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Seiberg-Witten Theory and Random Partitions

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The Unity of Mathematics

Part of the book series: Progress in Mathematics ((PM,volume 244))

Summary

We study \( \mathcal{N} = 2 \) supersymmetric four-dimensional gauge theories, in a certain 525-02 = 2 supergravity background, called theΩ-background. The partition function of the theory in the Ω-background can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.

These representations allow us to derive rigorously the Seiberg-Witten geometry, the curves, the differentials, and the prepotential.

We study pure 525-03 = 2 theory, as well as the theory with matter hypermultiplets in the fundamental or adjoint representations, and the five-dimensional theory compactified on a circle.

ITEP Moscow 117259 Russia (on leave of absence)

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References

  1. V. Novikov, M. Shifman, A. Vainshtein, and V. Zakharov, Nuclear Phys. B, 229 (1983), 381; Nuclear Phys. B, 260 (1985), 157–181; Phys. Lett. B, 217 (1989), 103–106.

    Article  Google Scholar 

  2. N. Seiberg, Phys. Lett. B, 206 (1988), 75–80.

    Article  MathSciNet  Google Scholar 

  3. N. Seiberg and E. Witten, hep-th/9407087; hep-th/9408099.

    Google Scholar 

  4. A. Klemm, W. Lerche, S. Theisen, and S. Yankielowicz, hep-th/9411048. P. Argyres and A. Faraggi, hep-th/9411057. A. Hanany and Y. Oz, hep-th/9505074.

    Google Scholar 

  5. A. Klemm, W. Lerche, P. Mayr, C. Vafa, and N. Warner, hep-th/9604034.

    Google Scholar 

  6. S. Katz, A. Klemm, and C. Vafa, hep-th/9609239.

    Google Scholar 

  7. E. Witten, hep-th/9703166.

    Google Scholar 

  8. R. Dijkgraaf and C. Vafa, hep-th/0206255; hep-th/0207106; hep-th/0208048. R. Dijkgraaf, S. Gukov, V. Kazakov, and C. Vafa, hep-th/0210238. R. Dijkgraaf, M. Grisaru, C. Lam, C. Vafa, and D. Zanon, hep-th/0211017. M. Aganagic, M. Marino, A. Klemm, and C. Vafa, hep-th/0211098. R. Dijkgraaf, A. Neitzke, and C. Vafa, hep-th/0211194.

    Google Scholar 

  9. N. Nekrasov, hep-th/0206161.

    Google Scholar 

  10. A. Losev, A. Marshakov, and N. Nekrasov, hep-th/0302191.

    Google Scholar 

  11. C. Vafa and E. Witten, hep-th/9408074.

    Google Scholar 

  12. J. A. Minahan, D. Nemeschansky, C. Vafa, N. P. Warner, hep-th/9802168. T. Eguchi and K. Sakai, hep-th/0203025; hep-th/0211213.

    Google Scholar 

  13. N. Seiberg and E. Witten, hep-th/9908142; J. High Energy Phys., 9909 (1999), 032.

    Google Scholar 

  14. N. Nekrasov, hep-th/9609219. A. Lawrence and N. Nekrasov, hep-th/9706025.

    Google Scholar 

  15. H. Ooguri and C. Vafa, hep-th/0302109; hep-th/0303063. N. Seiberg, hep-th/0305248.

    Google Scholar 

  16. M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, hep-th/9309140.

    Google Scholar 

  17. R. Gopakumar and C. Vafa, hep-th/9809187; hep-th/9812127.

    Google Scholar 

  18. A. Marshakov, Seiberg-Witten Theory and Integrable Systems, World Scientific, Singapore, 1999. H. Braden and I. Krichever, eds., Integrability: The Seiberg-Witten and Whitham Equations, Gordon and Breach, New York, 2000.

    MATH  Google Scholar 

  19. R. Donagi, alg-geom/9705010.

    Google Scholar 

  20. R. Donagi and E. Witten, hep-th/9510101.

    Google Scholar 

  21. H. Nakajima and K. Yoshioka, math.AG/0306198.

    Google Scholar 

  22. A. Belavin, A. Polyakov, A. Schwarz, and Yu. Tyupkin, Phys. Lett. B, 59 (1975), 85–87.

    Article  MathSciNet  Google Scholar 

  23. S. Coleman, The uses of instantons, in Aspects of Symmetry: Selected Erice Lectures, Cambridge University Press, Cambridge, UK, 1985, 265–350.

    Google Scholar 

  24. E. Witten, Comm. Math. Phys., 117 (1988), 353.

    Article  MATH  MathSciNet  Google Scholar 

  25. E. Witten, hep-th/9403195.

    Google Scholar 

  26. M. Atiyah and L. Jeffrey, J. Geom. Phys., 7 (1990), 119–136.

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Atiyah and R. Bott, Philos. Trans. Roy. Soc. London Ser. A, 308 (1982), 524–615. E. Witten, hep-th/9204083. S. Cordes, G. Moore, and S. Rangoolam, hep-th/9411210.

    MathSciNet  Google Scholar 

  28. N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, Nuclear Phys. B, 498 (1997), 467; hep-th/9612115.

    Article  MATH  MathSciNet  Google Scholar 

  29. N. Seiberg, hep-th/0008013.

    Google Scholar 

  30. S. Minwala, M. van Raamsdonk, and N. Seiberg, hep-th/9912072.

    Google Scholar 

  31. N. Nekrasov and A. S. Schwarz, hep-th/9802068; Comm. Math. Phys., 198 (1998), 689.

    Google Scholar 

  32. A. Okounkov, hep-th/9702001.

    Google Scholar 

  33. K. Johansson, The longest increasing subsequence in a random permutation and a unitary random matrix model, Math. Res. Lett., 5-1–2 (1998), 63–82. J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc., 12–4 (1999), 1119–1178. A. Okounkov, Random matrices and random permutations, Internat. Math. Res. Notices, 20 (2000), 1043–1095. A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc., 13-3 (2000), 481–515. K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math. (2), 153-1 (2001), 259–296.

    MATH  MathSciNet  Google Scholar 

  34. A. Okounkov, Infinite wedge and random partitions, Selecta Math. (N.S.), 7-1 (2001), 57–81.

    Article  MATH  MathSciNet  Google Scholar 

  35. P. van Moerbeke, Integrable lattices: random matrices and random permutations, in P. M. Bleher and A. R. Its, eds., Random Matrix Models and Their Applications, Mathematical Sciences Research Institute Publications 40, Cambridge University, Press, Cambridge, UK, 2001, 321–406.

    Google Scholar 

  36. A. Okounkov and R. Pandharipande, math.AG/0207233; math.AG/0204305.

    Google Scholar 

  37. S.V. Kerov, Interlacing measures, in Kirillov’s Seminar on Representation Theory, American Mathematical Society Translations Series 2, American Mathematical Society, Providence, RI, 1998, 35–83. S.V. Kerov, Anisotropic Young diagrams and symmetric Jack functions, Функцuонaл. Анал. u Прuложен., 34-1 (2000), 51–64, 96 (in Russian); Functional Anal. Appl., 34-1 (2000), 41–51. A. M. Vershik, Hook formulae and related identities, Эanuскu сем. ЛОМИ, 172 (1989), 3–20 (in Russian). S.V. Kerov, Random Young tableaux, Теор. верояm. u ее nрuмененuя, 3 (1986), 627–628 (in Russian).

    Google Scholar 

  38. N. Dorey, T. J. Hollowood, V. Khoze, and M. Mattis, hep-th/0206063 and references therein.

    Google Scholar 

  39. N. Dorey, V. V. Khoze, and M. P. Mattis, hep-th/9607066.

    Google Scholar 

  40. A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 355 (1995), 466–474; hep-th/9505035.

    Article  MATH  MathSciNet  Google Scholar 

  41. G. Chan and E. D’Hoker, hep-th/9906193. E. D’Hoker, I. Krichever, and D. Phong, hep-th/9609041.

    Google Scholar 

  42. B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Adv. Math., 26-2 (1977), 206–222.

    Article  MATH  MathSciNet  Google Scholar 

  43. S. V. Kerov and A. M. Vershik, Asymptotics of the Plancherel measure of the symmetric group and the limit shape of the Young diagrams, ДАН СССР, 233-6 (1977), 1024–1027 (in Russian).

    Google Scholar 

  44. S.V. Kerov and A. M. Vershik, Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group, Функцuонaл. Анал. u Прuложен., 19-1 (1985), 25–36 (in Russian).

    MathSciNet  Google Scholar 

  45. S.V. Kerov, Gaussian limit for the Plancherel measure of the symmetric group, C. R. Acad. Sci. Paris Sér. I Math., 316-4 (1993), 303–308.

    MATH  MathSciNet  Google Scholar 

  46. G. Moore, N. Nekrasov, and S. Shatashvili, hep-th/9712241; hep-th/9803265.

    Google Scholar 

  47. R. Kenyon, A. Okounkov, and S. Sheffield, Dimers and amoebae, to appear. R. Kenyon and A. Okounkov, in preparation.

    Google Scholar 

  48. I. Krichever, hep-th/9205110; Comm. Math. Phys., 143 (1992), 415.

    Google Scholar 

  49. A. Gorsky, A. Marshakov, A. Mironov, and A. Morozov, Nuclear Phys. B, B527 (1998), 690–716; hep-th/9802007.

    Article  MathSciNet  Google Scholar 

  50. M. Mariño and G. Moore, hep-th/9802185. J. Edelstein, M. Mariño, and J. Mas, hep-th/9805172.

    Google Scholar 

  51. K. Takasaki, hep-th/9901120.

    Google Scholar 

  52. R. Gopakumar and C. Vafa, hep-th/9802016; hep-th/9811131.

    Google Scholar 

  53. M. Douglas, hep-th/9311130; hep-th/9303159.

    Google Scholar 

  54. H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, University Lecture Series, American Mathematical Society, Providence, RI, 1999.

    MATH  Google Scholar 

  55. M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. RIMS Kyoto Univ., 19 (1983), 943–1001.

    Article  MATH  MathSciNet  Google Scholar 

  56. K. Ueno and K. Takasaki, Adv. Stud. Pure Math., 4 (1984), 1.

    MathSciNet  Google Scholar 

  57. A. Orlov, nlin.SI/0207030; nlin.SI/0305001.

    Google Scholar 

  58. T. Takebe, Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy I, Lett. Math. Phys., 21-1 (1991), 77–84.

    Article  MATH  MathSciNet  Google Scholar 

  59. T. Hollowood, hep-th/0201075; hep-th/0202197.

    Google Scholar 

  60. F. Calogero, J. Math Phys., 12 (1971), 419. J. Moser, Adv. Math., 16 (1975), 197–220. M. Olshanetsky and A. Perelomov, Invent. Math., 31 (1976), 93; Phys. Rev., 71 (1981), 313. I. Krichever, Funk. An. Appl., 14 (1980), 45.

    Article  MathSciNet  Google Scholar 

  61. N. Hitchin, Duke Math. J., 54-1 (1987).

    Google Scholar 

  62. N. Nekrasov and A. Gorsky, hep-th/9401021. N. Nekrasov, hep-th/9503157.

    Google Scholar 

  63. J. Hurtubise and E. Markman, math.AG/9912161.

    Google Scholar 

  64. H. Awata, M. Fukuma, S. Odake, and Y.-H. Quano, hep-th/9312208. H. Awata, M. Fukuma, Y. Matsuo, and S. Odake, hep-th/9408158. R. Dijkgraaf, hep-th/9609022.

    Google Scholar 

  65. S. Bloch and A. Okounkov, alg-geom/9712009.

    Google Scholar 

  66. R. Dijkgraaf, Mirror symmetry and elliptic curves, in R. H. Dijkgraaf, C. Faber, and G. B. M. van der Geer, eds., The Moduli Space of Curves, Progress in Mathematics, Birkhäuser Boston, Cambridge, MA, 1994, 149–163.

    Google Scholar 

  67. J. Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Elementary Particle Res. J. (Kyoto), 80 (1989).

    Google Scholar 

  68. V. Kazakov, I. Kostov, and N. Nekrasov, D-particles, matrix integrals and KP hierarchy, Nuclear Phys. B, 557 (1999), 413–442; hep-th/9810035.

    Article  MATH  MathSciNet  Google Scholar 

  69. A. M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics 3, Harwood Academic Publishers, Chur, Switzerland, 1987.

    Google Scholar 

  70. H. Braden and N. Nekrasov, hep-th/9912019.

    Google Scholar 

  71. E. Shuryak, Z. Phys. C, 38 (1988), 141–145, 165–172; Nuclear Phys. B, 319 (1989), 521–541; Nuclear Phys. B, 328 (1989), 85–102.

    Article  Google Scholar 

  72. D. Gross and R. Gopakumar, hep-th/9411021. R. Gopakumar, hep-th/0211100.

    Google Scholar 

  73. G. Moore and E. Witten, hep-th/9709193.

    Google Scholar 

  74. A. Losev, N. Nekrasov, and S. Shatashvili, hep-th/9711108; hep-th/9801061.

    Google Scholar 

  75. H. Kawai, T. Kuroki, and T. Morita, hep-th/0303210.

    Google Scholar 

  76. F. Cachazo, M. Douglas, N. Seiberg, and E. Witten, hep-th/0211170.

    Google Scholar 

  77. N. Nekrasov and S. Shatashvili, in preparation.

    Google Scholar 

  78. M. Douglas and V. Kazakov, hep-th/9305047.

    Google Scholar 

  79. D. Gross, hep-th/9212149. D. Gross and W. Taylor, hep-th/9301068; hep-th/9303046.

    Google Scholar 

  80. A. Iqbal, hep-th/0212279.

    Google Scholar 

  81. M. Aganagic, M. Mariño, and C. Vafa, hep-th/0206164.

    Google Scholar 

  82. P. Wiegmann and A. Zabrodin, hep-th/9909147. I. K. Kostov, I. Krichever, M. Mineev-Weinstein, P. Wiegmann, and A. Zabrodin, hepth/0005259.

    Google Scholar 

  83. V. Pasquier, hep-th/9405104. R. Caracciollo, A. Lerda, and G. R. Zemba, hep-th/9503229. J. Minahan, A. P. Polychronakos, hep-th/9404192; hep-th/9303153. A. P. Polychronakos, hep-th/9902157. E. Langmann, math-ph/0007036; math-ph/0102005.

    Google Scholar 

  84. S. Girvin, cond-mat/9907002.

    Google Scholar 

  85. V. Knizhnik and A. Zamolodchikov, Nuclear Phys. B, 247 (1984), 83–103.

    Article  MATH  MathSciNet  Google Scholar 

  86. D. Bernard, Nuclear Phys. B, 303 (1988), 77–93; Nuclear Phys. B, 309 (1988), 145–174.

    Article  MathSciNet  Google Scholar 

  87. A. S. Losev, Coset Construction and Bernard Equation, Preprint TH-6215-91, CERN, Geneva, 1991.

    Google Scholar 

  88. G. Felder, hep-th/9609153.

    Google Scholar 

  89. D. Ivanov, hep-th/9610207.

    Google Scholar 

  90. M. A. Olshanetsky, Painlevé type equations and Hitchin systems, in Seiberg-Witten Theory and Integrable Systems, World Scientific, Singapore, 1999.

    Google Scholar 

  91. C. Vafa, hep-th/0008142.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut des Hautes Études Scientifiques, 35 Route de Chartres, F-91440, Bures-sur-Yvette, France

    Nikita A. Nekrasov

  2. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Andrei Okounkov

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  1. Nikita A. Nekrasov
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  2. Andrei Okounkov
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Editors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Pavel Etingof  & I. M. Singer  & 

  2. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Vladimir Retakh

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Nekrasov, N.A., Okounkov, A. (2006). Seiberg-Witten Theory and Random Partitions. In: Etingof, P., Retakh, V., Singer, I.M. (eds) The Unity of Mathematics. Progress in Mathematics, vol 244. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4467-9_15

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