Chern-Simons gauge theory and Witten's invariant

introduction

• Topological quantum field theory(TQFT)
• CS is an invariant for 3-manifolds
  • Chern-Simons invariant
• action
  Let \(A\) be a SU(2)-connection on the trivicla C^2 bundle over S^3

\[S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A)=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{1}{3}A\wedge [A,A])\]

Chern-Simons partition function

• path integral defined by Witten

$$ Z_k(M)=\int e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA\ $$ where \(e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA\): formal probability measure on the space of all connections, coming from quantum field theory

asymptotic expansion

$$ Z_k(M)\approx \frac{1}{2}e^{-3\pi i/4}\sum_{\alpha}\sqrt{T_{\alpha}(M)} e^{-2\pi i I_{\alpha}/4} e^{2\pi (k+2) \operatorname{CS}(A)} $$ where the sum is over flat connections $\alpha$

Jones Polynomial

• path integral gives Jones polynomials

\[\langle K\rangle=\int {\operatorname{Tr}(\int_{K} A)}e^{2\pi i k \operatorname{CS}(A)}DA=(q^{1/2}+q^{-1/2})V(K,q^{-1})\] where the Chern-Simons action \({\operatorname{Tr}(\int_{K} A)}\) measures the twisting of the connection along the knot

 

 

 

• curvature and parallel transport
• Chern class
• vector valued differential forms

 

 

M : threefold

\(P\to M\) : principal G-bundle

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

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

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

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

\(\int_M c_3\)

 

 

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 invariant

• Chern-Simons invariant

 

 

memo

• Notes on flat connections loop groups

 

 

history

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

 

 

related items

• closely related to the Kashaev Volume conjecture
• WZW model
• quantum dilogarithm
• characteristic class

 

encyclopedia

• http://en.wikipedia.org/wiki/Chern–Simons_theory

question and answers(Math Overflow)

• http://mathoverflow.net/questions/31905/some-basic-questions-about-chern-simons-theory
• http://mathoverflow.net/questions/36178/what-is-the-trace-in-the-chern-simons-action

 

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.
• Curtis T. McMullen, The evolution of geometric structures on 3-manifolds Bull. Amer. Math. Soc. 48 (2011), 259-274.

 

articles

• http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf
• Rationality problems for K-theory and Chern-Simons invariants of hyperbolic 3-manifolds
  
  • Walter Neumann, 1995
• Witten, Edward. 1992. “Chern-Simons Gauge Theory As A String Theory”. hep-th/9207094 (7월 28). http://arxiv.org/abs/hep-th/9207094.
  

links

• Volume conjecture links and notes