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

2012년 8월 20일 (월) 04:40 판

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
\(<K>=\int {\operatorname{Tr}(\int_{K} A)}e^{2\pi i k \opratorname{CS}(A)}DA=(q^{1/2}+q^{-1/2})V(K,q^{-1})\)
\(e^{2\pi i k \opratorname{CS}(A)}DA\): formal probability measure on the space of all connections, coming grom quantum field theory and 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

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

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

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

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

\(\int_M c_3\)

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

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.

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 <math>P\to M</math> : principal G-bundle
 <math>F=A\wedge dA+A\wedge A</math>
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 <h5>Chern-Simons invariant</h5>
-* [[search?q=Chern-Simons%20invariant&parent id=5211693|Chern-Simons invariant]]
+* [[Chern-Simons invariant]]