Seiberg-Witten Theory and Random Partitions

  • Nikita A. Nekrasov
  • Andrei Okounkov
Part of the Progress in Mathematics book series (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.

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References

  1. [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.CrossRefGoogle Scholar
  2. [2]
    N. Seiberg, Phys. Lett. B, 206 (1988), 75–80.CrossRefMathSciNetGoogle Scholar
  3. [3]
    N. Seiberg and E. Witten, hep-th/9407087; hep-th/9408099.Google Scholar
  4. [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. [5]
    A. Klemm, W. Lerche, P. Mayr, C. Vafa, and N. Warner, hep-th/9604034.Google Scholar
  6. [6]
    S. Katz, A. Klemm, and C. Vafa, hep-th/9609239.Google Scholar
  7. [7]
    E. Witten, hep-th/9703166.Google Scholar
  8. [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. [9]
    N. Nekrasov, hep-th/0206161.Google Scholar
  10. [10]
    A. Losev, A. Marshakov, and N. Nekrasov, hep-th/0302191.Google Scholar
  11. [11]
    C. Vafa and E. Witten, hep-th/9408074.Google Scholar
  12. [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. [13]
    N. Seiberg and E. Witten, hep-th/9908142; J. High Energy Phys., 9909 (1999), 032.Google Scholar
  14. [14]
    N. Nekrasov, hep-th/9609219. A. Lawrence and N. Nekrasov, hep-th/9706025.Google Scholar
  15. [15]
    H. Ooguri and C. Vafa, hep-th/0302109; hep-th/0303063. N. Seiberg, hep-th/0305248.Google Scholar
  16. [16]
    M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, hep-th/9309140.Google Scholar
  17. [17]
    R. Gopakumar and C. Vafa, hep-th/9809187; hep-th/9812127.Google Scholar
  18. [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.MATHGoogle Scholar
  19. [19]
    R. Donagi, alg-geom/9705010.Google Scholar
  20. [20]
    R. Donagi and E. Witten, hep-th/9510101.Google Scholar
  21. [21]
    H. Nakajima and K. Yoshioka, math.AG/0306198.Google Scholar
  22. [22]
    A. Belavin, A. Polyakov, A. Schwarz, and Yu. Tyupkin, Phys. Lett. B, 59 (1975), 85–87.CrossRefMathSciNetGoogle Scholar
  23. [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. [24]
    E. Witten, Comm. Math. Phys., 117 (1988), 353.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    E. Witten, hep-th/9403195.Google Scholar
  26. [26]
    M. Atiyah and L. Jeffrey, J. Geom. Phys., 7 (1990), 119–136.MATHCrossRefMathSciNetGoogle Scholar
  27. [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.MathSciNetGoogle Scholar
  28. [28]
    N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, Nuclear Phys. B, 498 (1997), 467; hep-th/9612115.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    N. Seiberg, hep-th/0008013.Google Scholar
  30. [30]
    S. Minwala, M. van Raamsdonk, and N. Seiberg, hep-th/9912072.Google Scholar
  31. [31]
    N. Nekrasov and A. S. Schwarz, hep-th/9802068; Comm. Math. Phys., 198 (1998), 689.Google Scholar
  32. [32]
    A. Okounkov, hep-th/9702001.Google Scholar
  33. [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.MATHMathSciNetGoogle Scholar
  34. [34]
    A. Okounkov, Infinite wedge and random partitions, Selecta Math. (N.S.), 7-1 (2001), 57–81.MATHCrossRefMathSciNetGoogle Scholar
  35. [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. [36]
    A. Okounkov and R. Pandharipande, math.AG/0207233; math.AG/0204305.Google Scholar
  37. [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. [38]
    N. Dorey, T. J. Hollowood, V. Khoze, and M. Mattis, hep-th/0206063 and references therein.Google Scholar
  39. [39]
    N. Dorey, V. V. Khoze, and M. P. Mattis, hep-th/9607066.Google Scholar
  40. [40]
    A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 355 (1995), 466–474; hep-th/9505035.MATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    G. Chan and E. D’Hoker, hep-th/9906193. E. D’Hoker, I. Krichever, and D. Phong, hep-th/9609041.Google Scholar
  42. [42]
    B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Adv. Math., 26-2 (1977), 206–222.MATHCrossRefMathSciNetGoogle Scholar
  43. [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. [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).MathSciNetGoogle Scholar
  45. [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.MATHMathSciNetGoogle Scholar
  46. [46]
    G. Moore, N. Nekrasov, and S. Shatashvili, hep-th/9712241; hep-th/9803265.Google Scholar
  47. [47]
    R. Kenyon, A. Okounkov, and S. Sheffield, Dimers and amoebae, to appear. R. Kenyon and A. Okounkov, in preparation.Google Scholar
  48. [48]
    I. Krichever, hep-th/9205110; Comm. Math. Phys., 143 (1992), 415.Google Scholar
  49. [49]
    A. Gorsky, A. Marshakov, A. Mironov, and A. Morozov, Nuclear Phys. B, B527 (1998), 690–716; hep-th/9802007.CrossRefMathSciNetGoogle Scholar
  50. [50]
    M. Mariño and G. Moore, hep-th/9802185. J. Edelstein, M. Mariño, and J. Mas, hep-th/9805172.Google Scholar
  51. [51]
    K. Takasaki, hep-th/9901120.Google Scholar
  52. [52]
    R. Gopakumar and C. Vafa, hep-th/9802016; hep-th/9811131.Google Scholar
  53. [53]
    M. Douglas, hep-th/9311130; hep-th/9303159.Google Scholar
  54. [54]
    H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, University Lecture Series, American Mathematical Society, Providence, RI, 1999.MATHGoogle Scholar
  55. [55]
    M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. RIMS Kyoto Univ., 19 (1983), 943–1001.MATHMathSciNetCrossRefGoogle Scholar
  56. [56]
    K. Ueno and K. Takasaki, Adv. Stud. Pure Math., 4 (1984), 1.MathSciNetGoogle Scholar
  57. [57]
    A. Orlov, nlin.SI/0207030; nlin.SI/0305001.Google Scholar
  58. [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.MATHCrossRefMathSciNetGoogle Scholar
  59. [59]
    T. Hollowood, hep-th/0201075; hep-th/0202197.Google Scholar
  60. [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.CrossRefMathSciNetGoogle Scholar
  61. [61]
    N. Hitchin, Duke Math. J., 54-1 (1987).Google Scholar
  62. [62]
    N. Nekrasov and A. Gorsky, hep-th/9401021. N. Nekrasov, hep-th/9503157.Google Scholar
  63. [63]
    J. Hurtubise and E. Markman, math.AG/9912161.Google Scholar
  64. [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. [65]
    S. Bloch and A. Okounkov, alg-geom/9712009.Google Scholar
  66. [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. [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. [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.MATHCrossRefMathSciNetGoogle Scholar
  69. [69]
    A. M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics 3, Harwood Academic Publishers, Chur, Switzerland, 1987.Google Scholar
  70. [70]
    H. Braden and N. Nekrasov, hep-th/9912019.Google Scholar
  71. [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.CrossRefGoogle Scholar
  72. [72]
    D. Gross and R. Gopakumar, hep-th/9411021. R. Gopakumar, hep-th/0211100.Google Scholar
  73. [73]
    G. Moore and E. Witten, hep-th/9709193.Google Scholar
  74. [74]
    A. Losev, N. Nekrasov, and S. Shatashvili, hep-th/9711108; hep-th/9801061.Google Scholar
  75. [75]
    H. Kawai, T. Kuroki, and T. Morita, hep-th/0303210.Google Scholar
  76. [76]
    F. Cachazo, M. Douglas, N. Seiberg, and E. Witten, hep-th/0211170.Google Scholar
  77. [77]
    N. Nekrasov and S. Shatashvili, in preparation.Google Scholar
  78. [78]
    M. Douglas and V. Kazakov, hep-th/9305047.Google Scholar
  79. [79]
    D. Gross, hep-th/9212149. D. Gross and W. Taylor, hep-th/9301068; hep-th/9303046.Google Scholar
  80. [80]
    A. Iqbal, hep-th/0212279.Google Scholar
  81. [81]
    M. Aganagic, M. Mariño, and C. Vafa, hep-th/0206164.Google Scholar
  82. [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. [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. [84]
    S. Girvin, cond-mat/9907002.Google Scholar
  85. [85]
    V. Knizhnik and A. Zamolodchikov, Nuclear Phys. B, 247 (1984), 83–103.MATHCrossRefMathSciNetGoogle Scholar
  86. [86]
    D. Bernard, Nuclear Phys. B, 303 (1988), 77–93; Nuclear Phys. B, 309 (1988), 145–174.CrossRefMathSciNetGoogle Scholar
  87. [87]
    A. S. Losev, Coset Construction and Bernard Equation, Preprint TH-6215-91, CERN, Geneva, 1991.Google Scholar
  88. [88]
    G. Felder, hep-th/9609153.Google Scholar
  89. [89]
    D. Ivanov, hep-th/9610207.Google Scholar
  90. [90]
    M. A. Olshanetsky, Painlevé type equations and Hitchin systems, in Seiberg-Witten Theory and Integrable Systems, World Scientific, Singapore, 1999.Google Scholar
  91. [91]
    C. Vafa, hep-th/0008142.Google Scholar

Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • Nikita A. Nekrasov
    • 1
  • Andrei Okounkov
    • 2
  1. 1.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUSA

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