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

2021년 2월 17일 (수) 02:54 기준 최신판

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 \(\lambda :\{1,\cdots,m\} \to P_{+}(k)\)

\(Z_k(S^3)=S_{00}=\sqrt{\frac{2}{k+2}}\sin( \frac{\pi }{k+2})\sim \sqrt{2}\pi k^{-3/2}\)
\(Z_k(S^1\times S^2)=S_{00}\sum_{\mu} S_{0\mu}\frac{S_{0\mu}}{S_{00}}=1\)
Here \(S\) denotes the entries of Kac-Peterson modular S-matrix

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 \(\alpha\)

Borot, Gaëtan, Bertrand Eynard, and Alexander Weiße. “Root Systems, Spectral Curves, and Analysis of a Chern-Simons Matrix Model for Seifert Fibered Spaces.” arXiv:1407.4500 [math-Ph], July 16, 2014. http://arxiv.org/abs/1407.4500.

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

related items

Complex Chern-Simons theory
closely related to the Kashaev Volume conjecture
WZW (Wess-Zumino-Witten) model
quantum dilogarithm
characteristic class
Morse theory
Arithmetic Chern-Simons Theory

encyclopedia

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

question and answers(Math Overflow)

expositions

Freed, Daniel S., and Robert E. Gompf. 1991. “Computer Tests of Witten’s Chern-Simons Theory against the Theory of Three-Manifolds.” Physical Review Letters 66 (10): 1255–1258. doi:10.1103/PhysRevLett.66.1255.
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

Hahn, Atle. “Infinite Dimensional Analysis and the Chern-Simons Path Integral.” arXiv:1506.06809 [math-Ph], June 22, 2015. http://arxiv.org/abs/1506.06809.
Mittal, Sunil, Sriram Ganeshan, Jingyun Fan, Abolhassan Vaezi, and Mohammad Hafezi. “Observation of the Chern-Simons Gauge Anomaly.” arXiv:1504.00369 [cond-Mat, Physics:hep-Th], April 1, 2015. http://arxiv.org/abs/1504.00369.
Henriques, Andre. ‘What Chern-Simons Theory Assigns to a Point’. arXiv:1503.06254 [math-Ph], 20 March 2015. http://arxiv.org/abs/1503.06254.
Fiorenza, Domenico, Urs Schreiber, and Alessandro Valentino. “Central Extensions of Mapping Class Groups from Characteristic Classes.” arXiv:1503.00888 [math], March 3, 2015. http://arxiv.org/abs/1503.00888.
Mkrtchyan, R. L. “On a Gopakumar-Vafa Form of Partition Function of Chern-Simons Theory on Classical and Exceptional Lines.” arXiv:1410.0376 [hep-Th, Physics:math-Ph], October 1, 2014. http://arxiv.org/abs/1410.0376.
Gelca, Razvan, and Alastair Hamilton. 2014. “The Topological Quantum Field Theory of Riemann’s Theta Functions.” arXiv:1406.4269 [math-Ph], June. http://arxiv.org/abs/1406.4269.
Bytsenko, A. A., A. E. Gon\ccalves, and W. da Cruz. 1998. “Torsion on Hyperbolic Manifolds and the Semiclassical Limit for Chern-Simons Theory.” Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics 13 (30): 2453–2461. doi:10.1142/S0217732398002618.
Adams, David H. 1998. “The Semiclassical Approximation for the Chern-Simons Partition Function.” Physics Letters. B 417 (1-2): 53–60. doi:10.1016/S0370-2693(97)01343-9.
Bytsenko, Andrei A., Luciano Vanzo, and Sergio Zerbini. 1997. “Ray-Singer Torsion for a Hyperbolic \(3\)-Manifold and Asymptotics of Chern-Simons-Witten Invariant.” Nuclear Physics. B 505 (3): 641–659. doi:10.1016/S0550-3213(97)00566-X.
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.” arXiv:hep-th/9207094, July. 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.
Freed, Daniel S., and Robert E. Gompf. 1991. “Computer Calculation of Witten’s \(3\)-Manifold Invariant.” Communications in Mathematical Physics 141 (1): 79–117.
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

메타데이터

위키데이터

ID : Q1528019

Spacy 패턴 목록

[{'LOWER': 'chern'}, {'OP': '*'}, {'LOWER': 'simons'}, {'LEMMA': 'theory'}]

@@ 2번째 줄: / 2번째 줄: @@
 * 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
+* <math>M</math> : compact oriented 3-manifold
-* $G=SU(2)$
+* <math>G=SU(2)</math>
-* <math>P\to M</math> : principal G-bundle, trivial $SU(2)$ bundle over $M$ since $SU(2)$ is simply connected
+* <math>P\to M</math> : principal G-bundle, trivial <math>SU(2)</math> bundle over <math>M</math> since <math>SU(2)</math> is simply connected
-* $\mathcal{A}_M$ : the space of connections on $P$
+* <math>\mathcal{A}_M</math> : the space of connections on <math>P</math>
 ** 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 <math>\Omega^{1}(M,\mathfrak{g})</math>, the space of 1-forms on <math>M</math> with values in <math>\mathfrak{g}</math>
-* $A\in \mathcal{A}_M$ : connection
+* <math>A\in \mathcal{A}_M</math> : connection
-* <math>F=A\wedge dA+A\wedge A\in \Omega^{2}(M,\mathfrak{g})</math> : the curvature of connection $A$
+* <math>F=A\wedge dA+A\wedge A\in \Omega^{2}(M,\mathfrak{g})</math> : the curvature of connection <math>A</math>
-* $\mathcal{G}=\operatorname{Map}(M,G)$ : the gauge group acting on $\mathcal{A}_M$ by
+* <math>\mathcal{G}=\operatorname{Map}(M,G)</math> : the gauge group acting on <math>\mathcal{A}_M</math> by
-$$
+:<math>
 g^{*}A=g^{-1}Ag+g^{-1}dg, g\in \mathcal{G}
-$$
+</math>
 * 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>
@@ 33번째 줄: / 36번째 줄: @@
 * [[Feynman diagrams and path integral]]
 * The path integral defined by Witten
-$$
+:<math>
 Z_k(M)=\int_{\mathcal{A}_M/\mathcal{G}} e^{2\pi \sqrt{-1} k \operatorname{CS}(A)}DA\
-$$
+</math>
 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
+* M obtained as Dehn surgery on a framed link L with m components <math>L_j\, , 1\leq j \leq m</math> in <math>S^3</math>. Then
-$$
+:<math>
 Z_k(M)=S_{00}C^{\sigma(L)}\sum_{\lambda}S_{0\lambda_1}\cdots S_{0\lambda_m}J(L;\lambda_1,\cdots,\lambda_m)
-$$
+</math>
-is a topological invariant of $M$ and does not depend on the choice of $L$
+is a topological invariant of <math>M</math> and does not depend on the choice of <math>L</math>
-where them sum is for any coloring $\lambda :\{1,\cdots,m\} \to P_{+}(k)$
+where them sum is for any coloring <math>\lambda :\{1,\cdots,m\} \to P_{+}(k)</math>
-* $Z_k(S^3)=S_{00}$
+* <math>Z_k(S^3)=S_{00}=\sqrt{\frac{2}{k+2}}\sin( \frac{\pi }{k+2})\sim \sqrt{2}\pi k^{-3/2}</math>
-* $Z_k(S^1\times S^2)=S_{00}\sum_{\mu} S_{0\mu}\frac{S_{0\mu}}{S_{00}}=1$
+* <math>Z_k(S^1\times S^2)=S_{00}\sum_{\mu} S_{0\mu}\frac{S_{0\mu}}{S_{00}}=1</math>
+* Here <math>S</math> denotes the entries of [[Kac-Peterson modular S-matrix]]
 ===asymptotic expansion===
-* As $k\to \infty$,
+* As <math>k\to \infty</math>,
-$$
+:<math>
 Z_k(M)\approx \frac{1}{2}e^{-3\pi i/4}\sum_{\alpha}\sqrt{T_{\alpha}(M)} e^{-2\pi i I_{\alpha}/4} e^{2\pi (k+2) \operatorname{CS}(A)}
-$$
+</math>
-where the sum is over flat connections $\alpha$
+where the sum is over flat connections <math>\alpha</math>
+* Borot, Gaëtan, Bertrand Eynard, and Alexander Weiße. “Root Systems, Spectral Curves, and Analysis of a Chern-Simons Matrix Model for Seifert Fibered Spaces.” arXiv:1407.4500 [math-Ph], July 16, 2014. http://arxiv.org/abs/1407.4500.
 ===examples===
 * [[Quantum modular forms]]
 ==Jones Polynomial==
@@ 69번째 줄: / 71번째 줄: @@
 ==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==
@@ 85번째 줄: / 87번째 줄: @@
-==history==
-* http://www.google.com/search?hl=en&tbs=tl:1&q=
 ==related items==
 * [[Complex Chern-Simons theory]]
 * closely related to the [[Kashaev's volume conjecture|Kashaev Volume conjecture]]
-* [[WZW (Wess-Zumino-Witten) model and its central charge]]
+* [[WZW (Wess-Zumino-Witten) model]]
 * [[quantum dilogarithm]]
 * [[characteristic class]]
+* [[Morse theory]]
+* [[Arithmetic Chern-Simons Theory]]
 ==encyclopedia==
@@ 111번째 줄: / 105번째 줄: @@
 * http://mathoverflow.net/questions/36178/what-is-the-trace-in-the-chern-simons-action
 ==expositions==
+* Freed, Daniel S., and Robert E. Gompf. 1991. “Computer Tests of Witten’s Chern-Simons Theory against the Theory of Three-Manifolds.” Physical Review Letters 66 (10): 1255–1258. doi:10.1103/PhysRevLett.66.1255.
 * [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://131.220.77.51/event/2009/gauge_theory/ Chern-Simons Gauge Theory: 20 years after]
+** conference
 * 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.
 * Freed, Daniel S. 1992. “Classical Chern-Simons Theory, Part 1.” arXiv:hep-th/9206021 (June 4). http://arxiv.org/abs/hep-th/9206021.
 ==articles==
+* Hahn, Atle. “Infinite Dimensional Analysis and the Chern-Simons Path Integral.” arXiv:1506.06809 [math-Ph], June 22, 2015. http://arxiv.org/abs/1506.06809.
+* Mittal, Sunil, Sriram Ganeshan, Jingyun Fan, Abolhassan Vaezi, and Mohammad Hafezi. “Observation of the Chern-Simons Gauge Anomaly.” arXiv:1504.00369 [cond-Mat, Physics:hep-Th], April 1, 2015. http://arxiv.org/abs/1504.00369.
+* Henriques, Andre. ‘What Chern-Simons Theory Assigns to a Point’. arXiv:1503.06254 [math-Ph], 20 March 2015. http://arxiv.org/abs/1503.06254.
+* Fiorenza, Domenico, Urs Schreiber, and Alessandro Valentino. “Central Extensions of Mapping Class Groups from Characteristic Classes.” arXiv:1503.00888 [math], March 3, 2015. http://arxiv.org/abs/1503.00888.
+* Mkrtchyan, R. L. “On a Gopakumar-Vafa Form of Partition Function of Chern-Simons Theory on Classical and Exceptional Lines.” arXiv:1410.0376 [hep-Th, Physics:math-Ph], October 1, 2014. http://arxiv.org/abs/1410.0376.
+* Gelca, Razvan, and Alastair Hamilton. 2014. “The Topological Quantum Field Theory of Riemann’s Theta Functions.” arXiv:1406.4269 [math-Ph], June. http://arxiv.org/abs/1406.4269.
+* Bytsenko, A. A., A. E. Gon\ccalves, and W. da Cruz. 1998. “Torsion on Hyperbolic Manifolds and the Semiclassical Limit for Chern-Simons Theory.” Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics 13 (30): 2453–2461. doi:10.1142/S0217732398002618.
+* Adams, David H. 1998. “The Semiclassical Approximation for the Chern-Simons Partition Function.” Physics Letters. B 417 (1-2): 53–60. doi:10.1016/S0370-2693(97)01343-9.
+* Bytsenko, Andrei A., Luciano Vanzo, and Sergio Zerbini. 1997. “Ray-Singer Torsion for a Hyperbolic <math>3</math>-Manifold and Asymptotics of Chern-Simons-Witten Invariant.” Nuclear Physics. B 505 (3): 641–659. doi:10.1016/S0550-3213(97)00566-X.
 * 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”. <em>hep-th/9207094</em> (7월 28). http://arxiv.org/abs/hep-th/9207094.
+* Witten, Edward. 1992. “Chern-Simons Gauge Theory As A String Theory.” arXiv:hep-th/9207094, July. 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.
+* Freed, Daniel S., and Robert E. Gompf. 1991. “Computer Calculation of Witten’s <math>3</math>-Manifold Invariant.” Communications in Mathematical Physics 141 (1): 79–117.
 * 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, [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104178138 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]]
 [[분류:math and physics]]
 [[분류:TQFT]]
 [[분류:Knot theory]]
+[[분류:migrate]]
+==메타데이터==
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q1528019 Q1528019]
+===Spacy 패턴 목록===
+* [{'LOWER': 'chern'}, {'OP': '*'}, {'LOWER': 'simons'}, {'LEMMA': 'theory'}]