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

2011년 9월 22일 (목) 11:27 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
\(<K>=\int {\operatorname{Tr}(\int_{A} A)}e^{2\pi i k \opratorname{CS}(A)}DA=(q^{1/2}+q^{-1/2})V(K,q^{-1})\)

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

Notes on flat connections loop groups

history

http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

books

encyclopedia

http://ko.wikipedia.org/wiki/
http://en.wikipedia.org/wiki/Chern–Simons_theory
http://en.wikipedia.org/wiki/
http://en.wikipedia.org/wiki/
Princeton companion to mathematics(Companion_to_Mathematics.pdf)

question and answers(Math Overflow)

blogs

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expositions

An Introduction to Chern-Simons Theory
Lie groups and Chern-Simons Theory Benjamin Himpel
Labastida, J. M. F. 1999. “Chern-Simons Gauge Theory: Ten Years After”. hep-th/9905057 (5월 8). doi:doi:10.1063/1.59663. http://arxiv.org/abs/hep-th/9905057.

@@ 4번째 줄: / 4번째 줄: @@
 * CS is an invariant for 3-manifolds
 * [[Kashaev's volume conjecture|Kashaev Volume conjecture]]
-*  action<br> Let <math>A</math> be a SU(2)-<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_{A} A)}e^{2\pi i k \opratorname{CS}(A)}DA=(q^{1/2}+q^{-1/2})V(K,q^{-1})</math><br>