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

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* $A\in \mathcal{A}_M$ : connection
 
* $A\in \mathcal{A}_M$ : connection
 
* <math>F=A\wedge dA+A\wedge A</math> : curvature
 
* <math>F=A\wedge dA+A\wedge A</math> : curvature
−
* $\mathcal{G}=\operatorname{Map}(M,G)$ : the gauge group acting on \mathcal{A}_M$ by
+
* $\mathcal{G}=\operatorname{Map}(M,G)$ : the gauge group acting on $\mathcal{A}_M$ by
 
$$
 
$$
 
g^{*}A=g^{-1}Ag+g^{-1}dg, g\in \mathcal{G}
 
g^{*}A=g^{-1}Ag+g^{-1}dg, g\in \mathcal{G}

2013년 2월 11일 (월) 09:48 판

introduction

  • prototypical example of Topological quantum field theory(TQFT)
  • Witten introduced classical Chern-Simons theory to topology
  • M : 3-manifold
  • Let \(A\) be a SU(2)-connection on the trivial C^2 bundle over S^3
  • $\operatorname{CS}(A)$ denotes the Chern-Simons functional
  • the Chern-Simons action is given by

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


setting

  • M : compact oriented 3-manifold
  • $G=SU(2)$
  • \(P\to M\) : principal G-bundle
  • $\mathcal{A}_M$ : 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
  • \(F=A\wedge dA+A\wedge A\) : curvature
  • $\mathcal{G}=\operatorname{Map}(M,G)$ : the gauge group acting on $\mathcal{A}_M$ by

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

Chern-Simons partition function

$$ Z_k(M)=\int e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA\ $$ 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

  • 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 $\alpha$


Jones Polynomial

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

 

 

memo


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