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String theory and math: Why this marriage may last. Mathematics and dualities of quantum physics


Author: Mina Aganagic
Journal: Bull. Amer. Math. Soc. 53 (2016), 93-115
MSC (2010): Primary 00-XX, 81-XX, 51-XX
DOI: https://doi.org/10.1090/bull/1517
Published electronically: August 31, 2015
MathSciNet review: 3403082
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Abstract | References | Similar Articles | Additional Information

Abstract: String theory is changing the relationship between mathematics and physics. The central role is played by the phenomenon of duality, which is intrinsic to quantum physics and abundant in string theory.


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  • [1] J. Polchinski, String duality: A colloquium, Rev. Mod. Phys. 68 (1996), 1245 [hep-th/9607050].
  • [2] C. Vafa, Geometric physics, hep-th/9810149.
  • [3] J. Polchinski, Dualities of Fields and Strings, arXiv:1412.5704 [hep-th].
  • [4] G. Moore, Physical Mathematics and the Future, http://www.physics.rutgers.edu/gmoore/PhysicalMathematicsAndFuture.pdf
  • [5] V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103-111. MR 766964 (86e:57006), https://doi.org/10.1090/S0273-0979-1985-15304-2
  • [6] V. F. R. Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math. 137 (1989), no. 2, 311-334. MR 990215 (89m:57005)
  • [7] Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351-399. MR 990772 (90h:57009)
  • [8] N. Reshetikhin and V. G. Turaev, Invariants of $ 3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547-597. MR 1091619 (92b:57024), https://doi.org/10.1007/BF01239527
  • [9] Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow, Mirror symmetry, Clay Mathematics Monographs, vol. 1, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2003. With a preface by Vafa. MR 2003030 (2004g:14042)
  • [10] Andrei Okounkov, Random surfaces enumerating algebraic curves, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 751-768. MR 2185779 (2007b:14122)
  • [11] R. Thomas, ``The geometry of mirror symmetry'', Encylopedia of mathematical physics. Francoise, Naber, and Tsou (editors), Elsevier, Oxford.
  • [12] Edward Witten, Topological sigma models, Comm. Math. Phys. 118 (1988), no. 3, 411-449. MR 958805 (90b:81080)
  • [13] Edward Witten, On the structure of the topological phase of two-dimensional gravity, Nuclear Phys. B 340 (1990), no. 2-3, 281-332. MR 1068086 (91m:32020), https://doi.org/10.1016/0550-3213(90)90449-N
  • [14] Edward Witten, Mirror manifolds and topological field theory, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 120-158. MR 1191422 (94c:81194)
  • [15] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys. 165 (1994), no. 2, 311-427. MR 1301851 (95f:32029)
  • [16] Wolfgang Lerche, Cumrun Vafa, and Nicholas P. Warner, Chiral rings in $ N=2$ superconformal theories, Nuclear Phys. B 324 (1989), no. 2, 427-474. MR 1025424 (91d:81132), https://doi.org/10.1016/0550-3213(89)90474-4
  • [17] Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21-74. MR 1115626 (93b:32029), https://doi.org/10.1016/0550-3213(91)90292-6
  • [18] K. J. Costello and S. Li, ``Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model,'' arXiv:1201.4501 [math.QA].
  • [19] E. Witten, Chern-Simons gauge theory as a string theory, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 637-678. MR 1362846 (97j:57052)
  • [20] Rajesh Gopakumar and Cumrun Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999), no. 5, 1415-1443. MR 1796682 (2001k:81272)
  • [21] C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173-199. MR 1728879 (2000m:14057), https://doi.org/10.1007/s002229900028
  • [22] Hirosi Ooguri and Cumrun Vafa, Knot invariants and topological strings, Nuclear Phys. B 577 (2000), no. 3, 419-438. MR 1765411 (2001i:81254), https://doi.org/10.1016/S0550-3213(00)00118-8
  • [23] Clifford Henry Taubes, Lagrangians for the Gopakumar-Vafa conjecture, Adv. Theor. Math. Phys. 5 (2001), no. 1, 139-163. MR 1894340 (2003f:53163)
  • [24] Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423-472. MR 1318886 (97d:57004), https://doi.org/10.1016/0040-9383(95)93237-2
  • [25] Marcos Mariño, Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants, Comm. Math. Phys. 253 (2005), no. 1, 25-49. MR 2105636 (2005k:81290), https://doi.org/10.1007/s00220-004-1194-4
  • [26] K. Fukaya, Talk at String-Math 2013, $ http://media.scgp.stonybrook.edu/presentations/$
    $ 20130618 2 fukaya.pdf$
  • [27] Sheldon Katz and Chiu-Chu Melissa Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys. 5 (2001), no. 1, 1-49. MR 1894336 (2003e:14047)
  • [28] D. Maulik, A. Oblomkov, A. Okounkov, and R. Pandharipande, Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds, Invent. Math. 186 (2011), no. 2, 435-479. MR 2845622 (2012h:14140), https://doi.org/10.1007/s00222-011-0322-y
  • [29] G. 't Hooft, ``A planar diagram theory for strong interactions,'' Nucl. Phys. B 72, 461 (1974).
  • [30] Mina Aganagic, Albrecht Klemm, Marcos Mariño, and Cumrun Vafa, The topological vertex, Comm. Math. Phys. 254 (2005), no. 2, 425-478. MR 2117633 (2006e:81263), https://doi.org/10.1007/s00220-004-1162-z
  • [31] S. K. Donaldson and R. P. Thomas, Gauge theory in higher dimensions, The geometric universe (Oxford, 1996) Oxford Univ. Press, Oxford, 1998, pp. 31-47. MR 1634503 (2000a:57085)
  • [32] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math. 142 (2006), no. 5, 1263-1285. MR 2264664 (2007i:14061), https://doi.org/10.1112/S0010437X06002302
  • [33] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. II, Compos. Math. 142 (2006), no. 5, 1286-1304. MR 2264665 (2007i:14062), https://doi.org/10.1112/S0010437X06002314
  • [34] Andrei Okounkov, Nikolai Reshetikhin, and Cumrun Vafa, Quantum Calabi-Yau and classical crystals, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 597-618. MR 2181817 (2006k:81342), https://doi.org/10.1007/0-8176-4467-9_16
  • [35] Amer Iqbal, Cumrun Vafa, Nikita Nekrasov, and Andrei Okounkov, Quantum foam and topological strings, J. High Energy Phys. 4 (2008), 011, 47. MR 2425292 (2009k:81210), https://doi.org/10.1088/1126-6708/2008/04/011
  • [36] R. Dijkgraaf, C. Vafa and E. Verlinde, ``M-theory and a topological string duality,'' hep-th/0602087.
  • [37] Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120-139. MR 1403918 (97f:32040)
  • [38] Yuichi Nohara and Kazushi Ueda, Homological mirror symmetry for the quintic 3-fold, Geom. Topol. 16 (2012), no. 4, 1967-2001. MR 2975297, https://doi.org/10.2140/gt.2012.16.1967
  • [39] Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, Mirror symmetry is $ T$-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243-259. MR 1429831 (97j:32022), https://doi.org/10.1016/0550-3213(96)00434-8
  • [40] Mark Gross, Mirror symmetry and the Strominger-Yau-Zaslow conjecture, Current developments in mathematics 2012, Int. Press, Somerville, MA, 2013, pp. 133-191. MR 3204345
  • [41] M. Aganagic and C. Vafa, ``Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots,'' arXiv:1204.4709;
  • [42] Mina Aganagic, Tobias Ekholm, Lenhard Ng, and Cumrun Vafa, Topological strings, D-model, and knot contact homology, Adv. Theor. Math. Phys. 18 (2014), no. 4, 827-956. MR 3277674
  • [43] Lenhard Ng, Combinatorial knot contact homology and transverse knots, Adv. Math. 227 (2011), no. 6, 2189-2219. MR 2807087 (2012j:53116), https://doi.org/10.1016/j.aim.2011.04.014
  • [44] Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359-426. MR 1740682 (2002j:57025), https://doi.org/10.1215/S0012-7094-00-10131-7
  • [45] Webster, B., ``Knot invariants and higher representation theory'' arXiv:1309.3796
  • [46] Webster, B., ``Tensor product algebras, Grassmannians and Khovanov homology'' arXiv:1312.7357
  • [47] Sergei Gukov, Albert Schwarz, and Cumrun Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005), no. 1, 53-74. MR 2193547 (2007a:57014), https://doi.org/10.1007/s11005-005-0008-8
  • [48] Edward Witten, Fivebranes and knots, Quantum Topol. 3 (2012), no. 1, 1-137. MR 2852941, https://doi.org/10.4171/QT/26
  • [49] Edward Witten, Khovanov homology and gauge theory, Proceedings of the Freedman Fest, Geom. Topol. Monogr., vol. 18, Geom. Topol. Publ., Coventry, 2012, pp. 291-308. MR 3084242, https://doi.org/10.2140/gtm.2012.18.291
  • [50] Davide Gaiotto and Edward Witten, Knot invariants from four-dimensional gauge theory, Adv. Theor. Math. Phys. 16 (2012), no. 3, 935-1086. MR 3024278
  • [51] M. Aganagic and S. Shakirov, ``Knot Homology from Refined Chern-Simons Theory,'' arXiv:1105.5117 [hep-th].
  • [52] M. Aganagic, N. Haouzi and S. Shakirov, ``$ A_n$-Triality,'' arXiv:1403.3657 [hep-th].
  • [53] Oblomkov, A., Rasmussen, J., & Shende, V., ``The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link'', arXiv:1201.2115
  • [54] Hiraku Nakajima, Refined Chern-Simons theory and Hilbert schemes of points on the plane, Perspectives in representation theory, Contemp. Math., vol. 610, Amer. Math. Soc., Providence, RI, 2014, pp. 305-331. MR 3220632, https://doi.org/10.1090/conm/610/12157
  • [55] Gorsky, E., & Negut, A., ``Refined Knot Invariants and Hilbert Schemes'', arXiv:1304.3328
  • [56] N. Nekrasov and A. Okounkov, ``Membranes and Sheaves,'' arXiv:1404.2323 [math.AG].
  • [57] R. Gopakumar and C. Vafa, ``M theory and topological strings. 1.,'' hep-th/9809187.
  • [58] R. Gopakumar and C. Vafa, ``M theory and topological strings. 2.,'' hep-th/9812127.
  • [59] Amihay Hanany and Edward Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nuclear Phys. B 492 (1997), no. 1-2, 152-190. MR 1451054 (98h:81096), https://doi.org/10.1016/S0550-3213(97)00157-0
  • [60] K. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996), no. 3, 513-519. MR 1413696 (97j:81261), https://doi.org/10.1016/0370-2693(96)01088-X
  • [61] Davide Gaiotto and Edward Witten, Supersymmetric boundary conditions in $ \mathcal {N}=4$ super Yang-Mills theory, J. Stat. Phys. 135 (2009), no. 5-6, 789-855. MR 2548595 (2011a:81238), https://doi.org/10.1007/s10955-009-9687-3
  • [62] Tom Braden, Nicholas Proudfoot, Ben Webster: ``Quantizations of conical symplectic resolutions I: local and global structure'', arXiv:1208.3863; ``Quantizations of conical symplectic resolutions II: category $ \mathcal O$ and symplectic duality'', arXiv: 1407.0964
  • [63] N. Seiberg and E. Witten, Erratum: ``Electric-magnetic duality, monopole condensation, and confinement in $ N=2$ supersymmetric Yang-Mills theory'', Nuclear Phys. B 430 (1994), no. 2, 485-486. MR 1303306 (95m:81202b), https://doi.org/10.1016/0550-3213(94)00449-8
  • [64] Edward Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), no. 6, 769-796. MR 1306021 (96d:57035), https://doi.org/10.4310/MRL.1994.v1.n6.a13
  • [65] Davide Gaiotto, $ N=2$ dualities, J. High Energy Phys. 8 (2012), 034, front matter + 57. MR 3006961
  • [66] Luis F. Alday, Davide Gaiotto, and Yuji Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010), no. 2, 167-197. MR 2586871 (2010k:81243), https://doi.org/10.1007/s11005-010-0369-5
  • [67] A. Braverman, M. Finkelberg, H. Nakajima, ``Instanton moduli spaces and W-algebras'', arxiv:1406.2381[math:QA]
  • [68] C. Montonen and D. Olive, Phys. Lett. B72 (1977) 117; P. Goddard, J. Nyuts and D. Olive, Nucl. Phys. B125 (1977) 1.
  • [69] Cumrun Vafa and Edward Witten, A strong coupling test of $ S$-duality, Nuclear Phys. B 431 (1994), no. 1-2, 3-77. MR 1305096 (95k:81138), https://doi.org/10.1016/0550-3213(94)90097-3
  • [70] Anton Kapustin and Edward Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), no. 1, 1-236. MR 2306566 (2008g:14018), https://doi.org/10.4310/CNTP.2007.v1.n1.a1
  • [71] R. Donagi and T. Pantev, Langlands duality for Hitchin systems, Invent. Math. 189 (2012), no. 3, 653-735. MR 2957305, https://doi.org/10.1007/s00222-012-0373-8
  • [72] E. Frenkel, ``Lectures on the Langlands program and conformal field theory,'' hep-th/0512172.

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Additional Information

Mina Aganagic
Affiliation: Departments of Mathematics and Physics, University of California, Berkeley, California
Email: aganacic@berkeley.edu

DOI: https://doi.org/10.1090/bull/1517
Received by editor(s): May 1, 2015
Published electronically: August 31, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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