"Chern-Simons gauge theory and Witten's invariant"의 두 판 사이의 차이
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imported>Pythagoras0 |
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==introduction== | ==introduction== | ||
− | * [[ | + | * [[Topological quantum field theory(TQFT)]] |
* CS is an invariant for 3-manifolds | * CS is an invariant for 3-manifolds | ||
− | * [[ | + | ** [[Chern-Simons invariant]] |
− | * | + | * action<br> Let <math>A</math> be a SU(2)-connection on the trivicla C^2 bundle over S^3 |
− | * | + | :<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><br> |
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+ | ==Chern-Simons partition function== | ||
+ | * path integral defined by Witten | ||
+ | $$ | ||
+ | Z_k(M)=\int e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA\ | ||
+ | $$ | ||
+ | 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 | ||
+ | ===asymptotic expansion=== | ||
+ | $$ | ||
+ | Z_k(M)\approx \frac{1}{2}e^{-3\pi i/4}\sum_{\alpha}\sqrt{T_{\alpha}(M)} e^{-2\pi i I_{\alpha}/4} e^{2\pi (k+2) \operatorname{CS}(A)} | ||
+ | $$ | ||
+ | where the sum is over flat connections $\alpha$ | ||
+ | |||
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+ | ==Jones Polynomial== | ||
+ | * path integral gives [[Jones polynomials]] | ||
+ | :<math>\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})</math> | ||
+ | where the Chern-Simons action <math>{\operatorname{Tr}(\int_{K} A)}</math> measures the twisting of the connection along the knot<br> | ||
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==related items== | ==related items== | ||
− | + | * closely related to the [[Kashaev's volume conjecture|Kashaev Volume conjecture]] | |
* [[WZW (Wess-Zumino-Witten) model and its central charge|WZW model]] | * [[WZW (Wess-Zumino-Witten) model and its central charge|WZW model]] | ||
* [[quantum dilogarithm]] | * [[quantum dilogarithm]] | ||
* [[characteristic class]] | * [[characteristic class]] | ||
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==encyclopedia== | ==encyclopedia== | ||
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* [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)== | ||
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* http://mathoverflow.net/questions/31905/some-basic-questions-about-chern-simons-theory | * 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 | * http://mathoverflow.net/questions/36178/what-is-the-trace-in-the-chern-simons-action | ||
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==expositions== | ==expositions== | ||
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* [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] | ||
* [http://www.math.uni-bonn.de/people/himpel/himpel_cstheory.pdf Lie groups and Chern-Simons Theory] Benjamin Himpel | * [http://www.math.uni-bonn.de/people/himpel/himpel_cstheory.pdf Lie groups and Chern-Simons Theory] Benjamin Himpel | ||
* Labastida, J. M. F. 1999. “Chern-Simons Gauge Theory: Ten Years After”. <em>hep-th/9905057</em> (5월 8). doi:doi:10.1063/1.59663. http://arxiv.org/abs/hep-th/9905057. | * Labastida, J. M. F. 1999. “Chern-Simons Gauge Theory: Ten Years After”. <em>hep-th/9905057</em> (5월 8). doi:doi:10.1063/1.59663. http://arxiv.org/abs/hep-th/9905057. | ||
* Curtis T. McMullen, [http://dx.doi.org/10.1090/S0273-0979-2011-01329-5%20 The evolution of geometric structures on 3-manifolds] Bull. Amer. Math. Soc. 48 (2011), 259-274. | * Curtis T. McMullen, [http://dx.doi.org/10.1090/S0273-0979-2011-01329-5%20 The evolution of geometric structures on 3-manifolds] Bull. Amer. Math. Soc. 48 (2011), 259-274. | ||
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==articles== | ==articles== | ||
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* http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf | * http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf | ||
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* [http://www.math.columbia.edu/%7Eneumann/preprints/cs2.pdf Rationality problems for K-theory and Chern-Simons invariants of hyperbolic 3-manifolds]<br> | * [http://www.math.columbia.edu/%7Eneumann/preprints/cs2.pdf Rationality problems for K-theory and Chern-Simons invariants of hyperbolic 3-manifolds]<br> | ||
** Walter Neumann, 1995 | ** Walter Neumann, 1995 | ||
* Witten, Edward. 1992. “Chern-Simons Gauge Theory As A String Theory”. <em>hep-th/9207094</em> (7월 28). http://arxiv.org/abs/hep-th/9207094.<br> | * Witten, Edward. 1992. “Chern-Simons Gauge Theory As A String Theory”. <em>hep-th/9207094</em> (7월 28). http://arxiv.org/abs/hep-th/9207094.<br> | ||
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==links== | ==links== | ||
+ | * [http://staff.science.uva.nl/%7Eriveen/volume_conjecture.htm Volume conjecture links and notes] | ||
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[[분류:math and physics]] | [[분류:math and physics]] |
2013년 2월 10일 (일) 04:20 판
introduction
- Topological quantum field theory(TQFT)
- CS is an invariant for 3-manifolds
- action
Let \(A\) be a SU(2)-connection on the trivicla C^2 bundle over S^3
\[S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A)=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{1}{3}A\wedge [A,A])\]
Chern-Simons partition function
- path integral defined by Witten
$$ Z_k(M)=\int e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA\ $$ 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
asymptotic expansion
$$ Z_k(M)\approx \frac{1}{2}e^{-3\pi i/4}\sum_{\alpha}\sqrt{T_{\alpha}(M)} e^{-2\pi i I_{\alpha}/4} e^{2\pi (k+2) \operatorname{CS}(A)} $$ where the sum is over flat connections $\alpha$
Jones Polynomial
- path integral gives Jones polynomials
\[\langle K\rangle=\int {\operatorname{Tr}(\int_{K} A)}e^{2\pi i k \operatorname{CS}(A)}DA=(q^{1/2}+q^{-1/2})V(K,q^{-1})\]
where 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
- closely related to the Kashaev Volume conjecture
- WZW model
- quantum dilogarithm
- characteristic class
encyclopedia
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
- 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.