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

수학노트
둘러보기로 가기 검색하러 가기
imported>Pythagoras0
imported>Pythagoras0
2번째 줄: 2번째 줄:
 
* prototypical example of [[Topological quantum field theory(TQFT)]]
 
* prototypical example of [[Topological quantum field theory(TQFT)]]
 
* Witten introduced classical Chern-Simons theory to topology
 
* Witten introduced classical Chern-Simons theory to topology
−
* M : 3-manifold
+
 
−
* Let <math>A</math> 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
 
−
:<math>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])</math>
 
  
  
13번째 줄: 9번째 줄:
 
* M : compact oriented 3-manifold
 
* M : compact oriented 3-manifold
 
* $G=SU(2)$
 
* $G=SU(2)$
−
* <math>P\to M</math> : principal G-bundle
+
* <math>P\to M</math> : principal G-bundle, trivial $SU(2)$ bundle over $M$ since $SU(2)$ is simply connected
 
* $\mathcal{A}_M$ : the space of connections on $P$
 
* $\mathcal{A}_M$ : the space of connections on $P$
 
** forms an affine space
 
** forms an affine space
 
** can be identified with $\Omega^{1}(M,\mathfrak{g})$, the space of 1-forms on $M$ with values in $\mathfrak{g}$
 
** 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
 
* $A\in \mathcal{A}_M$ : connection
−
* <math>F=A\wedge dA+A\wedge A</math> : curvature
+
* <math>F=A\wedge dA+A\wedge A\in \Omega^{2}(M,\mathfrak{g})</math> : the curvature of connection $A$
 
* $\mathcal{G}=\operatorname{Map}(M,G)$ : 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}
 
$$
 
$$
 +
* the Chern-Simons action functional is given by
 +
:<math>\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])</math>
 
* <math>\operatorname{det}(I+\frac{iF}{2\pi})= c_0+c_1+c_2</math>
 
* <math>\operatorname{det}(I+\frac{iF}{2\pi})= c_0+c_1+c_2</math>
 
* <math>c_3=A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A</math>
 
* <math>c_3=A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A</math>
32번째 줄: 30번째 줄:
  
  
−
==Chern-Simons partition function==
+
==WRT invariant==
 +
* Chern-Simons partition function?
 
* [[Feynman diagrams and path integral]]
 
* [[Feynman diagrams and path integral]]
 
* The path integral defined by Witten
 
* The path integral defined by Witten
 
$$
 
$$
−
Z_k(M)=\int e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA\
+
Z_k(M)=\int_{\mathcal{A}_M/\mathcal{G}} e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA\
 
$$
 
$$
 
where <math>e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA</math>: formal probability measure on the space of all connections, coming from quantum field theory
 
where <math>e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA</math>: 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===
 
===asymptotic expansion===
 
* As $k\to \infty$,
 
* As $k\to \infty$,
113번째 줄: 125번째 줄:
  
  
−
==links==
 
−
* [http://staff.science.uva.nl/%7Eriveen/volume_conjecture.htm Volume conjecture links and notes]
 
  
 
[[분류:math and physics]]
 
[[분류:math and physics]]
 
[[분류:TQFT]]
 
[[분류:TQFT]]

2013년 5월 30일 (목) 13:14 판

introduction



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 $\alpha$


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


memo


history

 

 

related items

 

encyclopedia


question and answers(Math Overflow)

 

expositions

 

articles

원본 주소 "https://wiki.mathnt.net/index.php?title=Chern-Simons_gauge_theory_and_Witten%27s_invariant&oldid=37560"