"Chern-Simons gauge theory and Witten's invariant"의 두 판 사이의 차이
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− | ==introduction | + | ==introduction== |
* [[topological quantum field theory(TQFT)|3D TQFT( Chern-Simons theory)]] | * [[topological quantum field theory(TQFT)|3D TQFT( Chern-Simons theory)]] | ||
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− | == | + | == == |
* [[curvature and parallel transport]] | * [[curvature and parallel transport]] | ||
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− | ==Morse theory approach | + | ==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 | * 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 | ||
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− | ==Chern-Simons invariant | + | ==Chern-Simons invariant== |
* [[Chern-Simons invariant]] | * [[Chern-Simons invariant]] | ||
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− | ==memo | + | ==memo== |
* [http://www.math.ethz.ch/%7Esalamon/PREPRINTS/loopgroup.pdf Notes on flat connections loop groups] | * [http://www.math.ethz.ch/%7Esalamon/PREPRINTS/loopgroup.pdf Notes on flat connections loop groups] | ||
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− | ==history | + | ==history== |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
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− | ==related items | + | ==related items== |
* [[WZW (Wess-Zumino-Witten) model and its central charge|WZW model]] | * [[WZW (Wess-Zumino-Witten) model and its central charge|WZW model]] | ||
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− | ==books | + | ==books== |
* [[4909919|찾아볼 수학책]] | * [[4909919|찾아볼 수학책]] | ||
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− | ==encyclopedia | + | ==encyclopedia== |
* [http://en.wikipedia.org/wiki/Chern%E2%80%93Simons_theory http://en.wikipedia.org/wiki/Chern–Simons_theory] | * [http://en.wikipedia.org/wiki/Chern%E2%80%93Simons_theory http://en.wikipedia.org/wiki/Chern–Simons_theory] | ||
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− | ==question and answers(Math Overflow) | + | ==question and answers(Math Overflow)== |
* http://mathoverflow.net/questions/31905/some-basic-questions-about-chern-simons-theory | * http://mathoverflow.net/questions/31905/some-basic-questions-about-chern-simons-theory | ||
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− | ==blogs | + | ==blogs== |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
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− | ==expositions | + | ==expositions== |
* [http://www.math.sunysb.edu/%7Ebasu/notes/GSS2.pdf An Introduction to Chern-Simons Theory] | * [http://www.math.sunysb.edu/%7Ebasu/notes/GSS2.pdf An Introduction to Chern-Simons Theory] | ||
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− | ==articles | + | ==articles== |
* http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf | * http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf | ||
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− | ==experts on the field | + | ==experts on the field== |
* http://arxiv.org/ | * http://arxiv.org/ | ||
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− | ==links | + | ==links== |
* [http://staff.science.uva.nl/%7Eriveen/volume_conjecture.htm Volume conjecture links and notes] | * [http://staff.science.uva.nl/%7Eriveen/volume_conjecture.htm Volume conjecture links and notes] |
2012년 10월 28일 (일) 15:24 판
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
\(\langle K\rangle=\int {\operatorname{Tr}(\int_{K} A)}e^{2\pi i k \operatorname{CS}(A)}DA=(q^{1/2}+q^{-1/2})V(K,q^{-1})\)
\(e^{2\pi i k \operatorname{CS}(A)}DA\): 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
\(F=A\wedge dA+A\wedge A\)
\(\operatorname{det}(I+\frac{iF}{2\pi})= c_0+c_1+c_2\)
\(c_3=A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A\)
\(c_2=\frac{1}{8\pi^2} \operatorname{tr} F\wedge F =dc_3\)
\(\int_M c_3\)
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
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
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
- 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