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(피타고라스님이 이 페이지의 이름을 Chern-Simons gauge theory and invariant로 바꾸었습니다.) |
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| (사용자 3명의 중간 판 60개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
| − | + | ==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 | ||
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| − | * | + | ==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]] | ||
| + | * [[Chern class]] | ||
| + | * [[vector valued differential forms]] | ||
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| − | + | ==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]] | ||
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| − | * | + | ===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. | ||
| − | + | ===examples=== | |
| + | * [[Quantum modular forms]] | ||
| − | + | ==Jones Polynomial== | |
| + | * path integral gives [[Jones polynomials]] | ||
| + | :<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 | ||
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| − | + | ||
| − | + | ==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 | |
| − | + | ||
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| − | + | ==Chern-Simons invariant== | |
| + | * [[Chern-Simons invariant]] | ||
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| + | ==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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| − | + | ==related items== | |
| − | + | * [[Complex Chern-Simons theory]] | |
| − | + | * closely related to the [[Kashaev's volume conjecture|Kashaev Volume conjecture]] | |
| − | + | * [[WZW (Wess-Zumino-Witten) model]] | |
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| − | * [[WZW (Wess-Zumino-Witten) | ||
* [[quantum dilogarithm]] | * [[quantum dilogarithm]] | ||
| + | * [[characteristic class]] | ||
| + | * [[Morse theory]] | ||
| + | * [[Arithmetic Chern-Simons Theory]] | ||
| − | + | ==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)== | |
| + | * 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 | ||
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| + | ==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. | |
| − | + | * 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. | |
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| + | ==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 | ||
| − | * | + | * 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 |
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| − | + | ==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'}] | ||
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
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
memo
- 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
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
- 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
메타데이터
위키데이터
- ID : Q1528019
Spacy 패턴 목록
- [{'LOWER': 'chern'}, {'OP': '*'}, {'LOWER': 'simons'}, {'LEMMA': 'theory'}]