Chern-Simons gauge theory and Witten's invariant

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imported>Pythagoras0님의 2013년 7월 27일 (토) 20:03 판
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introduction

  • prototypical example of Topological quantum field theory(TQFT)
  • Witten introduced classical Chern-Simons theory to topology
  • Witten's invariant : an invariant of 3-manifold originally defined as the partition function of the Chern-Simons functional on the space of connections via path integral formalism


setting

  • M : compact oriented 3-manifold
  • $G=SU(2)$
  • \(P\to M\) : principal G-bundle, trivial $SU(2)$ bundle over $M$ since $SU(2)$ is simply connected
  • $\mathcal{A}_M$ : the space of connections on $P$
    • forms an affine space
    • can be identified with $\Omega^{1}(M,\mathfrak{g})$, the space of 1-forms on $M$ with values in $\mathfrak{g}$
  • $A\in \mathcal{A}_M$ : connection
  • \(F=A\wedge dA+A\wedge A\in \Omega^{2}(M,\mathfrak{g})\) : the curvature of connection $A$
  • $\mathcal{G}=\operatorname{Map}(M,G)$ : the gauge group acting on $\mathcal{A}_M$ by

$$ g^{*}A=g^{-1}Ag+g^{-1}dg, g\in \mathcal{G} $$

  • the Chern-Simons action functional is given by

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


WRT invariant

$$ Z_k(M)=\int_{\mathcal{A}_M/\mathcal{G}} e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA\ $$ 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

Dehn surgery formula

  • first established by Turaev-Reshetikhin
  • M : cpt oriented 3-manifold without boundary
  • M obtained as Dehn surgery on a framed link L with m components $L_j\, , 1\leq j \leq m$ in $S^3$. Then

$$ Z_k(M)=S_{00}C^{\sigma(L)}\sum_{\lambda}S_{0\lambda_1}\cdots S_{0\lambda_m}J(L;\lambda_1,\cdots,\lambda_m) $$ is a topological invariant of $M$ and does not depend on the choice of $L$ where them sum is for any coloring $\lambda :\{1,\cdots,m\} \to P_{+}(k)$

  • $Z_k(S^3)=S_{00}$
  • $Z_k(S^1\times S^2)=S_{00}\sum_{\mu} S_{0\mu}\frac{S_{0\mu}}{S_{00}}=1$


asymptotic expansion

  • As $k\to \infty$,

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


examples


Jones Polynomial

\[\langle K\rangle=\int {\operatorname{Tr}\left(\int_{K} A\right)}e^{2\pi i k \operatorname{CS}(A)}DA=(q^{1/2}+q^{-1/2})V(K,q^{-1})\] where \({\operatorname{Tr}(\int_{K} A)}\) 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 invariant


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