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Nekrasov functions and exact Bohr-Sommerfeld integrals

  • A. MironovEmail author
  • A. Morozov
Article

Abstract

In the case of SU(2), associated by the AGT relation to the 2d Liouville theory, the Seiberg-Witten prepotential is constructed from the Bohr-Sommerfeld periods of 1d sine-Gordon model. If the same construction is literally applied to monodromies of exact wave functions, the prepotential turns into the one-parametric Nekrasov prepotential \(\mathcal{F} (a, \epsilon_{1}) \) with the other epsilon parameter vanishing, ϵ2 = 0, and ϵ1 playing the role of the Planck constant in the sine-Gordon Shrödinger equation, ℏ = ϵ1. This seems to be in accordance with the recent claim in [1] and poses a problem of describing the full Nekrasov function as a seemingly straightforward double-parametric quantization of sine-Gordon model. This also provides a new link between the Liouville and sine-Gordon theories.

Keywords

Supersymmetric gauge theory Integrable Hierarchies Topological Field Theories 

References

  1. [1]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [SPIRES].
  2. [2]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [SPIRES].zbMATHCrossRefADSGoogle Scholar
  3. [3]
    N. Wyllard, A N−1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].CrossRefADSGoogle Scholar
  4. [4]
    N. Drukker, D.R. Morrison and T. Okuda, Loop operators and S-duality from curves on Riemann surfaces, JHEP 09 (2009) 031 [arXiv:0907.2593] [SPIRES].CrossRefADSGoogle Scholar
  5. [5]
    A. Marshakov, A. Mironov and A. Morozov, On combinatorial expansions of conformal blocks, arXiv:0907.3946 [SPIRES].
  6. [6]
    D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, arXiv:0908.0307 [SPIRES].
  7. [7]
    A. Mironov, S. Mironov, A. Morozov and A. Morozov, CFT exercises for the needs of AGT, arXiv:0908.2064 [SPIRES].
  8. [8]
    A. Mironov and A. Morozov, The power of Nekrasov functions, Phys. Lett. B 680 (2009) 188 [arXiv:0908.2190] [SPIRES].MathSciNetADSGoogle Scholar
  9. [9]
    A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  10. [10]
    S.M. Iguri and C.A. Núñez, Coulomb integrals and conformal blocks in the AdS3-WZNW model, JHEP 11 (2009) 090 [arXiv:0908.3460] [SPIRES].CrossRefADSGoogle Scholar
  11. [11]
    D.V. Nanopoulos and D. Xie, On crossing symmmetry and modular invariance in conformal field theory and S duality in gauge theory, Phys. Rev. D 80 (2009) 105015 [arXiv:0908.4409] [SPIRES].ADSGoogle Scholar
  12. [12]
    L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [SPIRES].CrossRefGoogle Scholar
  13. [13]
    N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [SPIRES].CrossRefGoogle Scholar
  14. [14]
    A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [SPIRES].MathSciNetADSGoogle Scholar
  15. [15]
    R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [SPIRES].
  16. [16]
    A. Marshakov, A. Mironov and A. Morozov, Zamolodchikov asymptotic formula and instanton expansion in N = 2 SUSY N f = 2N c QCD, JHEP 11 (2009) 048 [arXiv:0909.3338] [SPIRES].CrossRefADSGoogle Scholar
  17. [17]
    A. Mironov and A. Morozov, Proving AGT relations in the large-c limit, Phys. Lett. B 682 (2009) 118 [arXiv:0909.3531] [SPIRES].MathSciNetADSGoogle Scholar
  18. [18]
    R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [SPIRES].CrossRefADSGoogle Scholar
  19. [19]
    G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, arXiv:0909.4031 [SPIRES].
  20. [20]
    L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, arXiv:0909.4776 [SPIRES].
  21. [21]
    H. Awata and Y. Yamada, Five-dimensional AGT conjecture and the deformed Virasoro algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [SPIRES].CrossRefGoogle Scholar
  22. [22]
    V. Alba and A. Morozov, Non-conformal limit of AGT relation from the 1-point torus conformal block, arXiv:0911.0363 [SPIRES].
  23. [23]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].MathSciNetGoogle Scholar
  24. [24]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].MathSciNetGoogle Scholar
  25. [25]
    A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, hep-th/9801061 [SPIRES].
  26. [26]
    A. Morozov, Integrability and matrix models, Phys. Usp. 37 (1994) 1 [hep-th/9303139] [SPIRES].CrossRefADSGoogle Scholar
  27. [27]
    A. Morozov, Matrix models as integrable systems, hep-th/9502091 [SPIRES].
  28. [28]
    A. Morozov, Challenges of matrix models, hep-th/0502010 [SPIRES].
  29. [29]
    A. Mironov, 2 − D gravity and matrix models. 1. 2 − D gravity, Int. J. Mod. Phys. A 9 (1994) 4355 [hep-th/9312212] [SPIRES].MathSciNetADSGoogle Scholar
  30. [30]
    A. Mironov, Matrix models of two-dimensional gravity, Phys. Part. Nucl. 33 (2002) 537.Google Scholar
  31. [31]
    A.S. Alexandrov, A. Mironov and A. Morozov, Partition functions of matrix models as the first special functions of string theory. I: finite size Hermitean 1-matrix model, Int. J. Mod. Phys. A 19 (2004) 4127 [Theor. Math. Phys. 142 (2005) 349] [hep-th/0310113] [SPIRES].MathSciNetADSGoogle Scholar
  32. [32]
    A.S. Alexandrov, A. Mironov and A. Morozov, Instantons and merons in matrix models, Physica D 235 (2007) 126 [hep-th/0608228] [SPIRES].MathSciNetADSGoogle Scholar
  33. [33]
    A.S. Alexandrov, A. Mironov and A. Morozov, M-theory of matrix models, Teor. Mat. Fiz. 150 (2007) 179 [hep-th/0605171] [SPIRES].Google Scholar
  34. [34]
    A. Alexandrov, A. Mironov and A. Morozov, BGWM as second constituent of complex matrix model, JHEP 12 (2009) 053 [arXiv:0906.3305] [SPIRES].CrossRefADSGoogle Scholar
  35. [35]
    A.S. Alexandrov, A. Mironov, A. Morozov and P. Putrov, Partition functions of matrix models as the first special functions of string theory. II. Kontsevich model, Int. J. Mod. Phys. A 24 (2009) 4939 [arXiv:0811.2825] [SPIRES].MathSciNetADSGoogle Scholar
  36. [36]
    N. Seiberg and E. Witten, Monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [hep-th/9407087] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  37. [37]
    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  38. [38]
    A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [SPIRES].MathSciNetADSGoogle Scholar
  39. [39]
    P.C. Argyres and A.D. Shapere, The Vacuum structure of N = 2 superQCD with classical gauge groups, Nucl. Phys. B 461 (1996) 437 [hep-th/9509175] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  40. [40]
    J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145 [hep-th/9510129] [SPIRES].MathSciNetADSGoogle Scholar
  41. [41]
    R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  42. [42]
    H. Itoyama and A. Morozov, Integrability and Seiberg-Witten theory: curves and periods, Nucl. Phys. B 477 (1996) 855 [hep-th/9511126] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  43. [43]
    H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529 [hep-th/9512161] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  44. [44]
    H. Itoyama and A. Morozov, Integrability and Seiberg-Witten theory, hep-th/9601168 [SPIRES].
  45. [45]
    A. Gorsky, A. Marshakov, A. Mironov and A. Morozov, N = 2 supersymmetric QCD and integrable spin chains: rational case N f < 2N c, Phys. Lett. B 380 (1996) 75 [hep-th/9603140] [SPIRES].MathSciNetADSGoogle Scholar
  46. [46]
    A.Marshakov, Seiberg-Witten theory and integrable systems, World Scientific, Singapore (1999).zbMATHGoogle Scholar
  47. [47]
    A. Gorsky and A. Mironov, Integrable many-body systems and gauge theories, hep-th/0011197 [SPIRES].
  48. [48]
    N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [SPIRES].CrossRefGoogle Scholar
  49. [49]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [SPIRES].
  50. [50]
    A. Mironov and A. Morozov, Virasoro constraints for Kontsevich-Hurwitz partition function, JHEP 02 (2009) 024 [arXiv:0807.2843] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  51. [51]
    A. Mironov, A. Morozov and S. Natanzon, Complete set of cut-and-join operators in Hurwitz-Kontsevich theory, arXiv:0904.4227 [SPIRES].
  52. [52]
    G. Wentzel, Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik, Z. Phys. 38 (1926) 518.CrossRefADSGoogle Scholar
  53. [53]
    L. Brillouin, La mécanique ondulatoire de Schrödinger; une méthode générale de resolution par approximations successives, Comptes Rendus 183 (1926) 24.Google Scholar
  54. [54]
    H.A. Kramers, Wellenmechanik und halbzahlige Quantisierung, Z. Phys. 39 (1926) 828.CrossRefADSGoogle Scholar
  55. [55]
    A. Zwaan, Arch. Neerl. des Sciences 12 (1929) 33.Google Scholar
  56. [56]
    J.L. Dunham, The Wentzel-Brillouin-Kramers method of solving the wave equation, Phys. Rev. 41 (1932) 713.zbMATHCrossRefADSGoogle Scholar
  57. [57]
    A. Klemm, W. Lerche and S. Theisen, Nonperturbative effective actions of N = 2 supersymmetric gauge theories, Int. J. Mod. Phys. A 11 (1996) 1929 [hep-th/9505150] [SPIRES].MathSciNetADSGoogle Scholar
  58. [58]
    A. Gerasimov, D. Lebedev and A. Morozov, On possible implications of integrable systems for string theory, Int. J. Mod. Phys. A 6 (1991) 977 [SPIRES].MathSciNetADSGoogle Scholar
  59. [59]
    A. Morozov, Integrable systems and double loop algebras in string theory, Sov. J. Nucl. Phys. 52 (1990) 755 [SPIRES].Google Scholar
  60. [60]
    M. Gutzwiller, The quantum mechanical Toda lattice II, Annals Phys. 133 (1981) 304 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  61. [61]
    M. Gutzwiller, he quantum mechanical Toda lattice, Annals Phys. 124 (1980) 347 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  62. [62]
    S. Kharchev and D. Lebedev, Integral representation for the eigenfunctions of quantum periodic Toda chain, Lett. Math. Phys. 50 (1999) 53 [hep-th/9910265] [SPIRES].zbMATHCrossRefMathSciNetGoogle Scholar
  63. [63]
    S. Kharchev and D. Lebedev, Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism, J. Phys. A 34 (2001) 2247 [hep-th/0007040] [SPIRES].MathSciNetADSGoogle Scholar
  64. [64]
    V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B 240 (1984) 312 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  65. [65]
    A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S.L. Shatashvili, Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys. A 5 (1990) 2495 [SPIRES].MathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.Theory DepartmentLebedev Physics InstituteMoscowRussia
  2. 2.ITEPMoscowRussia

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