"Chern-Simons gauge theory and Witten's invariant"의 두 판 사이의 차이

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5번째 줄: 5번째 줄:
 
* [[Kashaev's volume conjecture|Kashaev Volume conjecture]]
 
* [[Kashaev's volume conjecture|Kashaev Volume conjecture]]
 
*  action<br> Let <math>A</math> be a SU(2)-connection on the trivicla C^2 bundle over S^3<br><math>S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A)=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{1}{3}A\wedge [A,A])</math><br>
 
*  action<br> Let <math>A</math> be a SU(2)-connection on the trivicla C^2 bundle over S^3<br><math>S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A)=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{1}{3}A\wedge [A,A])</math><br>
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*  path integral gives [[Jones polynomials]]<br><math><K>=\int {\operatorname{Tr}(\int_{A} A)}e^{2\pi i k \opratorname{CS}(A)}DA=(q^{1/2}+q^{-1/2})V(K,q^{-1})</math><br>
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*  path integral gives [[Jones polynomials]]<br><math><K>=\int {\operatorname{Tr}(\int_{K} A)}e^{2\pi i k \opratorname{CS}(A)}DA=(q^{1/2}+q^{-1/2})V(K,q^{-1})</math><br><math>e^{2\pi i k \opratorname{CS}(A)}DA</math>: formal probability measure on the space of all connections, coming grom quantum field theory and the Chern-Simons action<br><math>{\operatorname{Tr}(\int_{K} A)}</math> : measures the twisting of the connection along the knot<br>
  
 
 
 
 
56번째 줄: 56번째 줄:
 
* [[WZW (Wess-Zumino-Witten) model and its central charge|WZW model]]
 
* [[WZW (Wess-Zumino-Witten) model and its central charge|WZW model]]
 
* [[quantum dilogarithm]]
 
* [[quantum dilogarithm]]
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* [[#]]
  
 
 
 
 
111번째 줄: 112번째 줄:
 
* [http://www.math.sunysb.edu/%7Ebasu/notes/GSS2.pdf An Introduction to Chern-Simons Theory]
 
* [http://www.math.sunysb.edu/%7Ebasu/notes/GSS2.pdf An Introduction to Chern-Simons Theory]
 
* [http://www.math.uni-bonn.de/people/himpel/himpel_cstheory.pdf Lie groups and Chern-Simons Theory] Benjamin Himpel
 
* [http://www.math.uni-bonn.de/people/himpel/himpel_cstheory.pdf Lie groups and Chern-Simons Theory] Benjamin Himpel
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* Labastida, J. M. F. 1999. “Chern-Simons Gauge Theory: Ten Years After”. <em>hep-th/9905057</em> (5월 8). doi:doi:10.1063/1.59663. http://arxiv.org/abs/hep-th/9905057.<br>
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* Labastida, J. M. F. 1999. “Chern-Simons Gauge Theory: Ten Years After”. <em>hep-th/9905057</em> (5월 8). doi:doi:10.1063/1.59663. http://arxiv.org/abs/hep-th/9905057.
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* Curtis T. McMullen, [http://dx.doi.org/10.1090/S0273-0979-2011-01329-5%20 The evolution of geometric structures on 3-manifolds] Bull. Amer. Math. Soc. 48 (2011), 259-274.
  
 
 
 
 
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<h5>articles</h5>
 
<h5>articles</h5>
  
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* [http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf ]http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf
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* http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf
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* [http://www.math.columbia.edu/%7Eneumann/preprints/cs2.pdf Rationality problems for K-theory and Chern-Simons invariants of hyperbolic 3-manifolds]<br>
 
* [http://www.math.columbia.edu/%7Eneumann/preprints/cs2.pdf Rationality problems for K-theory and Chern-Simons invariants of hyperbolic 3-manifolds]<br>

2011년 10월 11일 (화) 05:17 판

introduction
  • 3D TQFT( Chern-Simons theory)
  • CS is an invariant for 3-manifolds
  • Kashaev Volume conjecture
  • action
    Let \(A\) be a SU(2)-connection on the trivicla C^2 bundle over S^3
    \(S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A)=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{1}{3}A\wedge [A,A])\)
  • path integral gives Jones polynomials
    \(<K>=\int {\operatorname{Tr}(\int_{K} A)}e^{2\pi i k \opratorname{CS}(A)}DA=(q^{1/2}+q^{-1/2})V(K,q^{-1})\)
    \(e^{2\pi i k \opratorname{CS}(A)}DA\): formal probability measure on the space of all connections, coming grom quantum field theory and the Chern-Simons action
    \({\operatorname{Tr}(\int_{K} A)}\) : measures the twisting of the connection along the knot

 

 

Morse theory approach
  • Taubes, Floer interpret the Chern-Simons function as a Morse function on the space of all gauge fields modulo the action of the group of gauge transformations
  • analogous to Euler characteristic of a manifold can be computed as the signed count of Morse indices

 

 

 

Chern-Simons and arithmetic

The Chern-Simons invariants of a closed 3-manifold are secondary characteristic numbers that are given in terms of a finite set of phases in the unit circle. Hyperbolic 3-dimensional geometry links these phases with arithmetic, and identifies them with values of the Rogers dilogarithm at algebraic numbers. The quantization of these invariants are the famous Witten-Reshetikhin-Turaev invariants of 3-manifolds, constructed by the Jones polynomial. In the talk we will review conjectures regarding the asymptotics of the quantum 3-manifold invariants, and their relation to Chern-Simons theory and arithmetic. We will review progress on those conjectures, theoretical, and experimental.

Garoufalidis, Stavros. 2007. “Chern-Simons theory, analytic continuation and arithmetic”. 0711.1716 (11월 12). http://arxiv.org/abs/0711.1716

 

 

 

 

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