Chern-Simons gauge theory and Witten's invariant

introduction

• 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
  

 

 

 

• curvature and parallel transport
• Chern class
• vector valued differential forms

 

 

M : threefold

\(P\to M\) : principal G-bundle

 

 

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

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

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

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

\(\int_M c_3\)

 

 

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 and arithmetic

• The Chern-Simons invariants of a closed 3-manifold are secondary characteristic numbers that are given in terms of a finite set of phases in the unit circle.
• Hyperbolic 3-dimensional geometry links these phases with arithmetic, and identifies them with values of the Rogers dilogarithm at algebraic numbers.
• The quantization of these invariants are the famous Witten-Reshetikhin-Turaev invariants of 3-manifolds, constructed by the Jones polynomial.
• review conjectures regarding the asymptotics of the quantum 3-manifold invariants, and their relation to Chern-Simons theory and arithmetic.
• We will review progress on those conjectures, theoretical, and experimental.
• Garoufalidis, Stavros. 2007. “Chern-Simons theory, analytic continuation and arithmetic”. 0711.1716 (11월 12). http://arxiv.org/abs/0711.1716

 

 

memo

• Notes on flat connections loop groups

 

 

history

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

 

 

related items

• WZW model
• quantum dilogarithm
• characteristic class

 

 

books

• 찾아볼 수학책
• http://gigapedia.info/1/chern+simons
• http://gigapedia.info/1/wzw
• http://gigapedia.info/1/Wess+zumino
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

encyclopedia

• http://en.wikipedia.org/wiki/Chern–Simons_theory
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

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

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

 

expositions

• 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.

 

 

 

articles

• http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf

• Rationality problems for K-theory and Chern-Simons invariants of hyperbolic 3-manifolds
  
  • Walter Neumann, 1995
• Witten, Edward. 1992. “Chern-Simons Gauge Theory As A String Theory”. hep-th/9207094 (7월 28). http://arxiv.org/abs/hep-th/9207094.
  
• 논문정리
• http://www.ams.org/mathscinet/search/publications.html?pg4=ALLF&s4=
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• http://dx.doi.org/

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Volume conjecture links and notes