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

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1번째 줄: 1번째 줄:
 
==introduction==
 
==introduction==
−
* [[Topological quantum field theory(TQFT)]]
+
* prototypical example of [[Topological quantum field theory(TQFT)]]
−
* CS is an invariant for 3-manifolds
+
* Witten introduced classical Chern-Simons theory to topology
−
** [[Chern-Simons invariant]]
+
* Witten gave a prescription for obtaining exact expressions for
−
* Let <math>A</math> be a SU(2)-connection on the trivial C^2 bundle over S^3
+
** partition function : this becomes new topological invariant of the 3-manifold
−
* the Chern-Simons action is given by
+
** expectation values of Wilson loops : it leads to Jones polynomial
−
:<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>
+
* 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
  
−
==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 $\alpha$
 
−
 
−
 
−
==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 <math>{\operatorname{Tr}(\int_{K} A)}</math> measures the twisting of the connection along the knot
 
−
 
−
 
 
−
 
−
 
 
−
 
−
== ==
 
  
 +
==setting ==
 +
* <math>M</math> : compact oriented 3-manifold
 +
* <math>G=SU(2)</math>
 +
* <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
 +
* <math>\mathcal{A}_M</math> : the space of connections on <math>P</math>
 +
** forms an affine space
 +
** 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>
 +
* <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 <math>A</math>
 +
* <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>
 +
* <math>\operatorname{det}(I+\frac{iF}{2\pi})= c_0+c_1+c_2</math>
 +
* <math>c_3=A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A</math>
 +
* <math>c_2=\frac{1}{8\pi^2} \operatorname{tr} F\wedge F =dc_3</math>
 +
* <math>\int_M c_3</math>
 
* [[curvature and parallel transport]]
 
* [[curvature and parallel transport]]
 
* [[Chern class]]
 
* [[Chern class]]
 
* [[vector valued differential forms]]
 
* [[vector valued differential forms]]
  
−
 
 
  
−
 
+
==WRT invariant==
 +
* Chern-Simons partition function?
 +
* [[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 <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 <math>M</math> and does not depend on the choice of <math>L</math>
 +
where them sum is for any coloring <math>\lambda :\{1,\cdots,m\} \to P_{+}(k)</math>
 +
* <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>
 +
* <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]]
  
−
M : threefold
 
  
−
<math>P\to M</math> : principal G-bundle
+
===asymptotic expansion===
 +
* 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 <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.
  
−
<math>F=A\wedge dA+A\wedge A</math>
+
===examples===
 +
* [[Quantum modular forms]]
  
−
<math>\operatorname{det}(I+\frac{iF}{2\pi})= c_0+c_1+c_2</math>
+
==Jones Polynomial==
−
 
+
* path integral gives [[Jones polynomials]]
−
<math>c_3=A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A</math>
+
:<math>\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})</math>
 +
where <math>{\operatorname{Tr}(\int_{K} A)}</math> measures the twisting of the connection along the knot
  
−
<math>c_2=\frac{1}{8\pi^2} \operatorname{tr} F\wedge F =dc_3</math>
 
  
−
<math>\int_M c_3</math>
+
−
 
 
−
 
 
−
 
 
−
 
 
  
 
==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
 
* analogous to Euler characteristic of a manifold can be computed as the signed count of Morse indices
 
* analogous to Euler characteristic of a manifold can be computed as the signed count of Morse indices
−
 
+
−
 
 
−
 
 
−
 
 
  
 
==Chern-Simons invariant==
 
==Chern-Simons invariant==
−
 
 
* [[Chern-Simons invariant]]
 
* [[Chern-Simons invariant]]
  
−
 
 
  
−
 
 
  
 
==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]
  
−
 
 
  
−
 
 
−
 
−
==history==
 
−
 
−
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
−
 
−
 
 
−
 
−
 
 
  
 
==related items==
 
==related items==
 +
* [[Complex Chern-Simons theory]]
 
* closely related to the [[Kashaev's volume conjecture|Kashaev Volume conjecture]]
 
* 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]]
 
* [[quantum dilogarithm]]
 
* [[quantum dilogarithm]]
 
* [[characteristic class]]
 
* [[characteristic class]]
−
 
+
* [[Morse theory]]
−
 
+
* [[Arithmetic Chern-Simons Theory]]
  
 
==encyclopedia==
 
==encyclopedia==
106번째 줄: 105번째 줄:
 
* 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
  
−
 
+
  
 
==expositions==
 
==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.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
 +
* [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.
 
* 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.
−
 
+
* Freed, Daniel S. 1992. “Classical Chern-Simons Theory, Part 1.” arXiv:hep-th/9206021 (June 4). http://arxiv.org/abs/hep-th/9206021.
−
 
+
  
 
==articles==
 
==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
 
* http://journal.ms.u-tokyo.ac.jp/pdf/jms030310.pdf
−
* [http://www.math.columbia.edu/%7Eneumann/preprints/cs2.pdf Rationality problems for K-theory and Chern-Simons invariants of hyperbolic 3-manifolds]<br>
+
* Witten, Edward. 1992. “Chern-Simons Gauge Theory As A String Theory.” arXiv:hep-th/9207094, July. http://arxiv.org/abs/hep-th/9207094.
−
** Walter Neumann, 1995
+
* 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.
−
*  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>
+
* 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]]
  
−
==links==
+
==메타데이터==
−
* [http://staff.science.uva.nl/%7Eriveen/volume_conjecture.htm Volume conjecture links and notes]
+
===위키데이터===
−
 
+
* ID :  [https://www.wikidata.org/wiki/Q1528019 Q1528019]
−
[[분류:math and physics]]
+
===Spacy 패턴 목록===
 +
* [{'LOWER': 'chern'}, {'OP': '*'}, {'LOWER': 'simons'}, {'LEMMA': 'theory'}]

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

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


WRT invariant

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

Jones Polynomial

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


memo


related items

encyclopedia


question and answers(Math Overflow)


expositions


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

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'chern'}, {'OP': '*'}, {'LOWER': 'simons'}, {'LEMMA': 'theory'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Chern-Simons_gauge_theory_and_Witten%27s_invariant&oldid=51852"