Chern-Simons gauge theory and Witten's invariant

introduction

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

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

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

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