Chern-Simons gauge theory and Witten's invariant

introduction

• prototypical example of Topological quantum field theory(TQFT)
• Witten introduced classical Chern-Simons theory to topology
• Witten gave a prescription for obtaining exact expressions for
  • partition function : this becomes new topological invariant of the 3-manifold
  • expectation values of Wilson loops : it leads to Jones polynomial
• 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

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

• \(\operatorname{det}(I+\frac{iF}{2\pi})= c_0+c_1+c_2\)
• \(c_3=A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A\)
• \(c_2=\frac{1}{8\pi^2} \operatorname{tr} F\wedge F =dc_3\)
• \(\int_M c_3\)
• curvature and parallel transport
• Chern class
• vector valued differential forms

WRT invariant

• Chern-Simons partition function?
• Feynman diagrams and path integral
• The path integral defined by Witten

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

• Quantum modular forms

Jones Polynomial

• path integral gives Jones polynomials

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

• Chern-Simons invariant

memo

• Notes on flat connections loop groups

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• Complex Chern-Simons theory
• closely related to the Kashaev Volume conjecture
• WZW (Wess-Zumino-Witten) model and its central charge
• quantum dilogarithm
• characteristic class

 

encyclopedia

• http://en.wikipedia.org/wiki/Chern–Simons_theory

question and answers(Math Overflow)

• http://mathoverflow.net/questions/31905/some-basic-questions-about-chern-simons-theory
• http://mathoverflow.net/questions/36178/what-is-the-trace-in-the-chern-simons-action

 

expositions

• An Introduction to Chern-Simons Theory
• Lie groups and Chern-Simons Theory Benjamin Himpel
• Chern-Simons Gauge Theory: 20 years after
  • conference
• 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.
• Freed, Daniel S. 1992. “Classical Chern-Simons Theory, Part 1.” arXiv:hep-th/9206021 (June 4). http://arxiv.org/abs/hep-th/9206021.

 

articles

• Kohno, Toshitake, and Toshie Takata. "Level-Rank Duality of Witten's 3-Manifold Invariants." Progress in algebraic combinatorics 24 (1996): 243. http://tqft.net/other-papers/knot-theory/Level-rank%20duality%20-%20Kohno,%20Takata.pdf
• http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf
• Witten, Edward. 1992. “Chern-Simons Gauge Theory As A String Theory”. hep-th/9207094 (7월 28). http://arxiv.org/abs/hep-th/9207094.
• Kirby, Robion, and Paul Melvin. 1991. “The 3-manifold Invariants of Witten and Reshetikhin-Turaev for Sl(2, C).” Inventiones Mathematicae 105 (1) (December 1): 473–545. doi:10.1007/BF01232277.
• Reshetikhin, N.Yu., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math.103, 547–597 (1991)
• Kevin Walkter, On Witten’s 3-manifold Invariants http://tqft.net/other-papers/KevinWalkerTQFTNotes.pdf
• Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399

books

• Conformal Field Theory and Topology by Kohno