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

수학노트
둘러보기로 가기 검색하러 가기
6번째 줄: 6번째 줄:
 
*  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>
 
*  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>
 
*  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>
 +
 +
 
 +
 +
 
 +
 +
<h5> </h5>
 +
 +
* [[curvature and parallel transport]]
 +
 +
<math>F=A\wedge dA+A\wedge A</math>
 +
 +
<math>\operatorname{det}(I+\frac{iF}{2\pi})= c_0+c_1+c_2</math>
 +
 +
<math>c_3=A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A</math>
 +
 +
<math>c_2=\frac{1}{8\pi^2} \operatorname{tr} F\wedge F =dc_3</math>
 +
 +
 
 +
 +
 
  
 
 
 
 
56번째 줄: 76번째 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]]
−
* [[#]]
+
* [[characteristic class]]
  
 
 
 
 

2011년 10월 11일 (화) 05:22 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

 

 

 

\(F=A\wedge dA+A\wedge A\)

\(\operatorname{det}(I+\frac{iF}{2\pi})= c_0+c_1+c_2\)

\(c_3=A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A\)

\(c_2=\frac{1}{8\pi^2} \operatorname{tr} F\wedge F =dc_3\)

 

 

 

 

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

 

 

 

 

memo

 

 

history

 

 

related items

 

 

books

 

 

encyclopedia

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

expositions

 

 

 

articles

 

 

experts on the field

 

 

links
원본 주소 "https://wiki.mathnt.net/index.php?title=Chern-Simons_gauge_theory_and_Witten%27s_invariant&oldid=37529"