numberedEq"><span>(10)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>Q</mi> <mrow><mi>&#x03D5;</mi><mo>,</mo><mi>W</mi><mo>,</mo><mi>E</mi></mrow></msub><mo>:</mo><mo>=</mo><msup><mi>&#x03D5;</mi> <mo>*</mo></msup><mi mathvariant="normal">Ch</mi><mo stretchy="false">(</mo><msub><mi>&#x03D5;</mi> <mo>!</mo></msub><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>W</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1271.  Q_{\phi,W,E} := \phi^* \mathrm{Ch}(\phi_! E)[W]
  1272.  \,.  
  1273. </annotation></semantics></math></div>
  1274.  
  1275. <p>Understanding this formula in more detail crucially involves Poinca&#x000E9; duality. The goal is to generalize this notion of Poincar&#x000E9; duality to K-theory over noncommutative spaces. The authors of the above paper argue that the best language for that  is
  1276. <a href="http://golem.ph.utexas.edu/string/archives/000880.html">Kasparov&#8217;s KK-Theory</a>
  1277. It turns out that the notion of Poincar&#x000E9; duality we wish to find amounts KK bimodule composition with invertible <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi><mo>&#x02297;</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A \otimes B</annotation></semantics></math>-<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo lspace="0em" rspace="thinmathspace">C</mo></mrow><annotation encoding='application/x-tex'>\C</annotation></semantics></math>-bimodules.</p>
  1278.  
  1279. <p>The authors emphasize the usefulness of a certain diagrammatic notation for dealing with the bimodule-like product on Kasparov bimodules. To my mind, it seems that this notation is nothing but the obvious string diagram notation for dealing with categoriy of bimodules internal to braided categories. Essentially, all this lives in fact in a 3-category of the sort discussed <a href="http://golem.ph.utexas.edu/string/archives/000786.html">here</a>.</p>
  1280.  
  1281. <p><br/><em>[ more later&#8230;.]</em></p>
  1282.      </div>
  1283.    </content>
  1284.  </entry>
  1285.  <entry>
  1286.    <title type="html">2-Palatini</title>
  1287.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000876.html" />
  1288.    <updated>2007-01-17T20:47:17Z</updated>
  1289.    <published>2006-07-19T17:20:54+00:00</published>
  1290.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.876</id>
  1291.    <summary type="text">Some remarks on formulations of (super)gravity in terms of n-connections.</summary>
  1292.    <author>
  1293.      <name>urs</name>
  1294.      <uri>http://www.math.uni-hamburg.de/home/schreiber</uri>
  1295.      <email>[email protected]</email>
  1296.    </author>
  1297.    <category term="mathematical physics" />
  1298.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000876.html">
  1299.      <div xmlns="http://www.w3.org/1999/xhtml">
  1300. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1301. <p>A few entries ago, I was claiming that other people are implicitly claiming that the field content of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding='application/x-tex'>D=11</annotation></semantics></math> supergravity encodes precisely a 3-connection taking values in a certain Lie 3-algebra (<a href="http://golem.ph.utexas.edu/string/archives/000840.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).  In my first attempt to make a couple of remarks on that, I ran out of time (<a href="http://golem.ph.utexas.edu/string/archives/000840.html#c003819"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>). Here is the second attempt.</p>
  1302. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1303. <p>There are in particular two questions which I would like to comment on.  </p>
  1304.  
  1305. <p>1) What about the 3-form really being a Chern-Simons 3-form?   </p>
  1306.  
  1307. <p>2) What happens in degree 2?     </p>
  1308.  
  1309. <p><br/><strong>1) What about the 3-form really being a Chern-Simons 3-form?</strong>
  1310. We know that the 3-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> should really come from Chern-Simons 3-forms of a Lorentz and an <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding='application/x-tex'>E_8</annotation></semantics></math> connection. This information is not present in the classical FDA formulation of supergravity.  Here is a general observation on connections on Chern-Simons 2-gerbes, which might be relevant.  <br/>  Consider a Lie group <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>. Its Lie algebra  is encoded in the fda defined by</p>
  1311.  
  1312. <div class="numberedEq"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>a</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow/> <mi>bc</mi></msub><msup><mi>a</mi> <mi>b</mi></msup><msup><mi>a</mi> <mi>c</mi></msup><mo>=</mo><mn>0</mn><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>   \mathbf{d} a^a + \frac{1}{2}C^a{}_{bc}a^b a^c = 0   \,, </annotation></semantics></math></div>
  1313.  
  1314. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow/> <mi>bc</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{C^a{}_{bc}\}</annotation></semantics></math> are the structure constant in some chosen basis. Nilpotency of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{d}</annotation></semantics></math> is equivalent to the Jacobi identity.  Next, consider the crossed module <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi><mo>&#x02192;</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>G \to G</annotation></semantics></math>. The fda corresponding to its Lie 2-algebra is given by</p>
  1315.  
  1316. <div class="numberedEq"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd/> <mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>a</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow/> <mi>bc</mi></msub><msup><mi>a</mi> <mi>b</mi></msup><msup><mi>a</mi> <mi>c</mi></msup><mo>+</mo><msup><mi>b</mi> <mi>a</mi></msup><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd/> <mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>b</mi> <mi>a</mi></msup><mo>+</mo><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow/> <mi>bc</mi></msub><msup><mi>a</mi> <mi>b</mi></msup><msup><mi>b</mi> <mi>c</mi></msup><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned}   &amp;\mathbf{d} a^a + \frac{1}{2}C^a{}_{bc}a^b a^c + b^a = 0   \\   &amp; \mathbf{d} b^a + C^a{}_{bc}a^b b^c = 0 \end{aligned}   \,. </annotation></semantics></math></div>
  1317.  
  1318. <p>A 2-connection with values in this Lie 2-algebra is given by a 1-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>A</mi> <mi>a</mi></msup></mrow><annotation encoding='application/x-tex'>A^a</annotation></semantics></math> and a 2-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>B</mi> <mi>a</mi></msup></mrow><annotation encoding='application/x-tex'>B^a</annotation></semantics></math> and has curvatures</p>
  1319.  
  1320. <div class="numberedEq"><span>(3)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>F</mi> <mn>1</mn></msub></mtd> <mtd><mo>=</mo><msub><mi>F</mi> <mi>A</mi></msub><mo>+</mo><mi>B</mi></mtd></mtr> <mtr><mtd><msub><mi>F</mi> <mn>2</mn></msub></mtd> <mtd><mo>=</mo><msub><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mi>A</mi></msub><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned}   F_1 &amp;= F_A + B   \\   F_2 &amp;= \mathbf{d}_A B \end{aligned}   \,. </annotation></semantics></math></div>
  1321.  
  1322. <p> Without mentioning anything like 2-connections, but implicitly considering precisely this, such 2-connections have been studied for instance in <a href="http://arxiv.org/abs/hep-th/0204059">hep-th/0204059</a>. The 2-group aspect is discussed very nicely in <a href="http://arxiv.org/abs/hep-th/0206130">hep-th/0206130</a>.  Suppose we demand this 2-curvature to <em>vanish</em>. This is equivalent to </p>
  1323.  
  1324. <div class="numberedEq"><span>(4)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>B</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><msub><mi>F</mi> <mi>A</mi></msub><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>   B = &#x2d;F_A   \,. </annotation></semantics></math></div>
  1325.  
  1326. <p>Hence a <em>flat</em> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>&#x02192;</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(G\to G)</annotation></semantics></math>-2-connection is the same as an ordinary <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-connection. In fact, a trivial flat <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>&#x02192;</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(G\to G)</annotation></semantics></math>-2-bundle is the same as an ordinary <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-bundle.  Next, let&#8217;s also add a generator in degree 3 to our fda, to get the Lie 3-algebra encoded by</p>
  1327.  
  1328. <div class="numberedEq"><span>(5)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd/> <mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>a</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow/> <mi>bc</mi></msub><msup><mi>a</mi> <mi>b</mi></msup><msup><mi>a</mi> <mi>c</mi></msup><mo>+</mo><msup><mi>b</mi> <mi>a</mi></msup><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd/> <mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>b</mi> <mi>a</mi></msup><mo>+</mo><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow/> <mi>bc</mi></msub><msup><mi>a</mi> <mi>b</mi></msup><msup><mi>b</mi> <mi>c</mi></msup><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd/> <mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>c</mi><mo>+</mo><msub><mi>k</mi> <mi>ab</mi></msub><msup><mi>b</mi> <mi>a</mi></msup><msup><mi>b</mi> <mi>b</mi></msup><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned}   &amp;\mathbf{d} a^a + \frac{1}{2}C^a{}_{bc}a^b a^c + b^a = 0   \\   &amp; \mathbf{d} b^a + C^a{}_{bc}a^b b^c = 0   \\   &amp; \mathbf{d} c + k_{ab} b^a b^b = 0 \end{aligned}   \,, </annotation></semantics></math></div>
  1329.  
  1330. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>k</mi> <mi>ab</mi></msub></mrow><annotation encoding='application/x-tex'>k_{ab}</annotation></semantics></math> is proportional to the Killing form on the Lie algebra of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>. This ensures that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{d}</annotation></semantics></math> is still nilpotent.  A 3-connection with values in this Lie 3-algebra is a 1-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>A</mi> <mi>a</mi></msup></mrow><annotation encoding='application/x-tex'>A^a</annotation></semantics></math>, a 2-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>B</mi> <mi>a</mi></msup></mrow><annotation encoding='application/x-tex'>B^a</annotation></semantics></math> and a 3-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. Its curvature is</p>
  1331.  
  1332. <div class="numberedEq"><span>(6)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>F</mi> <mn>1</mn></msub></mtd> <mtd><mo>=</mo><msub><mi>F</mi> <mi>A</mi></msub><mo>+</mo><mi>B</mi></mtd></mtr> <mtr><mtd><msub><mi>F</mi> <mn>2</mn></msub></mtd> <mtd><mo>=</mo><msub><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mi>A</mi></msub><mi>B</mi></mtd></mtr> <mtr><mtd><msub><mi>F</mi> <mn>3</mn></msub></mtd> <mtd><mo>=</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>C</mi><mo>+</mo><mi mathvariant="normal">tr</mi><mo stretchy="false">(</mo><mi>B</mi><mo>&#x02227;</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned}   F_1 &amp;= F_A + B   \\   F_2 &amp;= \mathbf{d}_A B   \\   F_3 &amp;= \mathbf{d} C + \mathrm{tr}(B \wedge B) \end{aligned}   \,. </annotation></semantics></math></div>
  1333.  
  1334. <p>Assume again that the curvature <em>vanishes</em>. In the first two degrees this is, as before, equivalent to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><msub><mi>F</mi> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>B = &#x2d; F_A</annotation></semantics></math>. In third degree it now says that</p>
  1335.  
  1336. <div class="numberedEq"><span>(7)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>C</mi><mo>&#x0221D;</mo><mi mathvariant="normal">tr</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>A</mi></msub><mo>&#x02227;</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>   \mathbf{d} C \propto \mathrm{tr}(F_A \wedge F_A)   \,. </annotation></semantics></math></div>
  1337.  
  1338. <p>But this means that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> itself must be, up to a closed part, the Chern-Simons 3-form of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math></p>
  1339.  
  1340. <div class="numberedEq"><span>(8)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>C</mi><mo>&#x0221D;</mo><mi mathvariant="normal">CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>   C \propto \mathrm{CS}(A)   \,. </annotation></semantics></math></div>
  1341.  
  1342. <p>Therefore a <em></em> 3-connection of the above sort is  the local connection of a Chern-Simons gerbe corresponding to some <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-bundle with connection.  Now take <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> to be the product of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding='application/x-tex'>E_8</annotation></semantics></math> with the Lorentz group in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding='application/x-tex'>D=11</annotation></semantics></math> and there we go.  There is one problem. The known FDA description of SUGRA uses not the Lorentz group in lowest degree, but the Poincar&#x000E9; group. Plus, there is already something going on in degree 3. Both facts imply that applying the above construction to this case is not enirely straightforward.  Partly motivated by this, in the next subsection I make some comments on the Poincar&#x000E9; versus Lorentz issue.  </p>
  1343.  
  1344. <p><br/><strong>2) What happens in degree 2?</strong>  </p>
  1345.  
  1346. <p>The Poincar&#x000E9;-group happens to be a semidirect product of the Lorentz and the translation group, with one factor being abelian. Every such semidirect product group is already naturally a 2-group. It&#8217;s Lie 2-algebra corresponds to the fda defined by</p>
  1347.  
  1348. <div class="numberedEq"><span>(9)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>a</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow/> <mi>bc</mi></msub><msup><mi>a</mi> <mi>b</mi></msup><msup><mi>a</mi> <mi>c</mi></msup><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>t</mi> <mi>i</mi></msup><mo>+</mo><msup><mi>&#x003B1;</mi> <mi>i</mi></msup><msub><mrow/> <mi>aj</mi></msub><mspace width="thinmathspace"/><msup><mi>a</mi> <mi>a</mi></msup><msup><mi>t</mi> <mi>j</mi></msup><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned}   \mathbf{d}a^a + \frac{1}{2}C^a{}_{bc}a ^b a^c = 0   \\   \mathbf{d}t^i + \alpha^i{}_{aj}\, a^a t^j = 0 \end{aligned} \,, </annotation></semantics></math></div>
  1349.  
  1350. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><msup><mi>a</mi> <mi>a</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{a^a\}</annotation></semantics></math> are a basis of the dual of the Lorentz Lie algebra, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><msup><mi>t</mi> <mi>i</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{t^i\}</annotation></semantics></math> a basis of the dual of the translation algebra, and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B1;</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math> encodes the action of one on the other.  Motivated by this observation, there have been attempts to formulate gravity in terms of the Poincar&#x000E9; 2-group (<a href="http://golem.ph.utexas.edu/string/archives/000835.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).  But notice this: In the above fda, the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>t</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>t^i</annotation></semantics></math> live in degree 2. Hence a 2-connection with values in this guy contains a 2-form valued in the translation Lie algebra. What&#8217;s that?  On the other hand, in the known fda formulation of sugra, one regards the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>t</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>t^i</annotation></semantics></math> as living in degree 1. Accordingly, the connection with values in this fda has a 1-form with values in the translation Lie algebra. This 1-form gets interpreted as the vielbein, which makes the fda SUGRA formulation something like an example for a Palatini-formulation of gravity (<a href="http://en.wikipedia.org/wiki/Palatini_action"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).  Still, let&#8217;s assume we agree that it is unnatural to have the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>t</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>t^i</annotation></semantics></math> not in degree 2. But let&#8217;s also assume that we agree that we want a vielbein. There is one way out:  Just like <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>so</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>so(n,1)</annotation></semantics></math> acts on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x0211D;</mi> <mrow><mi>n</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^{n,1}</annotation></semantics></math>, it also acts on the abelian group <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mo lspace="thinmathspace" rspace="thinmathspace">&#x022C0;</mo> <mn>2</mn></msup><msup><mi>&#x0211D;</mi> <mrow><mi>n</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\bigwedge^2 \mathbb{R}^{n,1}</annotation></semantics></math> of bivectors, surely. Hence we can just as well consider the 2-group coming from the crossed module</p>
  1351.  
  1352. <div class="numberedEq"><span>(10)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">(</mo><mover><mo lspace="thinmathspace" rspace="thinmathspace">&#x022C0;</mo> <mn>2</mn></mover><msup><mi>&#x0211D;</mi> <mrow><mi>n</mi><mo>,</mo><mn>1</mn></mrow></msup><mover><mo>&#x02192;</mo><mi>t</mi></mover><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>   (\bigwedge^2\mathbb{R}^{n,1}  \stackrel{t}{\to} SO(n,1)) </annotation></semantics></math></div>
  1353.  
  1354. <p>with trivial <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math>.  The corresponding fda looks precisley like the one above, just with <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>t</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>t^i</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B1;</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math> suitably reinterpreted.  A 2-connection with values in this Lie 2-algebra now consists of an <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>so</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>so(n,1)</annotation></semantics></math>-valued 1-form, together with a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mo lspace="thinmathspace" rspace="thinmathspace">&#x022C0;</mo> <mn>2</mn></msup><msup><mi>&#x0211D;</mi> <mrow><mi>n</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\bigwedge^2 \mathbb{R}^{n,&#x2d;1}</annotation></semantics></math>-valued 2-form.  This now has a chance to encode a vielbein, since by wedging any vielbein with itself we do obtain such a 2-form.  There is apparently an even better solution, using spin groups instead. This idea can be found discussed in section IV and V of <a href="http://arxiv.org/abs/hep-th/0204059">hep-th/0204059</a> (see also <a href="http://arxiv.org/abs/gr-qc/9502037">gr-qc/9502037</a>), whose authors consider precisely this, albeit without identifying <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-connections explicitly (they call what they do &#8220;generalized differential calculus&#8221;).  So consider <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>D</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>D = n + 1 = 3+ 1</annotation></semantics></math> for a moment and pick the double cover of the Lorentz group group <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi><mo>=</mo><mi mathvariant="normal">SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>&#x02102;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>G = \mathrm{SL}(2,\mathbb{C})</annotation></semantics></math> and the translation group <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>T</mi><mo>&#x02243;</mo><msup><mi>&#x0211D;</mi> <mrow><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>T \simeq \mathbb{R}^{3,1}</annotation></semantics></math>. Using the 2-component spinorial language (and the Pauli matrices <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>&#x003C3;</mi> <mi>a</mi> <mrow><mi>AB</mi><mo>&#x02032;</mo></mrow></msubsup></mrow><annotation encoding='application/x-tex'>\sigma_a^{AB&apos;}</annotation></semantics></math>), we can write a vielbein 1-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>e</mi> <mi>a</mi></msup></mrow><annotation encoding='application/x-tex'>e^a</annotation></semantics></math> as <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>e</mi> <mrow><mi>AB</mi><mo>&#x02032;</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>e^{AB&apos;}</annotation></semantics></math> and wedge it with itself to form the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>&#x02102;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{SL}(2,\mathbb{C})</annotation></semantics></math>-valued 2-form</p>
  1355.  
  1356. <div class="numberedEq"><span>(11)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>B</mi> <mi>AB</mi></msup><mo>=</mo><msup><mi>e</mi> <mrow><mi>AA</mi><mo>&#x02032;</mo></mrow></msup><mo>&#x02227;</mo><msup><mi>e</mi> <mi>B</mi></msup><msub><mrow/> <mrow><mi>A</mi><mo>&#x02032;</mo></mrow></msub><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>   B^{AB} = e^{AA&apos;} \wedge e^{B}{}_{A&apos;}   \,. </annotation></semantics></math></div>
  1357.  
  1358. <p>So we get a 2-connection for our Lie 2-algebra given by an <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>&#x02102;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{SL}(2,\mathbb{C})</annotation></semantics></math>-valued connection 1-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x003C9;</mi> <mi>AB</mi></msup></mrow><annotation encoding='application/x-tex'>\omega^{AB}</annotation></semantics></math> and that &#8220;surface-vielbein&#8221; 2-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>B</mi> <mi>AB</mi></msup></mrow><annotation encoding='application/x-tex'>B^{AB}</annotation></semantics></math>.  In terms of these the Einstein-Hilbert action (in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>D</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding='application/x-tex'>D=4</annotation></semantics></math>) is of BF-theory form (<a href="http://golem.ph.utexas.edu/string/archives/000777.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>)</p>
  1359.  
  1360. <div class="numberedEq"><span>(12)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo>&#x0221D;</mo><msub><mo>&#x0222B;</mo> <mi>X</mi></msub><mspace width="thinmathspace"/><mi mathvariant="normal">tr</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>&#x003C9;</mi></msub><mspace width="thinmathspace"/><mo>&#x02227;</mo><mspace width="thinmathspace"/><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>   \propto \int_X \,\mathrm{tr}(F_\omega \,\wedge\, B)   \,. </annotation></semantics></math></div>
  1361.  
  1362. <p>Consider again the situation where the curvature of our 2-connection <em>vanishes</em> in degree 2. This now says that our 1-form connection is torsion free with respect to the metric encoded by the vielbein.  The vielbein-2-form <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> should be precisely the &#8220;area vielbein&#8221; corresponding to the <strong>area metrics</strong> which have been argued to encode the geometry of string backgrounds (<a href="http://golem.ph.utexas.edu/string/archives/000687.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).    </p>
  1363.  
  1364. <p><strong>Update.</strong> Something closely related is also discussed in <a href="http://arxiv.org/abs/gr-qc/9804061">gr-qc/9804061</a>.</p>
  1365.      </div>
  1366.    </content>
  1367.  </entry>
  1368.  <entry>
  1369.    <title type="html"><![CDATA[Herbst, Hori &amp; Page on Equivalence of LG and CY]]></title>
  1370.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000874.html" />
  1371.    <updated>2006-07-19T03:43:35Z</updated>
  1372.    <published>2006-07-18T15:52:58+00:00</published>
  1373.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.874</id>
  1374.    <summary type="text">Hori and Herbst on equivalence of Landau-Ginzburg and Calabi-Yau TFT models using gauged linear sigma models.</summary>
  1375.    <author>
  1376.      <name>urs</name>
  1377.      <uri>http://www.math.uni-hamburg.de/home/schreiber</uri>
  1378.      <email>[email protected]</email>
  1379.    </author>
  1380.    <category term="mathematical physics" />
  1381.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000874.html">
  1382.      <div xmlns="http://www.w3.org/1999/xhtml">
  1383. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1384. <p>Yesterday, Kentaro Hori gave a talk on (unpublished) joint work with Manfred Herbst and David Page, another version of which I had heard a while ago in Vienna (<a href="http://golem.ph.utexas.edu/string/archives/000832.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>), on </p>
  1385.  
  1386. <p>K. Hori, M. Herbst
  1387. <br/><em>Phases of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>N=2</annotation></semantics></math> theories in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>1+1</annotation></semantics></math> dimensions with boundary, I</em></p>
  1388. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1389. <p>Given some homogeneous polynomial </p>
  1390.  
  1391. <div class="numberedEq"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>&#x022EF;</mi><mo>,</mo><msub><mi>x</mi> <mi>N</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1392.  W(x_1,\cdots,x_N)
  1393. </annotation></semantics></math></div>
  1394.  
  1395. <p>of degree <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math>, one can, roughly, associate two different sorts of 2-dimensional field theories with it.</p>
  1396.  
  1397. <p>1) On the one hand we can consider sigma-models whose target is the projective variety (<a href="http://golem.ph.utexas.edu/string/archives/000849.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>)  <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> of zeros of this polynomial. If this happens to be a Calabi-Yau we can consider the A- or B-model topological string on that target.</p>
  1398.  
  1399. <p>2) On the other hand, one can regard <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> as the superpotential of a Landau-Ginzburg model (<a href="http://golem.ph.utexas.edu/string/archives/000692.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).</p>
  1400.  
  1401. <p>In the first case, for the B-model string, the corresponding category of branes is <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>D</mi> <mi>b</mi></msup><mo stretchy="false">(</mo><mi mathvariant="normal">Coh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>D^b(\mathrm{Coh}(X))</annotation></semantics></math>, the bounded derived category of coherent sheaves on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> (<a href="http://golem.ph.utexas.edu/string/archives/000538.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>).</p>
  1402.  
  1403. <p>In the second case, the category of branes looks superficially different. Let me just call this the category of Landau-Ginzburg B-branes.</p>
  1404.  
  1405. <p>Now, we can think of both these models as different points in one and the same moduli space of a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>N=(2,2)</annotation></semantics></math> gauged linear sigma-model (GLSM). There is a certain parameter, called <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>r</mi></mrow><annotation encoding='application/x-tex'>r</annotation></semantics></math>, parameterizing this model, and in the limit that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>r</mi></mrow><annotation encoding='application/x-tex'>r</annotation></semantics></math> tends to plus or minus infinity, the GLSM tends to the nonlinear <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C3;</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math>-model on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">&#x0007C;</mo><mi>W</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>&#x022EF;</mi><mo>,</mo><msub><mi>x</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>X = \{x_i| W(x_1,\cdots, x_N) = 0\}</annotation></semantics></math> or the Landau-Ginzburg model with superpotential <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math>, respectively.</p>
  1406.  
  1407. <p>What Hori and Herbst are trying to do is to use this gauged linear sigma model to flow the category of Landau-Ginzburg B-branes through moduli space to the derived category of coherent sheaves on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, thus realizing the equivalence of these two categories by means of a &#8220;physical&#8221; system.</p>
  1408.  
  1409. <p>That both categories are in fact equivalent (when suitable assumptions hold which I am glossing over), was shown
  1410. in</p>
  1411.  
  1412. <p>Dmitri Orlov
  1413. <br/><em>Derived categories of coherent sheaves and triangulated categories of singularities</em>
  1414. <br/><a href="http://arxiv.org/abs/math.AG/0503632">math.AG/0503632</a>,</p>
  1415.  
  1416. <p>theorem 3.11.</p>
  1417.  
  1418. <p>Related results have been discussed in</p>
  1419.  
  1420. <p>Yujiro Kawamata
  1421. <br/><em>Log Crepant Birational Maps and Derived Categories</em>
  1422. <br/><a href="http://arxiv.org/abs/math.RT/0510187">math.RT/0510187</a> .</p>
  1423.  
  1424. <p>As far as I understood, Hori and Herbst expected that in fact the category of branes of the full GLSM is, too, equivalent to both of the above categories. Hoewever, it turns out that this equivalence has so far only been shown for special choices of some other parameter, called <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B8;</mi></mrow><annotation encoding='application/x-tex'>\theta</annotation></semantics></math>.</p>
  1425.  
  1426. <p>I am wondering if this should be worrisome. Wouldn&#8217;t it be natural for the category of branes of the GLSM to be larger (and non-equivalent) to the category of branes obtained in the limiting case <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>r</mi><mo>&#x02192;</mo><mo>&#x000B1;</mo><mn>&#x0221E;</mn></mrow><annotation encoding='application/x-tex'>r \to \pm \infty</annotation></semantics></math>?</p>
  1427.  
  1428. <p>Hori proceeded by spelling out lots of details at the level of Lagrangians, which I won&#8217;t even try to reproduce in total.</p>
  1429.  
  1430. <p>I&#8217;ll just indicate enough details to see the two parameters <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>r</mi></mrow><annotation encoding='application/x-tex'>r</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B8;</mi></mrow><annotation encoding='application/x-tex'>\theta</annotation></semantics></math> appearing.</p>
  1431.  
  1432. <p>The GLSM involves a twisted chiral gauge superfield <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> (I think), which appears in the Lagrangian in terms of its superderivative</p>
  1433.  
  1434. <div class="numberedEq"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x003A3;</mi><mo>=</mo><msub><mover><mi>D</mi><mo stretchy="false">&#x000AF;</mo></mover> <mo>+</mo></msub><msub><mi>D</mi> <mo>&#x02212;</mo></msub><mi>V</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1435.  \Sigma = \bar D_+ D_&#x2d; V
  1436.  \,.
  1437. </annotation></semantics></math></div>
  1438.  
  1439. <p>The <strong>gauge kinetic term</strong> of this field in the Lagranian is</p>
  1440.  
  1441. <div class="numberedEq"><span>(3)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo>&#x0221D;</mo><mo>&#x0222B;</mo><msup><mi>d</mi> <mn>2</mn></msup><mi>&#x003B8;</mi><mrow><mo>(</mo><mover><mi>&#x003A3;</mi><mo stretchy="false">&#x000AF;</mo></mover><mi>&#x003A3;</mi><mo>)</mo></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1442.  \propto \int d^2 \theta \left(
  1443.    \bar \Sigma \Sigma    
  1444.  \right)
  1445.  \,.
  1446. </annotation></semantics></math></div>
  1447.  
  1448. <p>There is also a <strong>matter kinetic term</strong> of the form</p>
  1449.  
  1450. <div class="numberedEq"><span>(4)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo>&#x0221D;</mo><mo>&#x0222B;</mo><msup><mi>d</mi> <mn>4</mn></msup><mi>&#x003B8;</mi><mrow><mo>(</mo><mover><mi>P</mi><mo stretchy="false">&#x000AF;</mo></mover><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>nV</mi></mrow></msup><mi>P</mi><mo>+</mo><msub><mover><mi>X</mi><mo stretchy="false">&#x000AF;</mo></mover> <mn>1</mn></msub><msup><mi>e</mi> <mi>V</mi></msup><msub><mi>X</mi> <mn>1</mn></msub><mo>+</mo><mi>&#x022EF;</mi><mo>+</mo><msub><mover><mi>X</mi><mo stretchy="false">&#x000AF;</mo></mover> <mn>1</mn></msub><msup><mi>e</mi> <mi>V</mi></msup><msub><mi>X</mi> <mn>1</mn></msub><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>
  1451.  \propto
  1452.  \int d^4 \theta
  1453.  \left(
  1454.    \bar P e^{&#x2d;nV} P
  1455.    +
  1456.    \bar X_1 e^V X_1
  1457.    +
  1458.    \cdots
  1459.    +
  1460.    \bar X_1 e^V X_1  
  1461.  \right)
  1462. </annotation></semantics></math></div>
  1463.  
  1464. <p>and a <strong>superpotential</strong> term for these matter fields</p>
  1465.  
  1466. <div class="numberedEq"><span>(5)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo>&#x0221D;</mo><mi mathvariant="normal">Re</mi><mo>&#x0222B;</mo><msup><mi>d</mi> <mn>2</mn></msup><mi>&#x003B8;</mi><mi>P</mi><mi>W</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>&#x022EF;</mi><mo>,</mo><msub><mi>X</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1467.  \propto
  1468.  \mathrm{Re}
  1469.  \int d^2 \theta P W(X_1,\cdots,X_N)
  1470.  \,,
  1471. </annotation></semantics></math></div>
  1472.  
  1473. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> is our polynomial from above.</p>
  1474.  
  1475. <p>Finally, and that&#8217;s where the two parameters come in, there is an <strong>F-term</strong></p>
  1476.  
  1477. <div class="numberedEq"><span>(6)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo>&#x0221D;</mo><mi mathvariant="normal">Re</mi><mo>&#x0222B;</mo><msup><mi>d</mi> <mn>2</mn></msup><mover><mi>&#x003B8;</mi><mo stretchy="false">&#x002DC;</mo></mover><mspace width="thickmathspace"/><mi>t</mi><mi>&#x003A3;</mi><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1478.  \propto
  1479.  \mathrm{Re}
  1480.  \int
  1481.  d^2 \tilde \theta
  1482.  \;t\Sigma
  1483.  \,,
  1484. </annotation></semantics></math></div>
  1485.  
  1486. <p>where the complex parameter <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math> has real and imaginary part given by the two parameters in question</p>
  1487.  
  1488. <div class="numberedEq"><span>(7)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>t</mi><mo>=</mo><mi>r</mi><mo>&#x02212;</mo><mi>i</mi><mi>&#x003B8;</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1489.  t = r &#x2d; i \theta
  1490.  \,.
  1491. </annotation></semantics></math></div>
  1492.  
  1493. <p>One can see that for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>r</mi><mo>&#x02192;</mo><mn>&#x0221E;</mn></mrow><annotation encoding='application/x-tex'>r \to \infty</annotation></semantics></math> the matter fields <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>X_i</annotation></semantics></math> localize on the zeros of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math>, thus leading to a nonlinear sigma-model on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p>
  1494.      </div>
  1495.    </content>
  1496.  </entry>
  1497.  <entry>
  1498.    <title type="html">Gomi on Reps of p-Form Connection Quantum Algebras</title>
  1499.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000873.html" />
  1500.    <updated>2006-07-19T03:43:35Z</updated>
  1501.    <published>2006-07-14T13:10:15+00:00</published>
  1502.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.873</id>
  1503.    <summary type="text">Gomi on the construction of projective unitary reps of centrally extended groups of p-form connections.</summary>
  1504.    <author>
  1505.      <name>urs</name>
  1506.      <uri>http://www.math.uni-hamburg.de/home/schreiber</uri>
  1507.      <email>[email protected]</email>
  1508.    </author>
  1509.    <category term="mathematical physics" />
  1510.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000873.html">
  1511.      <div xmlns="http://www.w3.org/1999/xhtml">
  1512. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1513. <p>Quantizing abelian self-dual <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>-form connections on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>p</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(2p+2)</annotation></semantics></math>-dimensional spaces gives rise to quantum observable algebras which are Heisenberg central extensions of the group of gauge equivalence classes of these connections, with the cocycle given by the Chern-Simons term in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>2p+1</annotation></semantics></math> dimensions (<a href="http://golem.ph.utexas.edu/string/archives/000868.html">I</a>, <a href="http://golem.ph.utexas.edu/string/archives/000871.html">II</a>, <a href="http://golem.ph.utexas.edu/string/archives/000866.html">III</a> ).</p>
  1514.  
  1515. <p>In</p>
  1516.  
  1517. <p>Kiyonori Gomi
  1518. <br/><em>Projective unitary representations of smooth Deligne cohomology groups</em>
  1519. <br/><a href="http://arxiv.org/abs/math.RT/0510187">math.RT/0510187</a></p>
  1520.  
  1521. <p>the author spells out the technical details of the construction of unitray representations for a certain (&#8220;level 2&#8221;) cases of these central extensions (compare the discussion in <a href="http://golem.ph.utexas.edu/string/archives/000871.html">II</a>), effectively generalizing the construction of positive energy reps of Kac-Moody groups <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>LU</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>&#x02124;</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\hat LU(1)/\mathbb{Z}_2</annotation></semantics></math> (corresponding to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>p=0</annotation></semantics></math>) to higher <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>.</p>
  1522.  
  1523. <p>These reps should be the Hilbert spaces of states of the quantum theory of self-dual <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>-form fields. Their irreps would correspond to the superselection sectors.
  1524. </p>
  1525. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  1526. <p>Fix once and for all</p>
  1527.  
  1528. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> an integer <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>k</mi><mo>&#x02208;</mo><mi>&#x02115;</mi></mrow><annotation encoding='application/x-tex'>k \in \mathbb{N}</annotation></semantics></math></p>
  1529.  
  1530. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mn>4</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(4k+1)</annotation></semantics></math>-dimensional manifold <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> .</p>
  1531.  
  1532. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> some Riemannian metric <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p>
  1533.  
  1534. <p>Denote</p>
  1535.  
  1536. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mover><mi>H</mi><mo stretchy="false">&#x5E;</mo></mover> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\hat H^n(X)</annotation></semantics></math> the group of gauge equivalence classes of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>&#x02212;</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(n&#x2d;1)</annotation></semantics></math>-form connections, here to be thought of as realized as the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th Deligne hypercohomology of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p>
  1537.  
  1538. <div class="numberedEq"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mover><mi>H</mi><mo stretchy="false">&#x5E;</mo></mover> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">(</mo><mi>n</mi><msubsup><mo stretchy="false">)</mo> <mi>D</mi> <mn>&#x0221E;</mn></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1539.  \hat H^n(X) = H^{n}(X,\mathbb{Z}(n)^\infty_D)
  1540.  \,.
  1541. </annotation></semantics></math></div>
  1542.  
  1543. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> by </p>
  1544.  
  1545. <div class="numberedEq"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><msubsup><mo stretchy="false">)</mo> <mi>D</mi> <mn>&#x0221E;</mn></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1546.  \mathbf{G}(X) = H^{2k+1}(X,\mathbb{Z}(2k+1)^\infty_D)
  1547. </annotation></semantics></math></div>
  1548.  
  1549. <p>the group of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2k</annotation></semantics></math>-form connections on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p>
  1550.  
  1551. <p><br/><strong>The strategy.</strong></p>
  1552.  
  1553. <p>The strategy is to decompose the space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mover><mi>H</mi><mo stretchy="false">&#x5E;</mo></mover> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\hat H^n</annotation></semantics></math> of all
  1554. connections into a product of subgroups, according to the
  1555. general analysis reviewed in the section &#8220;<em>The space
  1556. of all connections</em>&#8221; in <a href="http://golem.ph.utexas.edu/string/archives/000868.html">I</a>.
  1557. As described there, one finds</p>
  1558.  
  1559. <div class="numberedEq"><span>(3)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mover><mi>H</mi><mo stretchy="false">&#x5E;</mo></mover> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x02243;</mo><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>&#x02124;</mi></msub><mspace width="thickmathspace"/><mo>&#x000D7;</mo><mspace width="thickmathspace"/><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"/><mo>&#x000D7;</mo><mspace width="thickmathspace"/><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1560.  \hat H^{2k+1} \simeq
  1561.  \mathbb{H}^{2k}(X)/\mathbb{H}^{2k}(X)_\mathbb{Z}
  1562.  \;
  1563.  \times
  1564.  \;
  1565.  \mathbf{d}^*(A^{2k+1}(X))
  1566.  \;
  1567.  \times
  1568.  \;
  1569.  H^{2k+1}(X,\mathbb{Z})
  1570.  \,.
  1571. </annotation></semantics></math></div>
  1572.  
  1573. <p>The last factor contains the global gauge sectors and the first two factors encode the connections given by globally defined <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2k</annotation></semantics></math>-forms, the first factor containing the flat ones (compare the section &#8220;<em>The space of flat connections</em>&#8221; in <a href="http://golem.ph.utexas.edu/string/archives/000868.html">I</a>).</p>
  1574.  
  1575. <p>For the purpose of finding representations, Gomi goes one step further and also decomposes <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>H^{2k+1}(X,\mathbb{Z})</annotation></semantics></math> into a free part and a pure torsion part. </p>
  1576.  
  1577. <div class="numberedEq"><span>(4)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo><mo>&#x02243;</mo><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x000D7;</mo><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1578.  H^{2k+1}(X,\mathbb{Z}) \simeq F^{2k+1} \times T^{2k+1}
  1579.  \,.
  1580. </annotation></semantics></math></div>
  1581.  
  1582. <p>This is possible, but not canonical. Hence Gomi starts his construction by making a <em>choice</em></p>
  1583.  
  1584. <p><strong>choice number 1)</strong> a decomposition </p>
  1585.  
  1586. <div class="numberedEq"><span>(5)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo><mover><mo>&#x02243;</mo><mi>&#x003C9;</mi></mover><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x02295;</mo><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>
  1587.  H^{2k+1}(X,\mathbb{Z})
  1588.  \overset{\omega}{\simeq}
  1589.  F^{2k+1} \oplus T^{2k+1}
  1590. </annotation></semantics></math></div>
  1591.  
  1592. <p>of the integral cohomology group into a free and a torsion part.</p>
  1593.  
  1594. <p>With this choice performed, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{G}(X)</annotation></semantics></math> decomposes
  1595. (p. 17) into four groups as</p>
  1596.  
  1597. <div class="numberedEq"><span>(6)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x02243;</mo><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>&#x02124;</mi></msub><mspace width="thinmathspace"/><mo>&#x000D7;</mo><mspace width="thinmathspace"/><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>&#x000D7;</mo><mspace width="thinmathspace"/><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"/><mo>&#x000D7;</mo><mspace width="thinmathspace"/><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1598.  \mathbf{G}(X)
  1599.  \simeq
  1600.  \mathbb{H}^{2k}(X)/\mathbb{H}^{2k}(X)_\mathbb{Z}
  1601.  \,\times\,
  1602.  \mathbf{d}^*(A^{2k+1}(X))
  1603.  \,\times\,
  1604.  F^{2k+1}
  1605.  \,\times\,
  1606.  T^{2k+1}
  1607.  \,.
  1608. </annotation></semantics></math></div>
  1609.  
  1610. <p>The strategy is to represent this piecewise.</p>
  1611.  
  1612. <p>For that Gomi makes</p>
  1613.  
  1614. <p><strong> choice number 2</strong>:
  1615. some homomorphism</p>
  1616.  
  1617. <div class="numberedEq"><span>(7)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x003BB;</mi><mo>:</mo><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>&#x02124;</mi></msub><mo>&#x02192;</mo><mi>&#x0211D;</mi><mo stretchy="false">/</mo><mi>&#x02124;</mi><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1618.  \lambda : \mathbb{H}^{2k}(X)/\mathbb{H}^{2k}(X)_\mathbb{Z}
  1619.  \to
  1620.  \mathbb{R}/\mathbb{Z}
  1621.  \,,
  1622. </annotation></semantics></math></div>
  1623.  
  1624. <p>representing the group of harmonic <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2k</annotation></semantics></math>-forms up to integral harmonic <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2k</annotation></semantics></math>-forms </p>
  1625.  
  1626. <p>and</p>
  1627.  
  1628. <p><strong>choice number 3)</strong>
  1629. a finite dimensional projective unitary representation of the torsion subgroup</p>
  1630.  
  1631. <div class="numberedEq"><span>(8)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x003C0;</mi><mo>:</mo><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x02192;</mo><mi mathvariant="normal">Aut</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1632.  \pi : T^{2k+1} \to \mathrm{Aut}(V)
  1633. </annotation></semantics></math></div>
  1634.  
  1635. <p>coming from the restriction of the Chern-Simons cocycle on torsion connections</p>
  1636.  
  1637. <div class="numberedEq"><span>(9)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi mathvariant="normal">exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>&#x003C0;</mi><mi>i</mi><msub><mi>L</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>:</mo><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x000D7;</mo><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x02192;</mo><mi>&#x0211D;</mi><mo stretchy="false">/</mo><mi>&#x02124;</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1638.  \mathrm{exp}(2\pi i L_X) : T^{2k+1} \times T^{2k+1} \to
  1639.  \mathbb{R}/\mathbb{Z}
  1640.  \,.
  1641. </annotation></semantics></math></div>
  1642.  
  1643. <p>Here</p>
  1644.  
  1645. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>T^{2k+1} = T^{2k+1}(X)</annotation></semantics></math> is the <strong>torsion subgroup</strong> (p. 12) of the integral cohomology group
  1646. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>H^{2k+1}(X,\mathbb{Z})</annotation></semantics></math>.</p>
  1647.  
  1648. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>F^{2k+1}</annotation></semantics></math> is a group isomorphic to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">/</mo><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>H^{2k+1}/T^{2k+1}</annotation></semantics></math>, whose rank</p>
  1649.  
  1650. <div class="numberedEq"><span>(10)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>b</mi><mo>=</mo><msub><mi>b</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1651.  b = b_{2k+1}(X)
  1652. </annotation></semantics></math></div>
  1653.  
  1654. <p>is the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(2k+1)</annotation></semantics></math>st Betti number of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> (def 4.4, p. 13).</p>
  1655.  
  1656. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbb{H}^{2k}(X)</annotation></semantics></math> is the group of <strong>harmonic</strong> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2k</annotation></semantics></math>-forms on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> (prop. 3.1, p. 8) with respect to the chosen metric <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math></p>
  1657.  
  1658. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>&#x02124;</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{H}^{2k}(X)_\mathbb{Z}</annotation></semantics></math> is accordingly the group of <strong>harmonic</strong> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2k</annotation></semantics></math>-forms on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with <strong>integral periods</strong></p>
  1659.  
  1660. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>A^{2k+1}(X)</annotation></semantics></math> is the group of <strong>globally defined</strong> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(2k+1)</annotation></semantics></math>-forms on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p>
  1661.  
  1662. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math>  <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{d}^*(A^{2k+1}(X))</annotation></semantics></math> is accordingly the image under <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo>&#x0221D;</mo><mo>*</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo>*</mo></mrow><annotation encoding='application/x-tex'>\mathbf{d}^* \propto * \mathbf{d} * </annotation></semantics></math> of all <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(2k+1)</annotation></semantics></math>-forms.</p>
  1663.  
  1664. <p><br/>The strategy is then to </p>
  1665.  
  1666. <p><strong>1)</strong> construct the Heisenberg extension first only on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{d}^*(A^{2k+1})</annotation></semantics></math>,</p>
  1667.  
  1668. <p><strong>2)</strong> then tensor the result with the 1-dimensional rep of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">/</mo><msubsup><mi>&#x0210D;</mi> <mi>&#x02124;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msubsup></mrow><annotation encoding='application/x-tex'>\mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z}</annotation></semantics></math> using <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math>,</p>
  1669.  
  1670. <p><strong>3)</strong> then obtain from this an induced representation of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{G}_\omega(X)</annotation></semantics></math>, which is the group of connections whose characteristic class is in the free part <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>F^{2k+1}</annotation></semantics></math> of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>H^{2k+1}(X,\mathbb{Z})</annotation></semantics></math></p>
  1671.  
  1672. <p><strong>4)</strong> finally tensor this result with the finite dimensional rep of the extension of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>T^{2k+1}</annotation></semantics></math>.</p>
  1673.  
  1674. <p>This strategy is built on the known example <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>k=0</annotation></semantics></math>.</p>
  1675.  
  1676. <p><br/><strong>Example.</strong> Let <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>k=0</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>X = S^1</annotation></semantics></math>, the circle. An abelian <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-form connection on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is now simply a circle valued function on the circle - in the context of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>-form gauge theory to be thought of as the worldsheet boson of a string compactified on a circle.</p>
  1677.  
  1678. <p>Accordingly, the group of all these 0-form connections is simply the loop group of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>U(1)</annotation></semantics></math></p>
  1679.  
  1680. <div class="numberedEq"><span>(11)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>&#x02243;</mo><mi>LU</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1681.  \mathbf{G}(S^1) \simeq LU(1)
  1682.  \,.
  1683. </annotation></semantics></math></div>
  1684.  
  1685. <p>This decomposes as above, with</p>
  1686.  
  1687. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> </p>
  1688.  
  1689. <div class="numberedEq"><span>(12)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>&#x0210D;</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msubsup><mi>&#x0210D;</mi> <mi>&#x02124;</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>&#x02243;</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1690.  \mathbb{H}^0(S^1)/\mathbb{H}^0_\mathbb{Z}(S^1)
  1691.  \simeq U(1)
  1692. </annotation></semantics></math></div>
  1693.  
  1694. <p>the space of <strong>constant</strong> 0-forms on the circle, i.e. maps that send the entire string to a single point</p>
  1695.  
  1696. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> </p>
  1697.  
  1698. <div class="numberedEq"><span>(13)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>A</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>&#x02243;</mo><mrow><mo>{</mo><mi>&#x03D5;</mi><mo>:</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>&#x02192;</mo><mi>&#x0211D;</mi><mo stretchy="false">&#x0007C;</mo><msub><mo>&#x0222B;</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><mi>&#x03D5;</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mi>dt</mi><mo>=</mo><mn>0</mn><mo>}</mo></mrow></mrow><annotation encoding='application/x-tex'>
  1699.  \mathbf{d}^*(A^1) \simeq
  1700.  \left\{
  1701.    \phi : S^1 \to \mathbb{R}
  1702.    |
  1703.    \int_{S^1} \phi(t)\, dt = 0
  1704.  \right\}
  1705. </annotation></semantics></math></div>
  1706.  
  1707. <p>the space of maps that are total derivatives and hence have <strong>no winding</strong> around the circle</p>
  1708.  
  1709. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math></p>
  1710.  
  1711. <div class="numberedEq"><span>(14)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>F</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>&#x02243;</mo><mi>&#x02124;</mi></mrow><annotation encoding='application/x-tex'>
  1712.  F^{1}(S^1) \simeq \mathbb{Z}
  1713. </annotation></semantics></math></div>
  1714.  
  1715. <p>the space of linear maps <strong>with winding</strong></p>
  1716.  
  1717. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math></p>
  1718.  
  1719. <div class="numberedEq"><span>(15)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>T</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1720.  T^1(S^1) = 0
  1721.  \,.
  1722. </annotation></semantics></math></div>
  1723.  
  1724. <p>Applied to this motivating example, Gomi&#8217;s more general construction is supposed to reproduce the well-known positive energy reps of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>LU</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>&#x02124;</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\hat LU(1)/\mathbb{Z}_2</annotation></semantics></math>.</p>
  1725.  
  1726. <p><br/> So we need to understand the four steps of the above strategy</p>
  1727.  
  1728. <p><strong>Step 1) Projective irrrep of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{d}^*(A^{2k+1})</annotation></semantics></math> </strong></p>
  1729.  
  1730. <p>On the globally defined <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2k</annotation></semantics></math>-forms in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\mathbf{d}^*A^{2k+1}</annotation></semantics></math>, the Chern-Simons-like cocycle that controls the entire business is simply the integral</p>
  1731.  
  1732. <div class="numberedEq"><span>(16)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>&#x003BD;</mi><mo>,</mo><mi>&#x003BD;</mi><mo>&#x02032;</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mo>&#x0222B;</mo> <mi>X</mi></msub><mi>&#x003BD;</mi><mo>&#x02227;</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>&#x003BD;</mi><mo>&#x02032;</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mi mathvariant="normal">mod</mi><mi>&#x02124;</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1733.  S_X(\nu,\nu&apos;) = \int_X \nu \wedge \mathbf{d}\nu&apos; \;\;\mathrm{mod}
  1734.  \mathbb{Z}
  1735.  \,.
  1736. </annotation></semantics></math></div>
  1737.  
  1738. <p>There is a general theorem (from Pressley/Segal, prop. 5, p. 18 in Gomi&#8217;s paper) that</p>
  1739.  
  1740. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> given a group cocycle on a vector space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>, regarded as an abelian group</p>
  1741.  
  1742. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> and given a <em>continuous</em> complex structure <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>J</mi><mo>:</mo><mi>V</mi><mo>&#x02192;</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>J : V \to V</annotation></semantics></math> on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> which is compatible with the cocycle and turns it into a positive definite form</p>
  1743.  
  1744. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> then there is a continuous unitary rep </p>
  1745.  
  1746. <div class="numberedEq"><span>(17)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x003C1;</mi><mo>:</mo><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover><mo>&#x02192;</mo><mi mathvariant="normal">Aut</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1747.  \rho : \tilde V \to \mathrm{Aut}(H)
  1748. </annotation></semantics></math></div>
  1749.  
  1750. <p>of the central extension <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover></mrow><annotation encoding='application/x-tex'>\tilde V</annotation></semantics></math> </p>
  1751.  
  1752. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> such that the center acts by scalar multiplication</p>
  1753.  
  1754. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> which is an irrep if V is seperable and complete with respect to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>J</mi><mo>&#x022C5;</mo><mo>,</mo><mo>&#x022C5;</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>S_X(J\cdot,\cdot)</annotation></semantics></math>.</p>
  1755.  
  1756. <p>One of the more technical problems that Gomi indicates how to solve (prop. 3.1, p. 8) is the </p>
  1757.  
  1758. <p><strong>construction of such a complex structure</strong> on the space of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2k</annotation></semantics></math>-forms.</p>
  1759.  
  1760. <p>The idea is to start with the Hodge inner product</p>
  1761.  
  1762. <div class="numberedEq"><span>(18)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">(</mo><mi>&#x003B1;</mi><mo>,</mo><mi>&#x003B2;</mi><msub><mo stretchy="false">)</mo> <mrow><msup><mi>L</mi> <mn>2</mn></msup></mrow></msub><mo>:</mo><mo>=</mo><msub><mo>&#x0222B;</mo> <mi>X</mi></msub><mi>&#x003B1;</mi><mspace width="thinmathspace"/><mo>&#x02227;</mo><mspace width="thinmathspace"/><mo>*</mo><mspace width="thinmathspace"/><mi>&#x003B2;</mi></mrow><annotation encoding='application/x-tex'>
  1763.  (\alpha,\beta)_{L^2}
  1764.  :=
  1765.  \int_X \alpha \, \wedge \, * \, \beta
  1766. </annotation></semantics></math></div>
  1767.  
  1768. <p>and find a complex structure <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> such that we get the
  1769. Chern-Simons cocycle term from this, i.e. such that</p>
  1770.  
  1771. <div class="numberedEq"><span>(19)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">(</mo><mi>&#x003B1;</mi><mo>,</mo><mi>J</mi><mi>&#x003B2;</mi><msub><mo stretchy="false">)</mo> <mrow><msup><mi>L</mi> <mn>2</mn></msup></mrow></msub><mo>=</mo><msub><mo>&#x0222B;</mo> <mi>X</mi></msub><mi>&#x003B1;</mi><mspace width="thinmathspace"/><mstyle mathvariant="bold"><mi>d</mi></mstyle><mspace width="thinmathspace"/><mi>&#x003B2;</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1772.  (\alpha,J \beta)_{L^2}
  1773.  =
  1774.  \int_X \alpha\, \mathbf{d}\,\beta
  1775.  \,.
  1776. </annotation></semantics></math></div>
  1777.  
  1778. <p>Clearly, this would require setting <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> equal to
  1779. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>*</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo>*</mo></mrow><annotation encoding='application/x-tex'>* \mathbf{d} = &#x2d;\mathbf{d}^* *</annotation></semantics></math>. However, this operator
  1780. is not a complex structure, since it does not square to
  1781. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>&#x2d;1</annotation></semantics></math>.</p>
  1782.  
  1783. <p>In order for this guy to square to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>&#x2d;1</annotation></semantics></math> we need to divide it (using functional calculus) by its norm. So we set</p>
  1784.  
  1785. <div class="numberedEq"><span>(20)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>J</mi><mo>=</mo><mfrac><mrow><mo>*</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><mrow><mo stretchy="false">&#x0007C;</mo><mo>*</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">&#x0007C;</mo></mrow></mfrac><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1786.  J = \frac{*\mathbf{d}}{|*\mathbf{d}|}
  1787.  \,.
  1788. </annotation></semantics></math></div>
  1789.  
  1790. <p>This now is a complex structure, but no longer relates the inner product with the cocycle. In order to remedy this we need to modify the inner product, too.</p>
  1791.  
  1792. <p>As Gomi describes (p. 9) the right solution is to take the inner product <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mo>&#x022C5;</mo><mo>,</mo><mo>&#x022C5;</mo><msub><mo stretchy="false">)</mo> <mi>V</mi></msub></mrow><annotation encoding='application/x-tex'>(\cdot,\cdot)_V</annotation></semantics></math> induced from the modified norm</p>
  1793.  
  1794. <div class="numberedEq"><span>(21)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">&#x02016;</mo><mi>&#x003B1;</mi><msubsup><mo stretchy="false">&#x02016;</mo> <mi>V</mi> <mn>2</mn></msubsup><mo>:</mo><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">&#x02211;</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>b</mi></munderover><mo stretchy="false">(</mo><mi>&#x003B1;</mi><mo>,</mo><msub><mi>&#x003C8;</mi> <mi>i</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><msup><mi>L</mi> <mn>2</mn></msup></mrow> <mn>2</mn></msubsup><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">&#x02211;</mo> <mrow><mi>i</mi><mo>=</mo><mi>b</mi><mo>+</mo><mn>1</mn></mrow> <mn>&#x0221E;</mn></munderover><msqrt><mrow><mo stretchy="false">&#x0007C;</mo><msub><mi>&#x02113;</mi> <mi>i</mi></msub><mo stretchy="false">&#x0007C;</mo></mrow></msqrt><mo stretchy="false">(</mo><mi>&#x003B1;</mi><mo>,</mo><msub><mi>&#x003C8;</mi> <mi>i</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><msup><mi>L</mi> <mn>2</mn></msup></mrow> <mn>2</mn></msubsup><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1795.  \Vert
  1796.    \alpha
  1797.  \Vert^2_V
  1798.  :=
  1799.  \sum_{i=1}^b
  1800.  (\alpha,\psi_i)^2_{L^2}
  1801.  +
  1802.  \sum_{i=b+1}^\infty
  1803.  \sqrt{|\ell_i|}
  1804.  (\alpha,\psi_i)^2_{L^2}
  1805.  \,,
  1806. </annotation></semantics></math></div>
  1807.  
  1808. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><msub><mi>&#x003C8;</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{\psi_i\}</annotation></semantics></math> is an orthonormal system of eigenvectors of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x00394;</mi><mo>=</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo lspace="verythinmathspace" rspace="0em">+</mo><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding='application/x-tex'>\Delta = \mathbf{d}\mathbf{d}^* + \mathbf{d}^* \mathbf{d}</annotation></semantics></math> with eigenvalues <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><msub><mi>&#x02113;</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{\ell_i\}</annotation></semantics></math>, of which <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>b</mi><mo>=</mo><msub><mi>b</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>b = b_{2k}(X)</annotation></semantics></math> vanish.</p>
  1809.  
  1810. <p>It is at this point that use is made of the fact that we are currently restricting attention to the space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{d}^* A^{2k+1}(X)</annotation></semantics></math>. For, on this space the operator <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>*</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding='application/x-tex'>*\mathbf{d}</annotation></semantics></math> squares to minus the Laplace operator</p>
  1811.  
  1812. <div class="numberedEq"><span>(22)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mi>&#x00394;</mi><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mtext>on</mtext><mspace width="thinmathspace"/><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1813.  (*\mathbf{d})^2
  1814.  =
  1815.  &#x2d;\Delta
  1816.  \;\;
  1817.  \text{on}
  1818.  \,
  1819.  \mathbf{d}^*A^{2k+1}
  1820.  \,.
  1821. </annotation></semantics></math></div>
  1822.  
  1823. <p>Therefore, if we restrict the eigenbasis <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><msub><mi>&#x003C8;</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{\psi_i\}</annotation></semantics></math> to an eigenbasis <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><msub><mi>&#x03D5;</mi> <mi>j</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{\phi_j\}</annotation></semantics></math> only of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{d}^* A^{2k+1}(X)</annotation></semantics></math>, it becomes a simultaneous eigenbasis of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x00394;</mi></mrow><annotation encoding='application/x-tex'>\Delta</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>*</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding='application/x-tex'>*\mathbf{d}</annotation></semantics></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>*</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding='application/x-tex'>*\mathbf{d}</annotation></semantics></math> has eigenvalues <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">{</mo><msqrt><mrow><mo stretchy="false">&#x0007C;</mo><msub><mi>&#x02113;</mi> <mi>i</mi></msub><mo stretchy="false">&#x0007C;</mo></mrow></msqrt><mo stretchy="false">}</mo></mrow><annotation encoding='application/x-tex'>\{\sqrt{|\ell_i|}\}</annotation></semantics></math>.</p>
  1824.  
  1825. <p>The norm <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">&#x02016;</mo><mo>&#x022C5;</mo><msub><mo stretchy="false">&#x02016;</mo> <mi>V</mi></msub></mrow><annotation encoding='application/x-tex'>\Vert\cdot\Vert_V</annotation></semantics></math> is built in such a way that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>J</mi><mo>=</mo><mfrac><mrow><mo>*</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><mrow><mo stretchy="false">&#x0007C;</mo><mo>*</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">&#x0007C;</mo></mrow></mfrac></mrow><annotation encoding='application/x-tex'>J = \frac{*\mathbf{d}}{|*\mathbf{d}|}</annotation></semantics></math> is an isometry on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\mathbf{d}A^{2k+1}</annotation></semantics></math> and such that it relates the inner product with the cocycle</p>
  1826.  
  1827. <div class="numberedEq"><span>(23)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">(</mo><mi>&#x003B1;</mi><mo>,</mo><mi>J</mi><mi>&#x003B2;</mi><msub><mo stretchy="false">)</mo> <mi>V</mi></msub><mo>=</mo><msub><mo>&#x0222B;</mo> <mi>X</mi></msub><mi>&#x003B1;</mi><mspace width="thinmathspace"/><mstyle mathvariant="bold"><mi>d</mi></mstyle><mspace width="thinmathspace"/><mi>&#x003B2;</mi><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1828.  (\alpha,J\beta)_V
  1829.  =
  1830.  \int_X \alpha\, \mathbf{d}\, \beta
  1831.  \,,
  1832. </annotation></semantics></math></div>
  1833.  
  1834. <p>as desired. Hence if we take </p>
  1835.  
  1836. <div class="numberedEq"><span>(24)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>V</mi><mo>:</mo><mo>=</mo><mover><mrow><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mo>&#x000AF;</mo></mover></mrow><annotation encoding='application/x-tex'>
  1837. V := \widebar{\mathbf{d}^*(A^{2k+1})}
  1838. </annotation></semantics></math></div>
  1839.  
  1840. <p>to be the completion of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\mathbf{d}^*A^{2k+1}</annotation></semantics></math> with respect to this norm, Pressly and Segal guarantee us a representation</p>
  1841.  
  1842. <div class="numberedEq"><span>(25)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><mi>&#x003C1;</mi></mtd> <mtd><mo>:</mo></mtd> <mtd><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover></mtd> <mtd><mo>&#x02192;</mo></mtd> <mtd><mi mathvariant="normal">Aut</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  1843. \array{
  1844.  \rho &amp;:&amp; \tilde V &amp;\to&amp; \mathrm{Aut}(H)
  1845. }
  1846. </annotation></semantics></math></div>
  1847.  
  1848. <p>of the centrally extended group <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>V</mi><mo stretchy="false">&#x002DC;</mo></mover></mrow><annotation encoding='application/x-tex'>\tilde V</annotation></semantics></math>. Here the representation space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is (prop. 5.1, p. 18, due to Pressley-Segal) essentially the symmetric algebra of the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(+i)</annotation></semantics></math>-eigenspace of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> (the &#8220;positive energy eigenspace&#8221;).</p>
  1849.  
  1850. <p><br/><strong>Step 2) tensor the result with the 1-dimensional rep <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math> of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">/</mo><msubsup><mi>&#x0210D;</mi> <mi>&#x02124;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msubsup></mrow><annotation encoding='application/x-tex'>\mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z}</annotation></semantics></math></strong></p>
  1851.  
  1852. <p>Simply set (proof of lemma 5.3, p. 19)</p>
  1853.  
  1854. <div class="numberedEq"><span>(26)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>&#x003C1;</mi> <mi>&#x003BB;</mi></msub></mtd> <mtd><mo>:</mo></mtd> <mtd><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">/</mo><msubsup><mi>&#x0210D;</mi> <mi>&#x02124;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msubsup><mspace width="thickmathspace"/><mo>&#x000D7;</mo><mspace width="thickmathspace"/><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo>&#x02192;</mo></mtd> <mtd><mi mathvariant="normal">Aut</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd/> <mtd/> <mtd><msub><mi>&#x003C1;</mi> <mi>&#x003BB;</mi></msub><mo stretchy="false">(</mo><mi>&#x003B7;</mi><mo>,</mo><mi>&#x003BD;</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>&#x021A6;</mo></mtd> <mtd><mi>&#x003BB;</mi><mo stretchy="false">(</mo><mi>&#x003B7;</mi><mo stretchy="false">)</mo><mi>&#x003C1;</mi><mo stretchy="false">(</mo><mi>&#x003BD;</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1855. \array{
  1856.  \rho_\lambda &amp;:&amp;
  1857.  \mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z}
  1858.  \;\times\;
  1859.  \mathbf{d}^* A^{2k+1}
  1860.  &amp;\to&amp;
  1861.  \mathrm{Aut}(H)
  1862.  \\
  1863.  &amp;&amp;
  1864.  \rho_\lambda(\eta,\nu)
  1865.  &amp;\mapsto&amp;
  1866.  \lambda(\eta)\rho(\nu)
  1867. }
  1868. \,.
  1869. </annotation></semantics></math></div>
  1870.  
  1871. <p>Then use the fact that the cocycle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>S_X</annotation></semantics></math> vanishes on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>\mathbb{H}^{2k}</annotation></semantics></math> to deduce that this is still irreducible if <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C1;</mi></mrow><annotation encoding='application/x-tex'>\rho</annotation></semantics></math> is.</p>
  1872.  
  1873. <p><br/><strong>Step 3) form the unitary irrep induced from the subgroup rep </strong></p>
  1874.  
  1875. <p>We obtained above a projective irrep for</p>
  1876.  
  1877. <div class="numberedEq"><span>(27)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>=</mo><mtext>p.19</mtext></mover><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>&#x02124;</mi></msub><mover><mo>&#x02243;</mo><mtext>p. 19</mtext></mover><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">/</mo><msubsup><mi>&#x0210D;</mi> <mi>&#x02124;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msubsup><mspace width="thickmathspace"/><mo>&#x000D7;</mo><mspace width="thickmathspace"/><msup><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>A</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1878.  \mathbf{G}^0(X)
  1879.    \overset{\text{p.19}}{=} A^{2k}(X)/A^{2k}(X)_\mathbb{Z}
  1880.  \overset{\text{p. 19}}{\simeq}
  1881.   \mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z}
  1882.  \;\times\;
  1883.   \mathbf{d}^*(A^{2k+1})
  1884.  \,.
  1885. </annotation></semantics></math></div>
  1886.  
  1887. <p>This is a group of globally defined <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2k</annotation></semantics></math>-form connections. Hence the corresponding characteristic classes all vanish. In particular, they are not torsion. So this is a subgroup of</p>
  1888.  
  1889. <div class="numberedEq"><span>(28)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  1890.  \mathbf{G}_\omega(X)
  1891.  \,,
  1892. </annotation></semantics></math></div>
  1893.  
  1894. <p>the froup of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2k</annotation></semantics></math>-form connections whose characteristic class is in the free part <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x02282;</mo><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>F^{2k+1} \subset H^{2k+1}</annotation></semantics></math> (with respect to the choice of isomorphism <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C9;</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>).</p>
  1895.  
  1896. <p>We now <em>induce</em> on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{G}_\omega(X)</annotation></semantics></math> a projective irrep from the projective irrep of the subgroup <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{G}^0(X)</annotation></semantics></math>.</p>
  1897.  
  1898. <p>This works by </p>
  1899.  
  1900. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> forming the bundle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">&#x002DC;</mo></mover> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">&#x002DC;</mo></mover> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">&#x002DC;</mo></mover> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\tilde \mathbf{G}_\omega(X) \to \tilde\mathbf{G}_\omega(X)/\tilde\mathbf{G}_0(X)</annotation></semantics></math></p>
  1901.  
  1902. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> using the subgroup rep <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003C1;</mi> <mi>&#x003BB;</mi></msub><mo>:</mo><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">&#x002DC;</mo></mover> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><mi mathvariant="normal">Aut</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\rho_\lambda : \tilde \mathbf{G}_0(X) \to \mathrm{Aut}(H)</annotation></semantics></math> to associate a vector bundle
  1903. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">&#x002DC;</mo></mover> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>&#x000D7;</mo> <mrow><msub><mi>&#x003C1;</mi> <mi>&#x003BB;</mi></msub></mrow></msub><mi>H</mi><mo>&#x02192;</mo><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">&#x002DC;</mo></mover> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">&#x002DC;</mo></mover> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\tilde \mathbf{G}_\omega(X) \times_{\rho_\lambda} H \to \tilde\mathbf{G}_\omega(X)/\tilde\mathbf{G}_0(X)</annotation></semantics></math></p>
  1904.  
  1905. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> noticing that (because all groups are abelian) this still has a left <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">&#x002DC;</mo></mover> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\tilde \mathbf{G}_\omega(X)</annotation></semantics></math>-action</p>
  1906.  
  1907. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> hence finding a rep of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">&#x002DC;</mo></mover> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\tilde \mathbf{G}_\omega(X)</annotation></semantics></math> on the <strong>space</strong> </p>
  1908.  
  1909. <div class="numberedEq"><span>(29)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msubsup><mi>&#x0210B;</mi> <mi>&#x003BB;</mi> <mi>&#x003C9;</mi></msubsup></mrow><annotation encoding='application/x-tex'>
  1910.  \mathcal{H}_\lambda^\omega
  1911. </annotation></semantics></math></div>
  1912.  
  1913. <p><strong>of sections of this vector bundle</strong>.</p>
  1914.  
  1915. <p>Gomi gives a more explicit description of this induced rep for the present case (below equation (13), p. 20), which plays the crucial role for the classification of irreps later. I&#8217;ll discuss this below.</p>
  1916.  
  1917. <p><br/><strong>4) finally add the rep of the torsion part</strong></p>
  1918.  
  1919. <p>The above establishes a unitary projective rep of all but the torsion part <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>T^{2k+1}</annotation></semantics></math> of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{G}(X)</annotation></semantics></math>. It turns out one can construct lifts <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C3;</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> of every torsion class to a representing connection. (I&#8217;ll discuss these <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C3;</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> below). Using this we can pull back the Chern-Simons cocycle on connections to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>T^{2k+1}</annotation></semantics></math> (lemma 4.1, p. 12) and get the finite dimensional irrep</p>
  1920.  
  1921. <div class="numberedEq"><span>(30)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x003C0;</mi><mo>:</mo><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x02192;</mo><mi mathvariant="normal">Aut</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  1922.  \pi : T^{2k+1} \to \mathrm{Aut}(V)
  1923. </annotation></semantics></math></div>
  1924.  
  1925. <p>which we already assumed to be given above.</p>
  1926.  
  1927. <p>We combine this with the rep described above to a projective rep of all of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{G}(X)</annotation></semantics></math> by tensoring the two representation spaces and sending every connection <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover></mrow><annotation encoding='application/x-tex'>\hat A</annotation></semantics></math> to the rep of its torsion part times the rep of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover></mrow><annotation encoding='application/x-tex'>\hat A</annotation></semantics></math> minus the lift <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C3;</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\sigma(t(\hat A))</annotation></semantics></math> of the torsion part <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>t(\hat A)</annotation></semantics></math> of its characteristic class:</p>
  1928.  
  1929. <div class="numberedEq"><span>(31)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>&#x003C1;</mi> <mrow><mi>&#x003BB;</mi><mo>,</mo><mi>&#x003C0;</mi></mrow> <mi>&#x003C9;</mi></msubsup></mtd> <mtd><mo>:</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>&#x02192;</mo></mtd> <mtd><mi mathvariant="normal">Aut</mi><mo stretchy="false">(</mo><msubsup><mi>&#x0210B;</mi> <mi>&#x003BB;</mi> <mi>&#x003C9;</mi></msubsup><mo>&#x02297;</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd/> <mtd/> <mtd><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover></mtd> <mtd><mo>&#x021A6;</mo></mtd> <mtd><msubsup><mi>&#x003C1;</mi> <mi>&#x003BB;</mi> <mi>&#x003C9;</mi></msubsup><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo>&#x02212;</mo><mi>&#x003C3;</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>&#x02297;</mo><mi>&#x003C0;</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1930. \array{
  1931.  \rho_{\lambda,\pi}^\omega
  1932.  &amp;:&amp;
  1933.  \mathbf{G}(X)
  1934.  &amp;\to&amp;
  1935.  \mathrm{Aut}(\mathcal{H}_\lambda^\omega \otimes V)
  1936.  \\
  1937.  &amp;&amp;
  1938.  \hat A
  1939.  &amp;\mapsto&amp;
  1940.  \rho_\lambda^\omega(\hat A &#x2d; \sigma(t(\hat A)))
  1941.  \otimes
  1942.  \pi(t(\hat A))
  1943. }
  1944.  \,.
  1945. </annotation></semantics></math></div>
  1946.  
  1947. <p>That&#8217;s, in outline, the construction of the projective unitary rep of all of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{G}(X)</annotation></semantics></math>.</p>
  1948.  
  1949. <p>It is claimed to be independent of the choice of lift <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C3;</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> (remark 3, p. 21) and that different choices of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C9;</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math> lead to unitarily equivalent reps (prop. 6.2, p. 21).</p>
  1950.  
  1951. <p>It is also claimed that the rep obtained this way, when restricted to connections on trivial gerbes, decomposes nicely into a direct sum of  reps <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>H</mi> <mi>&#x003BB;</mi></msub></mrow><annotation encoding='application/x-tex'>H_\lambda</annotation></semantics></math> of these. This property is called <strong>admissability</strong> (def. 1.2, p. 3). This is supposed to make contact with Freed-Hopkins-Telemann.</p>
  1952.  
  1953. <p><br/>The <strong>main result</strong> (theorem 1.3, p. 3, proven in section 6) obtained from this construction is a classification of the reps constructed above, in particular a characterization of their irreps (hence, physically, of the superselection sectors of the theory).</p>
  1954.  
  1955. <p>The claim is that</p>
  1956.  
  1957. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> all these reps decompose into irreps;</p>
  1958.  
  1959. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> and there are (up to equivalence)</p>
  1960.  
  1961. <div class="numberedEq"><span>(32)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mn>2</mn> <mi>b</mi></msup><mi>r</mi></mrow><annotation encoding='application/x-tex'>
  1962.  2^b r
  1963. </annotation></semantics></math></div>
  1964.  
  1965. <p>different irreps, where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>b</mi><mo>=</mo><msub><mi>b</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>b = b_{2k+1}(X)</annotation></semantics></math> is the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(2k+1)</annotation></semantics></math>th Betti number of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>r</mi></mrow><annotation encoding='application/x-tex'>r</annotation></semantics></math> is the number of elements in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>H^{2k+1}</annotation></semantics></math> which are their own inverse.</p>
  1966.  
  1967. <p>As Gomi notes (p. 4) this coincides (as a special case) in particular with the result of a quantization analysis of 2-form gauge theory obtained by Henningson (see end of <a href="http://golem.ph.utexas.edu/string/archives/000871.html">II</a>).  </p>
  1968.  
  1969. <p>While in view of the claims and results of Freed-Moore-Segal there seems to be room for discussion as to what the correct quantization procedure for chiral <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>-form field theories is (I talked about that in  <a href="http://golem.ph.utexas.edu/string/archives/000871.html">II</a>), this does not affect the mathematical result presented. The precise relation to the quantization performed by Henningson would however seem like an interesting question.</p>
  1970.  
  1971. <p>Anyway, in order to understand the factor <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mn>2</mn> <mi>b</mi></msup></mrow><annotation encoding='application/x-tex'>2^b</annotation></semantics></math> in the above number of irreps, one needs the explit form of the induced irrep <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>&#x003C1;</mi> <mi>&#x003BB;</mi> <mi>&#x003C9;</mi></msubsup></mrow><annotation encoding='application/x-tex'>\rho_\lambda^\omega</annotation></semantics></math> of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{G}_\omega(X)</annotation></semantics></math> on the space of sections of that vector bundle, which I mentioned above.</p>
  1972.  
  1973. <p><strong>The appearance of the Betti numbers.</strong></p>
  1974.  
  1975. <p>First of all, it is shown that this vector bundle is in fact trivial and equivalent to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x000D7;</mo><msub><mi>H</mi> <mi>&#x003BB;</mi></msub></mrow><annotation encoding='application/x-tex'>F^{2k+1}\times H_\lambda</annotation></semantics></math>. Hence sections are just functions</p>
  1976.  
  1977. <div class="numberedEq"><span>(33)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x003A6;</mi><mo>:</mo><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>&#x02192;</mo><msub><mi>H</mi> <mi>&#x003BB;</mi></msub><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1978.  \Phi : F^{2k+1} \to H_\lambda
  1979.  \,.
  1980. </annotation></semantics></math></div>
  1981.  
  1982. <p>Gomi finds (p. 20), for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo>&#x02208;</mo><msub><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mi>&#x003C9;</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\hat A \in \mathbf{G}_\omega(X)</annotation></semantics></math> the action of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover></mrow><annotation encoding='application/x-tex'>\hat A</annotation></semantics></math> on such a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003A6;</mi></mrow><annotation encoding='application/x-tex'>\Phi</annotation></semantics></math> is given by</p>
  1983.  
  1984. <div class="numberedEq"><span>(34)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msubsup><mi>&#x003C1;</mi> <mi>&#x003BB;</mi> <mi>&#x003C9;</mi></msubsup><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>&#x003A6;</mi><mo stretchy="false">)</mo><mo>:</mo><mi>&#x003BE;</mi><mo>&#x021A6;</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>&#x022EF;</mi><mo stretchy="false">)</mo><msub><mi>&#x003C1;</mi> <mi>&#x003BB;</mi></msub><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo>&#x02212;</mo><mi>&#x003C3;</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>&#x003A6;</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo>&#x02212;</mo><mo stretchy="false">[</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  1985.  \rho_\lambda^\omega(\hat A)(\Phi)
  1986.  :
  1987.  \xi
  1988.  \mapsto
  1989.  \exp(\cdots) \rho_\lambda(\hat A &#x2d; \sigma([\hat A]))
  1990.  \Phi(\xi &#x2d; [\hat A])
  1991.  \,.
  1992. </annotation></semantics></math></div>
  1993.  
  1994. <p>Here, as before, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C3;</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> is a lift that associates to characteristic classes a connection on a gerbe with that class. So in particular
  1995. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo>&#x02212;</mo><mi>&#x003C3;</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\hat A &#x2d; \sigma([\hat A])</annotation></semantics></math> is a connection on a trivial gerbe.</p>
  1996.  
  1997. <p>The crucial part is the exponential term, which involves the Chern-Simons cocycle <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>S_X</annotation></semantics></math> evaluated on various objects:</p>
  1998.  
  1999. <div class="numberedEq"><span>(35)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>&#x022EF;</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mn>2</mn><mi>&#x003C0;</mi><mi>i</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>&#x003C3;</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><mi>&#x003C3;</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>&#x02212;</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>&#x003C3;</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><mi>&#x003C3;</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo>&#x02212;</mo><mi>&#x003C3;</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><mi>&#x003C3;</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2000.  \exp(\cdots)
  2001.  =
  2002.  \exp 2\pi i(
  2003.    S_X(\sigma([\hat A]),\sigma(\xi))
  2004.    &#x2d;
  2005.    S_X(\sigma([\hat A]),\sigma([\hat A]))
  2006.    +
  2007.    2 S_X(\hat A &#x2d; \sigma([\hat A]),\sigma(\xi))
  2008.  )
  2009.  \,.
  2010. </annotation></semantics></math></div>
  2011.  
  2012. <p>Related to the discussion reviewed in <a href="http://golem.ph.utexas.edu/string/archives/000871.html">II</a>, it is the factor of 2 in the last term which gives rise to certain degeneracies.</p>
  2013.  
  2014. <p>In order to make this more explicit, it is noted that if we choose the lift <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C3;</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> to take values in <em>harmonic</em> forms, then the last term is precisely a 1-dimensional rep of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">/</mo><msubsup><mi>&#x0210D;</mi> <mi>&#x02124;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msubsup></mrow><annotation encoding='application/x-tex'>\mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z}</annotation></semantics></math> that enters the definition of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003C1;</mi> <mi>&#x003BB;</mi></msub></mrow><annotation encoding='application/x-tex'>\rho_\lambda</annotation></semantics></math>. Hence we can absorb this term by replacing in the above formula</p>
  2015.  
  2016. <div class="numberedEq"><span>(36)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>&#x003C1;</mi> <mi>&#x003BB;</mi></msub><mo>&#x021A6;</mo><msub><mi>&#x003C1;</mi> <mrow><mi>&#x003BB;</mi><mo>+</mo><mn>2</mn><mi>s</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  2017.  \rho_\lambda \mapsto \rho_{\lambda + 2s(\xi)}
  2018.  \,,
  2019. </annotation></semantics></math></div>
  2020.  
  2021. <p>where </p>
  2022.  
  2023. <div class="numberedEq"><span>(37)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>s</mi><mo>:</mo><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">/</mo><msubsup><mi>&#x0210D;</mi> <mi>&#x02124;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msubsup><mo>&#x02192;</mo><mi mathvariant="normal">Hom</mi><mo stretchy="false">(</mo><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">/</mo><msubsup><mi>&#x0210D;</mi> <mi>&#x02124;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msubsup><mo>,</mo><mi>&#x0211D;</mi><mo stretchy="false">/</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  2024.  s : \mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z}
  2025.  \to
  2026.  \mathrm{Hom}(\mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z},
  2027.   \mathbb{R}/\mathbb{Z}
  2028.  )
  2029. </annotation></semantics></math></div>
  2030.  
  2031. <p>sends a harmonic form to the Chern-Simons pairing with that form (def. 4.11, p. 16).</p>
  2032.  
  2033. <p>Again, the crucial point is the factor of 2. </p>
  2034.  
  2035. <p>Namely (Lemma 4.12, p. 16), we can identify the space of all homomorphisms
  2036. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">Hom</mi><mo stretchy="false">(</mo><msup><mi>&#x0210D;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">/</mo><msubsup><mi>&#x0210D;</mi> <mi>&#x02124;</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msubsup><mo>,</mo><mi>&#x0211D;</mi><mo stretchy="false">/</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>  \mathrm{Hom}(\mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z},
  2037.   \mathbb{R}/\mathbb{Z}
  2038.  )</annotation></semantics></math>
  2039. with the free part <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">/</mo><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>H^{2k+1}/T^{2k+1}</annotation></semantics></math> of the integral cohomology, which looks like <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x02124;</mi> <mi>b</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{Z}^b</annotation></semantics></math>. This is where the Betti numbers come in.</p>
  2040.  
  2041. <p>Now, as we vary <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BE;</mi></mrow><annotation encoding='application/x-tex'>\xi</annotation></semantics></math> over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">/</mo><msup><mi>T</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>H^{2k+1}/T^{2k+1}</annotation></semantics></math> we hit with <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi><mo>+</mo><mn>2</mn><mi>s</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\lambda + 2s(\xi)</annotation></semantics></math> (for fixed <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math>) only every second homomorphism, due to the factor of 2. This means that there are <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mn>2</mn> <mi>b</mi></msup></mrow><annotation encoding='application/x-tex'>2^b</annotation></semantics></math> choices for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math> which lead to different reps <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003C1;</mi> <mrow><mi>&#x003BB;</mi><mo>+</mo><mn>2</mn><mi>s</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>\rho_{\lambda + 2s(\xi)}</annotation></semantics></math>, as <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BE;</mi></mrow><annotation encoding='application/x-tex'>\xi</annotation></semantics></math> varies. This is the key phenomenon that govers the classifications of irreps.</p>
  2042.  
  2043. <p><br/><strong>The number of irreps for topologically trivial connections.</strong></p>
  2044.  
  2045. <p>Let&#8217;s first again restrict attention to the globally defined connections <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">[</mo><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>[\hat A] = [A] = 0</annotation></semantics></math> in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{G}_0(X)</annotation></semantics></math>. Precisely for these, the funny shift in the last term in the above equation for
  2046. <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>&#x003C1;</mi> <mi>&#x003BB;</mi> <mi>&#x003C9;</mi></msubsup></mrow><annotation encoding='application/x-tex'>\rho_\lambda^\omega</annotation></semantics></math> vanishes. It follows that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>&#x003C1;</mi> <mi>&#x003BB;</mi> <mi>&#x003C9;</mi></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\rho_\lambda^\omega(A)</annotation></semantics></math> acts within each fiber of our bundle seperately as (proof of lemma 5.5, p. 20)</p>
  2047.  
  2048. <div class="numberedEq"><span>(38)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>&#x003C1;</mi> <mi>&#x003BB;</mi> <mi>&#x003C9;</mi></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>&#x003A6;</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>&#x003C1;</mi> <mrow><mi>&#x003BB;</mi><mo>+</mo><mn>2</mn><mi>s</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>&#x003A6;</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2049.  (\rho_\lambda^\omega(A)\Phi)(\xi)
  2050.  =
  2051.  \rho_{\lambda+2s(\xi)}(A)\Phi(\xi)
  2052.  \,.
  2053. </annotation></semantics></math></div>
  2054.  
  2055. <p>In other words, when restricted to connections with trivial classes in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{G}_0(X)</annotation></semantics></math>, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>&#x003C1;</mi> <mi>&#x003BB;</mi> <mi>&#x003C9;</mi></msubsup></mrow><annotation encoding='application/x-tex'>\rho_\lambda^\omega</annotation></semantics></math> sees the direct sum of all fibers of our bundle, and acts on the fiber over the class <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BE;</mi></mrow><annotation encoding='application/x-tex'>\xi</annotation></semantics></math> as the rep <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>&#x003C1;</mi> <mrow><mi>&#x003BB;</mi><mo>+</mo><mn>2</mn><mi>s</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>\rho_{\lambda+2s(\xi)}</annotation></semantics></math>. Hence (prop. 5.7, p.21)</p>
  2056.  
  2057. <div class="numberedEq"><span>(39)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msubsup><mi>&#x0210B;</mi> <mi>&#x003BB;</mi> <mi>&#x003C9;</mi></msubsup><msub><mo stretchy="false">&#x0007C;</mo> <mrow><msub><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"/><mo>&#x02243;</mo><mspace width="thickmathspace"/><msub><mover><mo>&#x02295;</mo><mo stretchy="false">&#x5E;</mo></mover> <mrow><mi>&#x003BE;</mi><mo>&#x02208;</mo><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msub><mspace width="thinmathspace"/><msub><mi>H</mi> <mrow><mi>&#x003BB;</mi><mo>+</mo><mn>2</mn><mi>s</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2058.  \mathcal{H}_\lambda^\omega |_{\mathbf{G}_0(X)}
  2059.  \;\simeq\;
  2060.  \hat \oplus_{\xi \in F^{2k+1}}
  2061.  \,
  2062.  H_{\lambda+2s(\xi)}
  2063.  \,.
  2064. </annotation></semantics></math></div>
  2065.  
  2066. <p>If we also take the rep <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> of the torsion part into account this reads</p>
  2067.  
  2068. <div class="numberedEq"><span>(40)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msubsup><mi>&#x0210B;</mi> <mrow><mi>&#x003BB;</mi><mo>,</mo><mi>V</mi></mrow> <mi>&#x003C9;</mi></msubsup><msub><mo stretchy="false">&#x0007C;</mo> <mrow><msub><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"/><mo>&#x02243;</mo><mspace width="thickmathspace"/><msub><mover><mo>&#x02295;</mo><mo stretchy="false">&#x5E;</mo></mover> <mrow><mi>&#x003BE;</mi><mo>&#x02208;</mo><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msub><mspace width="thinmathspace"/><mo stretchy="false">(</mo><msub><mi>H</mi> <mrow><mi>&#x003BB;</mi><mo>+</mo><mn>2</mn><mi>s</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mi mathvariant="normal">dim</mi><mi>V</mi></mrow></msup><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2069.  \mathcal{H}_{\lambda,V}^\omega |_{\mathbf{G}_0(X)}
  2070.  \;\simeq\;
  2071.  \hat \oplus_{\xi \in F^{2k+1}}
  2072.  \,
  2073.  (
  2074.  H_{\lambda+2s(\xi)}
  2075.  )^{\mathrm{dim}V}
  2076.  \,.
  2077. </annotation></semantics></math></div>
  2078.  
  2079. <p>It is in expressions like these that the remarks on Betti numbers in the previous section apply to: Clearly, there are <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mn>2</mn> <mi>b</mi></msup></mrow><annotation encoding='application/x-tex'>2^b</annotation></semantics></math> choices for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math> such that the representation spaces <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>&#x0210B;</mi> <mrow><mi>&#x003BB;</mi><mo>,</mo><mi>V</mi></mrow> <mi>&#x003C9;</mi></msubsup><msub><mo stretchy="false">&#x0007C;</mo> <mrow><msub><mstyle mathvariant="bold"><mi>G</mi></mstyle> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>  \mathcal{H}_{\lambda,V}^\omega |_{\mathbf{G}_0(X)}
  2080. </annotation></semantics></math> given above differ. In other words, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi><mo>&#x02032;</mo></mrow><annotation encoding='application/x-tex'>\lambda&apos;</annotation></semantics></math> lead to the same representation space iff they differ by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>2</mn><mi>s</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>2s(\xi)</annotation></semantics></math>, for some <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BE;</mi></mrow><annotation encoding='application/x-tex'>\xi</annotation></semantics></math>.</p>
  2081.  
  2082. <p><br/><strong>The number of irreps for the full group.</strong></p>
  2083.  
  2084. <p>Gomi now claims, that this situation essentially carries over to the full group of connections.</p>
  2085.  
  2086. <p>More precisely, he claims </p>
  2087.  
  2088. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math>
  2089. (theorem 6.11, p. 24) that the rep <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>&#x003C1;</mi> <mrow><mi>&#x003BB;</mi><mo>,</mo><mi>V</mi></mrow> <mi>&#x003C9;</mi></msubsup></mrow><annotation encoding='application/x-tex'>\rho_{\lambda,V}^\omega</annotation></semantics></math> is irreducible precisely if the finite dimensional rep on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> of the torsion part is;</p>
  2090.  
  2091. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02022;</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> and (theorem 6.10, p. 24) that two such irreps <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>&#x003C1;</mi> <mrow><mi>&#x003BB;</mi><mo>,</mo><mi>V</mi></mrow> <mi>&#x003C9;</mi></msubsup></mrow><annotation encoding='application/x-tex'>\rho_{\lambda,V}^\omega</annotation></semantics></math>, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>&#x003C1;</mi> <mrow><mi>&#x003BB;</mi><mo>&#x02032;</mo><mo>,</mo><mi>V</mi><mo>&#x02032;</mo></mrow> <mrow><mi>&#x003C9;</mi><mo>&#x02032;</mo></mrow></msubsup></mrow><annotation encoding='application/x-tex'>\rho_{\lambda&apos;,V&apos;}^{\omega&apos;}</annotation></semantics></math> are equivalent precisely if the reps of the torsion parts are and if there is <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BE;</mi><mo>&#x02208;</mo><msup><mi>F</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\xi \in F^{2k+1}</annotation></semantics></math> such that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi><mo>&#x02032;</mo><mo>=</mo><mi>&#x003BB;</mi><mo>+</mo><mn>2</mn><mi>s</mi><mo stretchy="false">(</mo><mi>&#x003BE;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\lambda&apos; = \lambda + 2s(\xi)</annotation></semantics></math>.</p>
  2092.  
  2093. <p>As discussed above, this means that for fixed <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> there are <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mn>2</mn> <mi>b</mi></msup></mrow><annotation encoding='application/x-tex'>2^b</annotation></semantics></math> inequivalent irreps. It hence remains to count the equivalence classes of irreps for the group of torsion connections.</p>
  2094.  
  2095. <p>Gomi (pp. 24-25) cites a standard fact about projective reps of finite groups, which says that their number is the same of the number of <em>regular elements</em> of the cocycle. In the present case an element is regular with respect to the Chern-Simons cocycle if an only if it is its own inverse. The number of these elements we called <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>r</mi></mrow><annotation encoding='application/x-tex'>r</annotation></semantics></math>. </p>
  2096.  
  2097. <p>Hence, in total, there are</p>
  2098.  
  2099. <div class="numberedEq"><span>(41)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mn>2</mn> <mi>b</mi></msup><mspace width="thinmathspace"/><mi>r</mi></mrow><annotation encoding='application/x-tex'>
  2100.  2^b\, r
  2101. </annotation></semantics></math></div>
  2102.  
  2103. <p>equivalence classes of irreps <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msubsup><mi>&#x003C1;</mi> <mrow><mi>&#x003BB;</mi><mo>,</mo><mi>V</mi></mrow> <mi>&#x003C9;</mi></msubsup></mrow><annotation encoding='application/x-tex'>\rho_{\lambda,V}^\omega</annotation></semantics></math> of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">&#x002DC;</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>\tilde \mathbf{G}(X)</annotation></semantics></math>.</p>
  2104.      </div>
  2105.    </content>
  2106.  </entry>
  2107.  <entry>
  2108.    <title type="html">Seminar on 2-Vector Bundles and Elliptic Cohomology, VI</title>
  2109.    <link rel="alternate" type="application/xhtml+xml" href="http://golem.ph.utexas.edu/string/archives/000872.html" />
  2110.    <updated>2006-07-19T03:43:35Z</updated>
  2111.    <published>2006-07-13T20:04:52+00:00</published>
  2112.    <id>tag:golem.ph.utexas.edu,2006:%2Fstring%2F2.872</id>
  2113.    <summary type="text">More detials on elliptic curves, formal groups and "classical" elliptic cohomology.</summary>
  2114.    <author>
  2115.      <name>urs</name>
  2116.      <uri>http://www.math.uni-hamburg.de/home/schreiber</uri>
  2117.      <email>[email protected]</email>
  2118.    </author>
  2119.    <category term="mathematical physics" />
  2120.    <content type="xhtml" xml:base="http://golem.ph.utexas.edu/string/archives/000872.html">
  2121.      <div xmlns="http://www.w3.org/1999/xhtml">
  2122. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  2123. <p>In the 4th (and probably last) session of our <a href="http://golem.ph.utexas.edu/string/archives/000737.html">seminar</a> Birgit Richter talked in more detail about</p>
  2124.  
  2125. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mrow><annotation encoding='application/x-tex'>\;\;</annotation></semantics></math> <strong>0)</strong> elliptic curves and formal groups</p>
  2126.  
  2127. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mrow><annotation encoding='application/x-tex'>\;\;</annotation></semantics></math><strong>1)</strong> &#8220;classical&#8221; elliptic cohomology (according to Landweber, Ochanine and Stong)</p>
  2128.  
  2129. <p>and a tiny bit about</p>
  2130.  
  2131. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mrow><annotation encoding='application/x-tex'>\;\;</annotation></semantics></math><strong>2)</strong> topological modular forms (due mainly to Hopkins)</p>
  2132.  
  2133. <p>and ran out of time before talking about</p>
  2134.  
  2135. <p><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mspace width="thickmathspace"/><mspace width="thickmathspace"/></mrow><annotation encoding='application/x-tex'>\;\;</annotation></semantics></math><strong>3)</strong> other forms of elliptic cohomology (e.g. Kriz-Sati) ,</p>
  2136.  
  2137. <p>complementing my rough outline <a href="http://golem.ph.utexas.edu/string/archives/000862.html">last time</a> with more technical details.</p>
  2138. <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div>
  2139. <p><strong>0) elliptic curves and formal groups</strong></p>
  2140.  
  2141. <p>The <strong>Weierstrass form of an elliptic curve </strong> is an equation in two variables <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> of the form</p>
  2142.  
  2143. <div class="numberedEq"><span>(1)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>E</mi><mo>:</mo><mspace width="thickmathspace"/><mspace width="thickmathspace"/><mspace width="thickmathspace"/><msup><mi>y</mi> <mn>2</mn></msup><mo>+</mo><msub><mi>a</mi> <mn>1</mn></msub><mi>xy</mi><mo>+</mo><msub><mi>a</mi> <mn>3</mn></msub><mi>y</mi><mo>=</mo><msup><mi>x</mi> <mn>3</mn></msup><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><mi>x</mi><mo>+</mo><msub><mi>a</mi> <mn>4</mn></msub><mi>y</mi><mo>+</mo><msub><mi>a</mi> <mn>6</mn></msub><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2144.  E :
  2145.  \;\;\;
  2146.  y^2 + a_1 xy + a_3y  = x^3 + a_2 x + a_4 y + a_6
  2147.  \,.
  2148. </annotation></semantics></math></div>
  2149.  
  2150. <p>There is something called the <strong>discriminant</strong> <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x00394;</mi></mrow><annotation encoding='application/x-tex'>\Delta</annotation></semantics></math> of  <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> and if it is nonvanishing we have a <em>smooth</em> curve.</p>
  2151.  
  2152. <p>Thinking of the above equation as living over the real numbers, such smooth curves are certain smooth curves in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x0211D;</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^2</annotation></semantics></math>. Straight lines in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x0211D;</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^2</annotation></semantics></math> which coincide with this curve in three point <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math>, <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> define an abelian group structure on points by setting</p>
  2153.  
  2154. <div class="numberedEq"><span>(2)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>P</mi><mo>+</mo><mi>Q</mi><mo>+</mo><mi>R</mi><mo>=</mo><mn>0</mn><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2155.  P + Q + R = 0
  2156.  \,.
  2157. </annotation></semantics></math></div>
  2158.  
  2159. <p>For many applications it is convenient to perform a coordinate transformation from <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(x,y)</annotation></semantics></math> to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(w,z)</annotation></semantics></math> with</p>
  2160.  
  2161. <div class="numberedEq"><span>(3)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>w</mi></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mfrac><mn>1</mn><mi>y</mi></mfrac></mtd></mtr> <mtr><mtd><mi>z</mi></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mfrac><mi>x</mi><mi>y</mi></mfrac><mspace width="thinmathspace"/><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  2162. \begin{aligned}
  2163.  w &amp;= &#x2d; \frac{1}{y}
  2164.  \\
  2165.  z &amp;= &#x2d;\frac{x}{y}
  2166.  \,.
  2167. \end{aligned}
  2168. </annotation></semantics></math></div>
  2169.  
  2170. <p>Then there is an <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> such that the above equation for the ellitptic curves reads equivalently</p>
  2171.  
  2172. <div class="numberedEq"><span>(4)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>w</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2173.  w = f(z,w)
  2174.  \,.
  2175. </annotation></semantics></math></div>
  2176.  
  2177. <p>By iteratively re-inserting <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> into itself according to this equation, we find that</p>
  2178.  
  2179. <div class="numberedEq"><span>(5)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>z</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>A</mi> <mn>1</mn></msub><mi>z</mi><mo>+</mo><msub><mi>A</mi> <mn>2</mn></msub><msup><mi>z</mi> <mn>2</mn></msup><mo>+</mo><mi>&#x022EF;</mi><mo stretchy="false">)</mo><mo>&#x02208;</mo><mi>&#x02124;</mi><mo stretchy="false">[</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>6</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  2180.  f(z,w)
  2181.  =
  2182.  z^3(
  2183.    1 + A_1 z + A_2 z^2 + \cdots
  2184.  )
  2185.  \in
  2186.  \mathbb{Z}[a_1,a_2,a_3,a_4,a_6][[z]]
  2187.  \,,
  2188. </annotation></semantics></math></div>
  2189.  
  2190. <p>which is a power series in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>z</mi></mrow><annotation encoding='application/x-tex'>z</annotation></semantics></math> starting in degree 3, with coefficients being polynomials in the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>a_i</annotation></semantics></math> over the integers.</p>
  2191.  
  2192. <p>Using this, we can understand the above mentioned addition on the elliptic curve as given by a power series in two variables. Namely, if <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><msub><mi>w</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(w_1,z_1)</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><msub><mi>w</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(w_2,z_2)</annotation></semantics></math> are two points on the smooth elliptic curve <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> (which means that the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>z</mi></mrow><annotation encoding='application/x-tex'>z</annotation></semantics></math> coordinate is determined by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>w</mi></mrow><annotation encoding='application/x-tex'>w</annotation></semantics></math>), then the result of adding them has a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>z</mi></mrow><annotation encoding='application/x-tex'>z</annotation></semantics></math>-coordinate which is given by a power series</p>
  2193.  
  2194. <div class="numberedEq"><span>(6)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>F</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>&#x02208;</mo><mi>&#x02124;</mi><mo stretchy="false">[</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>&#x022EF;</mi><mo>,</mo><msub><mi>a</mi> <mn>6</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2195.  F_E(z_1,z_2)
  2196.  \in \mathbb{Z}[a_1,\cdots, a_6][[z_1,z_2]]
  2197.  \,.
  2198. </annotation></semantics></math></div>
  2199.  
  2200. <p>This <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>F</mi> <mi>E</mi></msub></mrow><annotation encoding='application/x-tex'>F_E</annotation></semantics></math> is a <strong>formal group law</strong>, which implies (as I mentioned <a href="http://golem.ph.utexas.edu/string/archives/000862.html">last time</a>) that it satisfies equations</p>
  2201.  
  2202. <p>1) <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>z</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>F(z_1,0) = z_1</annotation></semantics></math></p>
  2203.  
  2204. <p>2) <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>F(z_1,z_2) = F(z_2,z_1)</annotation></semantics></math></p>
  2205.  
  2206. <p>3) <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>z</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>F(F(z_1,z_2),z_3) = F(z_1,F(z_2,z_3))</annotation></semantics></math> </p>
  2207.  
  2208. <p>for all <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>z</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>z_i</annotation></semantics></math>.</p>
  2209.  
  2210. <p>Form this one can show that inverses of all elements exist.</p>
  2211.  
  2212. <p>The <strong>prototypical example</strong> of such a formal group law is obtained by taking a 1-dimensional real Lie group, looking at the tangent space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>T</mi> <mi>e</mi></msub></mrow><annotation encoding='application/x-tex'>T_e</annotation></semantics></math> at a given point <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>e</mi></mrow><annotation encoding='application/x-tex'>e</annotation></semantics></math>, using the exponential map to identify a neighbourhood of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>e</mi></mrow><annotation encoding='application/x-tex'>e</annotation></semantics></math> in the group with the tangent space and expanding for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>&#x02208;</mo><msub><mi>T</mi> <mi>e</mi></msub></mrow><annotation encoding='application/x-tex'>x,y \in T_e</annotation></semantics></math> the multiplication in the group as</p>
  2213.  
  2214. <div class="numberedEq"><span>(7)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>&#x003BC;</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">&#x02211;</mo> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></munder><msub><mi>a</mi> <mi>nm</mi></msub><msup><mi>x</mi> <mi>n</mi></msup><msup><mi>y</mi> <mi>m</mi></msup></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mtext>higher terms</mtext><mo>&#x02208;</mo><mi>&#x0211D;</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2215. \begin{aligned}
  2216.  \mu(x,y) &amp;= \sum_{n,m} a_{nm} x^n y ^m
  2217.  \\
  2218.  &amp;=
  2219.  x + y + \text{higher terms} \in \mathbb{R}[[x,y]]
  2220. \end{aligned}
  2221.  \,.
  2222. </annotation></semantics></math></div>
  2223.  
  2224. <p>In general, formal groups are local expansions of group laws. The power series <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>F</mi> <mi>E</mi></msub></mrow><annotation encoding='application/x-tex'>F_E</annotation></semantics></math> associated to a smooth elliptic curve as described above is similarly the expansion of the additve group law defined by the elliptic curve.</p>
  2225.  
  2226. <p><strong>Quillen explained</strong> that formal groups are related to complex cobordisms.</p>
  2227.  
  2228. <p>Let <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>MU_*</annotation></semantics></math> be the complex bordsim ring, which is the ring whose elements are cobordism classes of (stably) complex manifolds with multiplication being cartesian product and additon being disjoint union. We write</p>
  2229.  
  2230. <div class="numberedEq"><span>(8)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msubsup><mi>&#x003A9;</mi> <mo>*</mo> <mi>U</mi></msubsup><mo>=</mo><msub><mi>MU</mi> <mo>*</mo></msub><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2231.  \Omega_*^U = MU_*
  2232.  \,.
  2233. </annotation></semantics></math></div>
  2234.  
  2235. <p>This ring is <strong>universal for formal group laws</strong> in the sense that there is a formal group law </p>
  2236.  
  2237. <div class="numberedEq"><span>(9)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>F</mi> <mi>MU</mi></msub></mrow><annotation encoding='application/x-tex'>
  2238.  F_{MU}
  2239. </annotation></semantics></math></div>
  2240.  
  2241. <p>over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>MU_*</annotation></semantics></math> such that for every formal group law <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> over any ring <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> there is a unique ring homomorphism</p>
  2242.  
  2243. <div class="numberedEq"><span>(10)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x003B8;</mi><mo>:</mo><msub><mi>MU</mi> <mo>*</mo></msub><mo>&#x02192;</mo><mi>R</mi></mrow><annotation encoding='application/x-tex'>
  2244.  \theta : MU_* \to R
  2245. </annotation></semantics></math></div>
  2246.  
  2247. <p>such that</p>
  2248.  
  2249. <div class="numberedEq"><span>(11)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>F</mi><mo>=</mo><msub><mi>&#x003B8;</mi> <mo>*</mo></msub><msub><mi>F</mi> <mi>MU</mi></msub></mrow><annotation encoding='application/x-tex'>
  2250.  F = \theta_* F_{MU}
  2251. </annotation></semantics></math></div>
  2252.  
  2253. <p>which means that if <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>F</mi> <mi>MU</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">&#x02211;</mo> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><msub><mi>a</mi> <mi>nm</mi></msub><msup><mi>x</mi> <mi>n</mi></msup><msup><mi>y</mi> <mi>m</mi></msup></mrow><annotation encoding='application/x-tex'>F_{MU}(x,y) = \sum_{n,m} a_{nm}x^n y^m</annotation></semantics></math>
  2254. then</p>
  2255.  
  2256. <div class="numberedEq"><span>(12)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">&#x02211;</mo> <mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></munder><mi>&#x003B8;</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mi>nm</mi></msub><mo stretchy="false">)</mo><msup><mi>x</mi> <mi>n</mi></msup><msup><mi>y</mi> <mi>m</mi></msup><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2257.  F(x,y) =   \sum_{n,m} \theta(a_{nm})x^n y^m
  2258.  \,.
  2259. </annotation></semantics></math></div>
  2260.  
  2261. <p>Recalling from above that every elliptic curve <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> gives rise to a formal group law <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>F</mi> <mi>E</mi></msub></mrow><annotation encoding='application/x-tex'>F_E</annotation></semantics></math> over the ring <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02124;</mi><mo stretchy="false">[</mo><msubsup><mi>a</mi> <mn>1</mn> <mo>,</mo></msubsup><mi>&#x022EF;</mi><mo>,</mo><msub><mi>a</mi> <mn>6</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>\mathbb{Z}[a_1^,\cdots,a_6]</annotation></semantics></math>, we find that for every elliptic curve there is a unique ring homomorphism</p>
  2262.  
  2263. <div class="numberedEq"><span>(13)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>&#x003B8;</mi> <mi>E</mi></msub><mo>:</mo><msub><mi>MU</mi> <mo>*</mo></msub><mo>&#x02192;</mo><mi>&#x02124;</mi><mo stretchy="false">[</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>&#x02026;</mi><mo>,</mo><msub><mi>a</mi> <mn>6</mn></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2264.  \theta_E : MU_* \to \mathbb{Z}[a_1,\dots,a_6]
  2265.  \,.
  2266. </annotation></semantics></math></div>
  2267.  
  2268. <p>Using this homomorphism we get an action of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>MU_*</annotation></semantics></math> on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02124;</mi><mo stretchy="false">[</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>&#x022EF;</mi><mo>,</mo><msub><mi>a</mi> <mn>6</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>\mathbb{Z}[a_1,\cdots,a_6]</annotation></semantics></math>. We want to use this to form a <strong>generalized cohomology theory</strong> (<a href="http://golem.ph.utexas.edu/string/archives/000862.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>) by tensoring <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02124;</mi><mo stretchy="false">[</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>&#x02026;</mi><mo>,</mo><msub><mi>a</mi> <mn>6</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>\mathbb{Z}[a_1,\dots,a_6]</annotation></semantics></math> with the universal cohomology <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>MU_*</annotation></semantics></math> theory defined by complex cobordisms.</p>
  2269.  
  2270. <p>Instead of describing the ring spectrum which represents <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>MU_*</annotation></semantics></math>, we here just say how the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>MU</mi></mrow><annotation encoding='application/x-tex'>MU</annotation></semantics></math> cohomology <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>MU_*(X)</annotation></semantics></math> of any space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> looks like.</p>
  2271.  
  2272. <p>We set <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>MU_n(X)</annotation></semantics></math> to be the ring of maps </p>
  2273.  
  2274. <div class="numberedEq"><span>(14)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x03D5;</mi><mo>:</mo><msup><mi>M</mi> <mi>n</mi></msup><mo>&#x02192;</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>
  2275.  \phi : M^n \to X
  2276. </annotation></semantics></math></div>
  2277.  
  2278. <p>from stably complex <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-manifolds <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>M</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>M^n</annotation></semantics></math> to <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, where we identitfy two maps if their domain manifolds are cobounded by a stably complex <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(n+1)</annotation></semantics></math>-manifold.</p>
  2279.  
  2280. <p>Here stably complex means that we can embed <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>M</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>M^n</annotation></semantics></math> in some <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi mathvariant="normal">R</mi> <mi>N</mi></msup></mrow><annotation encoding='application/x-tex'>\mathrm{R}^N</annotation></semantics></math> for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> sufficiently large, such that the normal bundle of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>M</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>M^n</annotation></semantics></math> in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msup><mi>&#x0211D;</mi> <mi>N</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^N</annotation></semantics></math> is a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02102;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{C}</annotation></semantics></math>-vector bundle.</p>
  2281.  
  2282. <p>The entire ring <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>MU_*(X)</annotation></semantics></math> is just the direct sum</p>
  2283.  
  2284. <div class="numberedEq"><span>(15)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>&#x02295;</mo> <mi>n</mi></msub><msub><mi>MU</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  2285.  MU_*(X) = \oplus_n MU_n(X)
  2286. </annotation></semantics></math></div>
  2287.  
  2288. <p>and in particular the bare <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>MU_*</annotation></semantics></math> from above is shorthand for the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>MU</mi></mrow><annotation encoding='application/x-tex'>MU</annotation></semantics></math>-cohomology of a point</p>
  2289.  
  2290. <div class="numberedEq"><span>(16)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub><mo>:</mo><mo>=</mo><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mtext>pt</mtext><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2291.  MU_* := MU_*(\text{pt})
  2292.  \,.
  2293. </annotation></semantics></math></div>
  2294.  
  2295. <p>It is important for the following construction that there is a natural graded action of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mtext>pt</mtext><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>MU_*(\text{pt})</annotation></semantics></math> on any <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>MU_*(X)</annotation></semantics></math></p>
  2296.  
  2297. <div class="numberedEq"><span>(17)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>MU</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mtext>pt</mtext><mo stretchy="false">)</mo><mo>&#x000D7;</mo><msub><mi>MU</mi> <mi>m</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x02192;</mo><msub><mi>MU</mi> <mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>
  2298.  MU_n(\text{pt}) \times MU_m(X) \to MU_{n+m}(X)
  2299. </annotation></semantics></math></div>
  2300.  
  2301. <p>simply given by taking a map</p>
  2302.  
  2303. <div class="numberedEq"><span>(18)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>M</mi> <mi>n</mi></msup><mo>&#x02192;</mo><mtext>pt</mtext></mrow><annotation encoding='application/x-tex'>
  2304.  M^n \to \text{pt}
  2305. </annotation></semantics></math></div>
  2306.  
  2307. <p>and </p>
  2308.  
  2309. <div class="numberedEq"><span>(19)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>M</mi> <mi>m</mi></msup><mo>&#x02192;</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>
  2310.  M^m \to X
  2311. </annotation></semantics></math></div>
  2312.  
  2313. <p>and forming the obvious map</p>
  2314.  
  2315. <div class="numberedEq"><span>(20)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>M</mi> <mi>n</mi></msup><mo>&#x000D7;</mo><msup><mi>M</mi> <mi>m</mi></msup><mo>&#x02192;</mo><mtext>pt</mtext><mo>&#x000D7;</mo><mi>X</mi><mo>&#x02243;</mo><mi>X</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2316.  M^n \times M^m \to \text{pt}\times X \simeq X
  2317.  \,.
  2318. </annotation></semantics></math></div>
  2319.  
  2320. <p>In summary, we have an action of the ring <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mtext>pt</mtext><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>MU_*(\text{pt})</annotation></semantics></math> both on the ring <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x02124;</mi><mo stretchy="false">[</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>&#x022EF;</mi><mo>,</mo><msub><mi>a</mi> <mn>6</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>\mathbb{Z}[a_1,\cdots,a_6]</annotation></semantics></math> and the ring <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>MU_*(X)</annotation></semantics></math>, for all <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. </p>
  2321.  
  2322. <p>Hence, for each elliptic curve <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> and each space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, we can form the graded ring</p>
  2323.  
  2324. <div class="numberedEq"><span>(21)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>&#x02297;</mo> <mrow><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mtext>pt</mtext><mo stretchy="false">)</mo></mrow></msub><mi>&#x02124;</mi><mo stretchy="false">[</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>&#x022EF;</mi><mo>,</mo><msub><mi>a</mi> <mn>6</mn></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2325.  E_*(X) := MU_*(X) \otimes_{MU_*(\text{pt})} \mathbb{Z}[a_1,\cdots,a_6]
  2326.  \,.
  2327. </annotation></semantics></math></div>
  2328.  
  2329. <p>This is the <strong>elliptic cohomology</strong> ring of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with respect to the elliptic curve <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>.</p>
  2330.  
  2331. <p>As an <strong>example</strong> for this we recover ordinary integral cohomology and K-theory as degenerate cases of elliptic cohomology.</p>
  2332.  
  2333. <p>Namely, if our elliptic curve happens to be</p>
  2334.  
  2335. <div class="numberedEq"><span>(22)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>x</mi> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>
  2336.  y^2 = x^3
  2337. </annotation></semantics></math></div>
  2338.  
  2339. <p>with a bad singularity at <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>(0,0)</annotation></semantics></math>, the corresponding group law is simply (this is not supposed to be obvious)</p>
  2340.  
  2341. <div class="numberedEq"><span>(23)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2342.  F(x,y) = x + y
  2343.  \,.
  2344. </annotation></semantics></math></div>
  2345.  
  2346. <p>As I reviewed <a href="http://golem.ph.utexas.edu/string/archives/000862.html">last time</a>, this is the group law which corresponds to ordinary integral cohomology.</p>
  2347.  
  2348. <p>The elliptic curve</p>
  2349.  
  2350. <div class="numberedEq"><span>(24)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>x</mi> <mn>3</mn></msup><mo>+</mo><msup><mi>x</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>
  2351.  y^2 = x^3 + x^2
  2352. </annotation></semantics></math></div>
  2353.  
  2354. <p>has a singularity which is not quite as bad. It gives rise to the group law</p>
  2355.  
  2356. <div class="numberedEq"><span>(25)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mn>1</mn><mo>&#x02212;</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>xy</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2357. \begin{aligned}
  2358.  F(x,y) &amp;= 1 &#x2d; (1&#x2d;x)(1&#x2d;y)
  2359.  \\
  2360.  &amp;= x + y + xy
  2361. \end{aligned}  
  2362. \,.
  2363. </annotation></semantics></math></div>
  2364.  
  2365. <p>As you can see from the table given <a href="http://golem.ph.utexas.edu/string/archives/000862.html">last time</a>, this is the group law which identifies complex K-theory.</p>
  2366.  
  2367. <p><br/><strong>1) classical elliptic cohomology</strong></p>
  2368.  
  2369. <p>A special case of elliptic curves are the <strong>Jacobi curves</strong>, which are of the form</p>
  2370.  
  2371. <div class="numberedEq"><span>(26)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>&#x02212;</mo><mn>2</mn><mi>&#x003B4;</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><mi>&#x003F5;</mi><msup><mi>x</mi> <mn>4</mn></msup><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  2372.  y^2 = 1 &#x2d; 2\delta x^2 + \epsilon x^4
  2373.  \,,
  2374. </annotation></semantics></math></div>
  2375.  
  2376. <p>depending on two parameters <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B4;</mi></mrow><annotation encoding='application/x-tex'>\delta</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003F5;</mi></mrow><annotation encoding='application/x-tex'>\epsilon</annotation></semantics></math>.</p>
  2377.  
  2378. <p>The discriminant of these is</p>
  2379.  
  2380. <div class="numberedEq"><span>(27)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x00394;</mi><mo>=</mo><mi>&#x003F5;</mi><mo stretchy="false">(</mo><msup><mi>&#x003B4;</mi> <mn>2</mn></msup><mo>&#x02212;</mo><mi>&#x003F5;</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2381.  \Delta = \epsilon(\delta^2 &#x2d; \epsilon)^2
  2382.  \,.
  2383. </annotation></semantics></math></div>
  2384.  
  2385. <p>Using </p>
  2386.  
  2387. <div class="numberedEq"><span>(28)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>g</mi> <mn>2</mn></msub></mtd> <mtd><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>&#x003B4;</mi> <mn>2</mn></msup><mo>&#x02212;</mo><mn>3</mn><mi>&#x003F5;</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mn>3</mn></mtd></mtr> <mtr><mtd><msub><mi>g</mi> <mn>3</mn></msub></mtd> <mtd><mo>=</mo><mi>&#x003B4;</mi><mo stretchy="false">(</mo><msup><mi>&#x003B4;</mi> <mn>2</mn></msup><mo>&#x02212;</mo><mn>9</mn><mi>&#x003F5;</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mn>27</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  2388. \begin{aligned}
  2389.  g_2 &amp;:= (\delta^2 &#x2d; 3\epsilon)/3
  2390.  \\
  2391.  g_3 &amp;= \delta(\delta^2 &#x2d; 9 \epsilon)/27
  2392. \end{aligned}
  2393. </annotation></semantics></math></div>
  2394.  
  2395. <p>we can alternatively write</p>
  2396.  
  2397. <div class="numberedEq"><span>(29)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mi>x</mi> <mn>3</mn></msup><mo>&#x02212;</mo><msub><mi>g</mi> <mn>2</mn></msub><mi>x</mi><mo>&#x02212;</mo><msub><mi>g</mi> <mn>3</mn></msub><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  2398.  y^2 = 4 x^3 &#x2d; g_2 x &#x2d; g_3
  2399.  \,,
  2400. </annotation></semantics></math></div>
  2401.  
  2402. <p>which however works only in characteristic <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&gt;</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>\gt 3</annotation></semantics></math>, which is problematic in particular when applied to speher spectra, cause homotopy classes there have lots of 2- and 3-torsion.</p>
  2403.  
  2404. <p>Anyway, the formal group law corresponding to these curves is</p>
  2405.  
  2406. <div class="numberedEq"><span>(30)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>x</mi><msqrt><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msqrt><mo>+</mo><mi>y</mi><msqrt><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msqrt></mrow><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  2407.  F(x,y) =
  2408.  \frac{x \sqrt{R(y)} + y \sqrt{R(x)}}{R(x)}
  2409.  \,,
  2410. </annotation></semantics></math></div>
  2411.  
  2412. <p>where</p>
  2413.  
  2414. <div class="numberedEq"><span>(31)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mn>1</mn><mo>&#x02212;</mo><mn>2</mn><mi>&#x003B4;</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><mi>&#x003F5;</mi><msup><mi>x</mi> <mn>4</mn></msup><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2415. R(x) := 1 &#x2d; 2\delta x^2 + \epsilon x^4
  2416. \,.
  2417. </annotation></semantics></math></div>
  2418.  
  2419. <p>This formula was originally found by Euler, even though he did not call it a formal group law.</p>
  2420.  
  2421. <p>We can rewrite <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>F(x,y)</annotation></semantics></math> as</p>
  2422.  
  2423. <div class="numberedEq"><span>(32)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>g</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  2424.  F(x,y)
  2425.  =
  2426.  g^{&#x2d;1}(g(x)+g(y))
  2427.  \,,
  2428. </annotation></semantics></math></div>
  2429.  
  2430. <p>where</p>
  2431.  
  2432. <div class="numberedEq"><span>(33)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>&#x0222B;</mo> <mn>0</mn> <mi>x</mi></msubsup><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mspace width="thinmathspace"/><mi>dt</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2433.  g(x) = \int_0^x R(t)^{&#x2d;1/2}\, dt
  2434.  \,.
  2435. </annotation></semantics></math></div>
  2436.  
  2437. <p>Now let <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mover><mi>M</mi><mo stretchy="false">&#x002DC;</mo></mover> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>\tilde M_*</annotation></semantics></math> be the ring of modular forms under the subgroup of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>&#x02124;</mi><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>SL(2,\mathbb{Z})</annotation></semantics></math> generated by <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C4;</mi><mo>&#x021A6;</mo><mi>&#x003C4;</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>\tau \mapsto \tau + 2</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003C4;</mi><mo>&#x021A6;</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mfrac><mn>1</mn><mi>&#x003C4;</mi></mfrac></mrow><annotation encoding='application/x-tex'>\tau \mapsto &#x2d;\frac{1}{\tau}</annotation></semantics></math>.</p>
  2438.  
  2439. <p>There is a <strong>theorem</strong> due to Landweber, Ravenal and Stong which says that for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B4;</mi></mrow><annotation encoding='application/x-tex'>\delta</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003F5;</mi></mrow><annotation encoding='application/x-tex'>\epsilon</annotation></semantics></math> algebraically independent over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0211A;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Q}</annotation></semantics></math> we have</p>
  2440.  
  2441. <div class="numberedEq"><span>(34)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mover><mi>M</mi><mo stretchy="false">&#x002DC;</mo></mover> <mo>*</mo></msub><mo>=</mo><mi>&#x02124;</mi><mo stretchy="false">[</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>&#x003B4;</mi><mo>,</mo><mi>&#x003F5;</mi><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>
  2442.  \tilde M_* = \mathbb{Z}[\frac{1}{2}][\delta,\epsilon]
  2443. </annotation></semantics></math></div>
  2444.  
  2445. <p>and for all of the rings in the diagram</p>
  2446.  
  2447. <div class="numberedEq"><span>(35)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mover><mi>M</mi><mo stretchy="false">&#x002DC;</mo></mover> <mo>*</mo></msub></mtd> <mtd><mo>&#x02192;</mo></mtd> <mtd><msub><mover><mi>M</mi><mo stretchy="false">&#x002DC;</mo></mover> <mo>*</mo></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msup><mi>&#x003B4;</mi> <mn>2</mn></msup><mo>&#x02212;</mo><mi>&#x003F5;</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">&#x02193;</mo></mtd> <mtd/> <mtd><mo stretchy="false">&#x02193;</mo></mtd></mtr> <mtr><mtd><msub><mover><mi>M</mi><mo stretchy="false">&#x002DC;</mo></mover> <mo>*</mo></msub><mo stretchy="false">[</mo><msup><mi>&#x003F5;</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mtd> <mtd><mo>&#x02192;</mo></mtd> <mtd><msub><mover><mi>M</mi><mo stretchy="false">&#x002DC;</mo></mover> <mo>*</mo></msub><mo stretchy="false">[</mo><msup><mi>&#x00394;</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  2448. \array{
  2449.  \tilde M_* &amp;\to&amp; \tilde M_*[(\delta^2 &#x2d; \epsilon)^{&#x2d;1}]
  2450.  \\
  2451.  \downarrow &amp;&amp; \downarrow
  2452.  \\
  2453.  \tilde M_*[\epsilon^{&#x2d;1}]
  2454.  &amp;\to&amp;
  2455.  \tilde M_*[\Delta^{&#x2d;1}]
  2456. }
  2457. </annotation></semantics></math></div>
  2458.  
  2459. <p>there exists a generalized homology theory <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>h</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>h_*</annotation></semantics></math> such that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>h</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mtext>pt</mtext><mo stretchy="false">)</mo></mrow><annotation encoding='application/x-tex'>h_*(\text{pt})</annotation></semantics></math> is that given ring.</p>
  2460.  
  2461. <p>This is constructed by noticing that the formal group law defes an action of the oriented cobordism ring on the given ring <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>R</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>R_*</annotation></semantics></math> from above, which allows us to form the homology ring of some space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> as</p>
  2462.  
  2463. <div class="numberedEq"><span>(36)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>X</mi><mo>&#x021A6;</mo><msubsup><mi>&#x003A9;</mi> <mo>*</mo> <mi mathvariant="normal">SO</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>&#x02297;</mo> <mrow><msubsup><mi>&#x003A9;</mi> <mo>*</mo> <mi mathvariant="normal">SO</mi></msubsup></mrow></msub><msub><mi>R</mi> <mo>*</mo></msub><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2464.  X \mapsto \Omega^\mathrm{SO}_*(X) \otimes_{\Omega_*^\mathrm{SO}}
  2465.  R_*
  2466.  \,.
  2467. </annotation></semantics></math></div>
  2468.  
  2469. <p>This is the homology which does the job.</p>
  2470.  
  2471. <p>The <strong>relation to genera</strong> is as follows. </p>
  2472.  
  2473. <p>A <strong>genus</strong> is a ring homomorphism </p>
  2474.  
  2475. <div class="numberedEq"><span>(37)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x03D5;</mi><mo>:</mo><msubsup><mi>&#x003A9;</mi> <mo>*</mo> <mi mathvariant="normal">SO</mi></msubsup><mo>&#x02192;</mo><mi>&#x0039B;</mi></mrow><annotation encoding='application/x-tex'>
  2476.  \phi : \Omega_*^\mathrm{SO} \to \Lambda
  2477. </annotation></semantics></math></div>
  2478.  
  2479. <p>from oriented cobordisms to any other ring <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0039B;</mi></mrow><annotation encoding='application/x-tex'>\Lambda</annotation></semantics></math> that is also a <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0211A;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Q}</annotation></semantics></math>-algebra.</p>
  2480.  
  2481. <p>Since </p>
  2482.  
  2483. <div class="numberedEq"><span>(38)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msubsup><mi>&#x003A9;</mi> <mo>*</mo> <mi mathvariant="normal">SO</mi></msubsup><mo>&#x02297;</mo><mi>&#x0211A;</mi><mo>&#x02243;</mo><mi>&#x0211A;</mi><mrow><mo>[</mo><mrow><mo>[</mo><mi>&#x02102;</mi><msup><mi>P</mi> <mn>2</mn></msup><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mi>&#x02102;</mi><msup><mi>P</mi> <mn>4</mn></msup><mo>]</mo></mrow><mo>,</mo><mi>&#x022EF;</mi><mo>]</mo></mrow></mrow><annotation encoding='application/x-tex'>
  2484.  \Omega_*^\mathrm{SO} \otimes \mathbb{Q}
  2485.  \simeq
  2486.  \mathbb{Q}
  2487.  \left[
  2488.    \left[\mathbb{C}P^2\right],
  2489.    \left[\mathbb{C}P^4\right],
  2490.    \cdots
  2491.  \right]
  2492. </annotation></semantics></math></div>
  2493.  
  2494. <p>it suffices to specify <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x03D5;</mi></mrow><annotation encoding='application/x-tex'>\phi</annotation></semantics></math> on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">[</mo><mi>&#x02102;</mi><msup><mi>P</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>[\mathbb{C}P^{2n}]</annotation></semantics></math>.</p>
  2495.  
  2496. <p>One calls the expression</p>
  2497.  
  2498. <div class="numberedEq"><span>(39)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>g</mi> <mi>&#x03D5;</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msubsup><mo>&#x0222B;</mo> <mn>0</mn> <mi>x</mi></msubsup><munder><mo lspace="thinmathspace" rspace="thinmathspace">&#x02211;</mo> <mrow><mi>n</mi><mo>&#x02265;</mo><mn>0</mn></mrow></munder><mi>&#x03D5;</mi><mo stretchy="false">[</mo><mi>&#x02102;</mi><msup><mi>P</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">]</mo><msup><mi>t</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mspace width="thinmathspace"/><mi>dt</mi><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">&#x02211;</mo> <mrow><mi>n</mi><mo>&#x02265;</mo><mn>0</mn></mrow></munder><mi>&#x03D5;</mi><mo stretchy="false">[</mo><mi>&#x02102;</mi><msup><mi>P</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">]</mo><mfrac><mrow><msup><mi>x</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding='application/x-tex'>
  2499.  g_\phi(x) :=
  2500.  \int_0^x
  2501.  \sum_{n \geq 0}
  2502.  \phi[\mathbb{C}P^{2n}]t^{2n}\, dt
  2503.  =
  2504.  \sum_{n\geq 0} \phi[\mathbb{C}P^{2n}]
  2505.  \frac{x^{2n+1}}{2n+1}
  2506. </annotation></semantics></math></div>
  2507.  
  2508. <p>the <strong>logarithm</strong> of the genus.</p>
  2509.  
  2510. <p><strong>Ochanine defined</strong> an genus to be <strong>elliptic</strong> if this logarithm is of the form</p>
  2511.  
  2512. <div class="numberedEq"><span>(40)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>g</mi> <mi>&#x03D5;</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>&#x0222B;</mo> <mn>0</mn> <mi>x</mi></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><mn>2</mn><mi>&#x003B4;</mi><msup><mi>t</mi> <mn>2</mn></msup><mo>+</mo><mi>&#x003F5;</mi><msup><mi>t</mi> <mn>4</mn></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mspace width="thickmathspace"/><mi>dt</mi></mrow><annotation encoding='application/x-tex'>
  2513.  g_\phi(x) = \int_0^x (1 &#x2d; 2\delta t^2 + \epsilon t^4)^{&#x2d;1/2}
  2514.  \; dt
  2515. </annotation></semantics></math></div>
  2516.  
  2517. <p>for suitable ring elements <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B4;</mi></mrow><annotation encoding='application/x-tex'>\delta</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003F5;</mi></mrow><annotation encoding='application/x-tex'>\epsilon</annotation></semantics></math>, algebraically independent over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x0211A;</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Q}</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x00394;</mi><mo>&#x02260;</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\Delta \neq 0</annotation></semantics></math>.</p>
  2518.  
  2519. <p>There is a <strong>theorem by Landweber, Ochanine and Stong</strong> which says that if a genus <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x03D5;</mi></mrow><annotation encoding='application/x-tex'>\phi</annotation></semantics></math> is elliptic, then its image is</p>
  2520.  
  2521. <div class="numberedEq"><span>(41)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x03D5;</mi><mo stretchy="false">(</mo><msubsup><mi>&#x003A9;</mi> <mo>*</mo> <mi mathvariant="normal">SO</mi></msubsup><mo stretchy="false">)</mo><mo>=</mo><mi>&#x02124;</mi><mo stretchy="false">[</mo><mi>&#x003B4;</mi><mo>,</mo><mn>2</mn><mi>&#x003B3;</mi><mo>,</mo><mn>2</mn><msup><mi>&#x003B3;</mi> <mn>2</mn></msup><mo>,</mo><mi>&#x022EF;</mi><mo>,</mo><mn>2</mn><msup><mi>&#x003B3;</mi> <mrow><msup><mn>2</mn> <mi>n</mi></msup></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>
  2522.  \phi(\Omega_*^\mathrm{SO})
  2523.  =
  2524.  \mathbb{Z}[\delta, 2\gamma, 2\gamma^2,\cdots, 2\gamma^{2^n}]
  2525. </annotation></semantics></math></div>
  2526.  
  2527. <p>with</p>
  2528.  
  2529. <div class="numberedEq"><span>(42)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x003B3;</mi><mo>=</mo><mfrac><mrow><msup><mi>&#x003B4;</mi> <mn>2</mn></msup><mo>&#x02212;</mo><mi>&#x003F5;</mi></mrow><mn>4</mn></mfrac><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2530.  \gamma = \frac{\delta^2 &#x2d; \epsilon}{4}
  2531.  \,.
  2532. </annotation></semantics></math></div>
  2533.  
  2534. <p>Furthermore, the image of spin cobordisms is</p>
  2535.  
  2536. <div class="numberedEq"><span>(43)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>&#x03D5;</mi><mo stretchy="false">(</mo><msubsup><mi>&#x003A9;</mi> <mo>*</mo> <mi mathvariant="normal">Spin</mi></msubsup><mo stretchy="false">)</mo><mo>=</mo><mi>&#x02124;</mi><mo stretchy="false">[</mo><mn>16</mn><mi>&#x003B4;</mi><mo>,</mo><mo stretchy="false">(</mo><mn>8</mn><mi>&#x003B4;</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>,</mo><mi>&#x003F5;</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2537.  \phi(\Omega_*^\mathrm{Spin})
  2538.  =
  2539.  \mathbb{Z}[16 \delta, (8\delta)^2, \epsilon]
  2540.  \,.
  2541. </annotation></semantics></math></div>
  2542.  
  2543. <p>Again, looking at degenerate cases we find famliar <strong>examples</strong>.</p>
  2544.  
  2545. <p>1) In the case that <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003F5;</mi><mo>=</mo><mi>&#x003B4;</mi></mrow><annotation encoding='application/x-tex'>\epsilon = \delta</annotation></semantics></math> we get</p>
  2546.  
  2547. <div class="numberedEq"><span>(44)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>g</mi> <mrow><mi>&#x003F5;</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>&#x003B4;</mi><mo>=</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msubsup><mo>&#x0222B;</mo> <mn>0</mn> <mi>x</mi></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><mn>2</mn><msup><mi>t</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>t</mi> <mn>4</mn></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mspace width="thinmathspace"/><mi>dt</mi></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><msubsup><mo>&#x0222B;</mo> <mn>0</mn> <mi>x</mi></msubsup><mfrac><mn>1</mn><mrow><mn>1</mn><mo>&#x02212;</mo><msup><mi>t</mi> <mn>2</mn></msup></mrow></mfrac><mspace width="thinmathspace"/><mi>dt</mi></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><msup><mi mathvariant="normal">tanh</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"/><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>
  2548. \begin{aligned}
  2549.  g_{\epsilon=1,\delta=1}(x)
  2550.  &amp;=
  2551.  \int_0^x (1&#x2d;2t^2 + t^4)^{&#x2d;1/2}
  2552.  \, dt
  2553.  \\
  2554.  &amp;=
  2555.  \int_0^x \frac{1}{1&#x2d;t^2}\,dt
  2556.  \\
  2557.  &amp;=
  2558.  \mathrm{tanh}^{&#x2d;1}(x)
  2559.  \,.
  2560. \end{aligned}
  2561. </annotation></semantics></math></div>
  2562.  
  2563. <p>This corresponds to the <strong>signature genus</strong> of <strong>L-genus</strong> (which I also mentioned <a href="http://golem.ph.utexas.edu/string/archives/000862.html">last time</a>).</p>
  2564.  
  2565. <p>Here, too, the corresponding Jacobi-curve is singular.</p>
  2566.  
  2567. <p><br/>
  2568. 2) Another example is <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003F5;</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\epsilon = 0</annotation></semantics></math> and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003B4;</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mfrac><mn>1</mn><mn>8</mn></mfrac></mrow><annotation encoding='application/x-tex'>\delta = &#x2d;\frac{1}{8}</annotation></semantics></math>.
  2569. Here one gets</p>
  2570.  
  2571. <div class="numberedEq"><span>(45)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>&#x0222B;</mo> <mn>0</mn> <mi>x</mi></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>&#x02212;</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>t</mi> <mn>2</mn></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">&#x02212;</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mspace width="thinmathspace"/><mi>dt</mi></mrow><annotation encoding='application/x-tex'>
  2572.  g_{&#x2d;\frac{1}{8},0}(x)
  2573.  =
  2574.  \int_0^x (1&#x2d;\frac{1}{4}t^2)^{&#x2d;1/2}\, dt
  2575. </annotation></semantics></math></div>
  2576.  
  2577. <p>and this corresponds to the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover></mrow><annotation encoding='application/x-tex'>\hat A</annotation></semantics></math>-genus.</p>
  2578.  
  2579. <p>As Atiyah and Singer found with their famous index theorem, for <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> compact and spin and <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">dim</mi><mi>M</mi><mo>=</mo><mn>2</mn><mi>n</mi></mrow><annotation encoding='application/x-tex'>\mathrm{dim} M = 2n</annotation></semantics></math> the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover></mrow><annotation encoding='application/x-tex'>\hat A</annotation></semantics></math>-genus</p>
  2580.  
  2581. <div class="numberedEq"><span>(46)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mover><mi>A</mi><mo stretchy="false">&#x5E;</mo></mover><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">ind</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>&#x02208;</mo><mi>&#x02124;</mi></mrow><annotation encoding='application/x-tex'>
  2582.  \hat A(M) = \mathrm{ind}(D) \in \mathbb{Z}
  2583. </annotation></semantics></math></div>
  2584.  
  2585. <p>is the index of a Dirac operator on <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>, taking values in the integers.</p>
  2586.  
  2587. <p>We want some lifting of this statement to the loop space of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>.</p>
  2588.  
  2589. <p>It turns out that this is possible if <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> is string (<a href="http://golem.ph.utexas.edu/string/archives/000571.html"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>), which means, according to a <strong>theorem by Laughlin</strong>, that it sfirst two Stieffel-Whitney classes and one half of the first Pontryagin class vanishes.</p>
  2590.  
  2591. <p>In this case there is something like a Dirac operator on loop space <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>LM</mi></mrow><annotation encoding='application/x-tex'>LM</annotation></semantics></math> and a <strong>theorem due to Witten and Zagier</strong> says that its index is a genus which is the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>q</mi></mrow><annotation encoding='application/x-tex'>q</annotation></semantics></math>-series of a modular form - the <strong>Witten genus</strong> (partition function of the heterotic string).</p>
  2592.  
  2593. <p>What we are after is the homology theory which corresponds to this genus.</p>
  2594.  
  2595. <p><br/><strong>tmf - topological modular forms</strong></p>
  2596.  
  2597. <p>According to Birgit Richer, in her experience it takes a group of experts a full week to discuss the construction of tmf. At that point 4 minutes time were left. </p>
  2598.  
  2599. <p>Apart from that, what is important about tmf is that, as Jacob Lurie describes on pp. 9-10 of his &#8220;survey&#8221; (<a href="http://www.math.harvard.edu/~lurie/papers/survey.pdf"><math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo>&#x02192;</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></a>), tmf is something like the universal elliptic cohomology.</p>
  2600.  
  2601. <p>We can get a glimpse of what this means by realizing that the way elliptic cohomology was defined above depended on a choice of coordinates (in the Weierstrass form) for an elliptic curve. In a vague sense tmf is the coordinate-free version of alliptic cohomology. Or something like that.</p>
  2602.  
  2603. <p>The point is that the Weierstrass form</p>
  2604.  
  2605. <div class="numberedEq"><span>(47)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msup><mi>y</mi> <mn>2</mn></msup><mo>+</mo><msub><mi>a</mi> <mn>1</mn></msub><mi>xy</mi><mo>+</mo><msub><mi>a</mi> <mn>3</mn></msub><mi>y</mi><mo>=</mo><msup><mi>x</mi> <mn>3</mn></msup><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msub><mi>a</mi> <mn>4</mn></msub><mi>x</mi><mo>+</mo><msub><mi>a</mi> <mn>6</mn></msub></mrow><annotation encoding='application/x-tex'>
  2606.  y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6
  2607. </annotation></semantics></math></div>
  2608.  
  2609. <p>is invariant under the coordinate transformations of the form</p>
  2610.  
  2611. <div class="numberedEq"><span>(48)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>x</mi></mtd> <mtd><mo>&#x021A6;</mo><msup><mi>&#x003BB;</mi> <mn>2</mn></msup><mo>+</mo><mi>r</mi></mtd></mtr> <mtr><mtd><mi>y</mi></mtd> <mtd><mo>&#x021A6;</mo><msup><mi>&#x003BB;</mi> <mn>3</mn></msup><mi>y</mi><mo>+</mo><msup><mi>&#x003BB;</mi> <mn>2</mn></msup><mi>sx</mi><mo>+</mo><mi>t</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"/><mo>,</mo></mrow><annotation encoding='application/x-tex'>
  2612. \begin{aligned}
  2613.  x &amp;\mapsto \lambda^2 + r
  2614.  \\
  2615.  y &amp;\mapsto \lambda^3 y + \lambda^2 sx + t
  2616. \end{aligned}
  2617. \,,
  2618. </annotation></semantics></math></div>
  2619.  
  2620. <p>where <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>&#x003BB;</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math> is a &#8220;unit&#8221; (invertible). Call the group of these transformations</p>
  2621.  
  2622. <div class="numberedEq"><span>(49)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>G</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2623.  G
  2624.  \,.
  2625. </annotation></semantics></math></div>
  2626.  
  2627. <p>Let </p>
  2628.  
  2629. <div class="numberedEq"><span>(50)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>A</mi><mo>=</mo><mi>&#x02124;</mi><mo stretchy="false">[</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>&#x022EF;</mi><mo>,</mo><msub><mi>a</mi> <mn>6</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">[</mo><msup><mi>u</mi> <mrow><mo>&#x000B1;</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>
  2630.  A = \mathbb{Z}[a_1,\cdots,a_6][u^{\pm 1}]
  2631. </annotation></semantics></math></div>
  2632.  
  2633. <p>and form the cohomology theory</p>
  2634.  
  2635. <div class="numberedEq"><span>(51)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>X</mi><mo>&#x021A6;</mo><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>A</mi></msub><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>&#x02297;</mo> <mrow><msub><mi>MU</mi> <mo>*</mo></msub></mrow></msub><mi>A</mi><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2636.  X \mapsto (E_A)_*(X) = MU_*(X) \otimes_{MU_*}A
  2637.  \,.
  2638. </annotation></semantics></math></div>
  2639.  
  2640. <p>Then, according to a <strong>theorem by Hopkins, Miller, Goerss</strong> which has been given a more conceptual proof by Jacob Lurie, the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-invariant part <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>A</mi></msub><msup><mo stretchy="false">)</mo> <mi>G</mi></msup></mrow><annotation encoding='application/x-tex'>(E_A)^G</annotation></semantics></math> of <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><msub><mi>E</mi> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>E_A</annotation></semantics></math> is a model for tmf.</p>
  2641.  
  2642. <p>(<em>I am only 30 per cent convinced that this statement makes good sense as stated. Need to check that.</em>)</p>
  2643.  
  2644. <p>There is a map from the tmf cohomology of a point to modular forms</p>
  2645.  
  2646. <div class="numberedEq"><span>(52)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi mathvariant="normal">tmf</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mtext>pt</mtext><mo stretchy="false">)</mo><mo>&#x02192;</mo><msub><mi>M</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>
  2647.  \mathrm{tmf}_*(\text{pt}) \to M_*
  2648. </annotation></semantics></math></div>
  2649.  
  2650. <p>whose kernel and cokernel are annihilated by 24. </p>
  2651.  
  2652. <p>Moreover, it is known that</p>
  2653.  
  2654. <div class="numberedEq"><span>(53)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><msub><mi mathvariant="normal">tmf</mi> <mo>*</mo></msub><mo stretchy="false">[</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mtext>pt</mtext><mo stretchy="false">]</mo><mo>=</mo><mi>&#x02124;</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mn>6</mn><mo>,</mo><msub><mi>c</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>6</mn></msub><mo stretchy="false">]</mo><mo>=</mo><msub><mi>M</mi> <mo>*</mo></msub><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mn>6</mn><mo stretchy="false">]</mo><mspace width="thinmathspace"/><mo>.</mo></mrow><annotation encoding='application/x-tex'>
  2655.  \mathrm{tmf}_*[\frac{1}{6}][\text{pt}]
  2656.  =
  2657.  \mathbb{Z}[1/6,c_4,c_6] = M_*[1/6]
  2658.  \,.
  2659. </annotation></semantics></math></div>
  2660.  
  2661. <p>Finally, the Witten genus is the composition map</p>
  2662.  
  2663. <div class="numberedEq"><span>(54)</span><math xmlns='http://www.w3.org/1998/Math/MathML' display='block'><semantics><mrow><mi>M</mi><mi mathvariant="normal">String</mi><mo>&#x02192;</mo><mi mathvariant="normal">tmf</mi><mo>&#x02192;</mo><mi>KU</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding='application/x-tex'>
  2664.  M\mathrm{String} \to \mathrm{tmf} \to KU[[q]]
  2665. </annotation></semantics></math></div>
  2666.  
  2667. <p>from string cobordisms over <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi mathvariant="normal">tmf</mi></mrow><annotation encoding='application/x-tex'>\mathrm{tmf}</annotation></semantics></math> to power series in <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mi>q</mi></mrow><annotation encoding='application/x-tex'>q</annotation></semantics></math> with coefficients in K-theory, restricted to the <math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-dimensional string manifold.</p>
  2668.  
  2669. <p><br/><em>Oh dear, you can tell that this last part was transmitted and received in mere 4 minutes.</em></p>
  2670.      </div>
  2671.    </content>
  2672.  </entry>
  2673.  
  2674. </feed>

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