Advertisement

Theoretical and Mathematical Physics

, Volume 181, Issue 1, pp 1206–1234 | Cite as

Darboux coordinates, Yang-Yang functional, and gauge theory

  • N. A. NekrasovEmail author
  • A. A. Rosly
  • S. L. Shatashvili
Article

Abstract

The moduli space of flat SL 2 connections on a punctured Riemann surface Σ with fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates in which the generating function of the variety of SL 2 -opers is identified with the universal part of the effective twisted superpotential of the corresponding four-dimensional N=2 supersymmetric theory subject to the two-dimensional Ω-deformation. This allows defining the Yang-Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group and relating it to the instanton counting of the four-dimensional gauge theories in the rank-one case.

Keywords

gauge theory supersymmetry Hitchin integrable system Darboux variable quantization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. D. Faddeev, “The Bethe Ansatz,” Preprint SFB-288-70, Sonderforschungsbereich 288, Berlin (1993).Google Scholar
  2. 2.
    C. N. Yang and C. P. Yang, J. Math. Phys., 10, 1115–1122 (1969).CrossRefADSzbMATHGoogle Scholar
  3. 3.
    C. N. Yang and C. P. Yang, Phys. Rev., 150, 321–327 (1966).CrossRefADSGoogle Scholar
  4. 4.
    E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, Theor. Math. Phys., 40, 688–706 (1979).CrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Jimbo, T. Miwa, and F. Smirnov, J. Phys. A, 42, 304018 (2009); arXiv:0811.0439v2 [math-ph] (2008).CrossRefMathSciNetGoogle Scholar
  6. 6.
    G. W. Moore, N. Nekrasov, and S. Shatashvili, Commun. Math. Phys., 209, 97–121 (2000); arXiv:hep-th/9712241v2 (1998).CrossRefADSzbMATHMathSciNetGoogle Scholar
  7. 7.
    A. A. Gerasimov and S. L. Shatashvili, Commun. Math. Phys., 277, 323–367 (2008); arXiv:hep-th/0609024v3 (2006).CrossRefADSzbMATHMathSciNetGoogle Scholar
  8. 8.
    A. A. Gerasimov and S. L. Shatashvili, “Two-dimensional gauge theories and quantum integrable systems,” in: From Hodge Theory to Integrability and TQFT: tt*-Geometry (Proc. Symp. Pure Math., Vol. 78, R. Y. Donagi and K. Wendland, eds.), Amer. Math. Soc., Providence, R. I. (2008), pp. 239–262; arXiv:0711.1472v1 [hep-th] (2007).CrossRefGoogle Scholar
  9. 9.
    E. Witten, J. Geom. Phys., 9, 303–368 (1992); arXiv:hep-th/9204083v1 (1992).CrossRefADSzbMATHMathSciNetGoogle Scholar
  10. 10.
    A. S. Gorsky and N. Nekrasov, Theor. Math. Phys., 100, 874–878 (1994).CrossRefMathSciNetGoogle Scholar
  11. 11.
    A. Gorsky and N. Nekrasov, Nucl. Phys. B, 436, 582–608 (1995); arXiv:hep-th/9401017v4 (1994).CrossRefADSzbMATHMathSciNetGoogle Scholar
  12. 12.
    A. Gorsky and N. Nekrasov, “Elliptic Calogero-Moser system from two dimensional current algebra,” arXiv:hep-th/9401021v1 (1994).Google Scholar
  13. 13.
    A. Gorsky and N. Nekrasov, Nucl. Phys. B, 414, 213–238 (1994); arXiv:hep-th/9304047v1 (1993).CrossRefADSzbMATHMathSciNetGoogle Scholar
  14. 14.
    N. A. Nekrasov and S. L. Shatashvili, Nucl. Phys. B Proc. Suppl., 192–193, 91–112 (2009); arXiv:0901.4744v2 [hep-th] (2009).CrossRefMathSciNetGoogle Scholar
  15. 15.
    N. Nekrasov and S. Shatashvili, AIP Conf. Proc., 1134, 154–169 (2009).CrossRefADSGoogle Scholar
  16. 16.
    N. A. Nekrasov and S. L. Shatashvili, Prog. Theor. Phys. Suppl., 177, 105–119 (2009); arXiv:0901.4748v2 [hep-th] (2009).CrossRefADSzbMATHGoogle Scholar
  17. 17.
    N. A. Nekrasov and S. L. Shatashvili, “Quantization of integrable systems and four dimensional gauge theories,” in: XVIth International Congress on Mathematical Physics (Prague, Czech Republic, 3–8 August 2009, P. Exne, ed.), World Scientific, Singapore (2010), pp. 265–289; arXiv:0908.4052v1 [hep-th] (2009).CrossRefGoogle Scholar
  18. 18.
    A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 355, 466–474 (1995); arXiv:hep-th/9505035v2 (1995).CrossRefADSzbMATHMathSciNetGoogle Scholar
  19. 19.
    E. Martinec and N. Warner, Nucl. Phys. B, 459, 97–112 (1996); arXiv:hep-th/9509161v2 (1995).CrossRefADSzbMATHMathSciNetGoogle Scholar
  20. 20.
    R. Donagi and E. Witten, Nucl. Phys. B, 460, 299–334 (1996); arXiv:hep-th/9510101v2 (1995).CrossRefADSzbMATHMathSciNetGoogle Scholar
  21. 21.
    R. Y. Donagi, “Seiberg-Witten integrable systems,” arXiv:alg-geom/9705010v1 (1997).Google Scholar
  22. 22.
    N. A. Nekrasov, Adv. Theor. Math. Phys., 7, 831–864 (2004); arXiv:hep-th/0206161v1 (2002).CrossRefMathSciNetGoogle Scholar
  23. 23.
    A. Losev, N. Nekrasov, and S. L. Shatashvili, “Testing Seiberg-Witten solution,” in: Strings, Branes and Dualities (Cargése France, 26 May–14 June 1997, L. Baulieu, P. Di Francesco, M. Douglas, V. Kazakov, M. Picco, and P. Windey, eds.), Kluwer, Dordrecht (1999), pp. 359–372; arXiv:hep-th/9801061v1 (1998).CrossRefGoogle Scholar
  24. 24.
    A. Losev, N. Nekrasov, and S. L. Shatashvili, Nucl. Phys. B, 534, 549–611 (1998); arXiv:hep-th/9711108v2 (1997).CrossRefADSzbMATHMathSciNetGoogle Scholar
  25. 25.
    E. Witten, “Some comments on string dynamics,” in: Proc. Strings’ 95: Future Perspectives in String Theory (USC, Los Angeles, 13–18 March 1995, I. Bars, P. Bouwknegt, J. Minahan, D. Nemeshansky, K. Pilch, H. Saleur, and N. Warner, eds.), World Scientific, Singapore (1996), pp. 501–523; arXiv:hep-th/9507121v1 (1995).Google Scholar
  26. 26.
    A. Strominger, Phys. Lett. B, 383, 44–47 (1996); arXiv:hep-th/9512059v1 (1995).CrossRefADSzbMATHMathSciNetGoogle Scholar
  27. 27.
    E. Witten, Nucl. Phys. B, 500, 3–42 (1997); arXiv:hep-th/9703166v1 (1997).CrossRefADSzbMATHMathSciNetGoogle Scholar
  28. 28.
    D. Gaiotto, “N=2 dualities,” arXiv:0904.2715v1 [hep-th] (2009).Google Scholar
  29. 29.
    D. Gaiotto, G. W. Moore, and A. Neitzke, “Wall-crossing, Hitchin systems, and the WKB approximation,” arXiv:0907.3987v2 [hep-th] (2009).Google Scholar
  30. 30.
    N. Seiberg and E. Witten, Nucl. Phys. B, 431, 484–550 (1994); arXiv:hep-th/9408099v1 (1994).CrossRefADSzbMATHMathSciNetGoogle Scholar
  31. 31.
    G. W. Moore, N. Nekrasov, and S. Shatashvili, Commun. Math. Phys., 209, 77–95 (2000); arXiv:hep-th/9803265v3 (1998).CrossRefADSzbMATHMathSciNetGoogle Scholar
  32. 32.
    E. Witten, Nucl. Phys. B, 443, 85–126 (1995); arXiv:hep-th/9503124v2 (1995).CrossRefADSzbMATHMathSciNetGoogle Scholar
  33. 33.
    M. Atiyah and R. Bott, Phil. Trans. Roy. Soc. London Ser. A, 308, 523–615 (1982).CrossRefADSMathSciNetGoogle Scholar
  34. 34.
    E. Witten, Commun. Math. Phys., 121, 351–399 (1989).CrossRefADSzbMATHMathSciNetGoogle Scholar
  35. 35.
    A. M. Polyakov, Modern Phys. Lett. A, 2, 893–898 (1987).CrossRefADSMathSciNetGoogle Scholar
  36. 36.
    A. Alekseev and S. L. Shatashvili, Nucl. Phys. B, 323, 719–733 (1989).CrossRefADSMathSciNetGoogle Scholar
  37. 37.
    E. Verlinde and H. Verlinde, Nucl. Phys. B, 348, 457–489 (1991).CrossRefADSMathSciNetGoogle Scholar
  38. 38.
    H. Verlinde, Nucl. Phys. B, 337, 652–680 (1990).CrossRefADSMathSciNetGoogle Scholar
  39. 39.
    E. Witten, Nucl. Phys. B, 311, 46–78 (1988).CrossRefADSzbMATHMathSciNetGoogle Scholar
  40. 40.
    L. Chekhov and V. V. Fock, Theor. Math. Phys., 120, 1245–1259 (1999).CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    L. O. Chekhov and V. V. Fock, Czech. J. Phys., 50, 1201–1208 (2000).CrossRefADSzbMATHMathSciNetGoogle Scholar
  42. 42.
    E. Witten, “Three-dimensional gravity revisited,” arXiv:0706.3359v1 [hep-th] (2007).Google Scholar
  43. 43.
    N. Seiberg and E. Witten, “Gauge dynamics and compactification to three dimensions,” in: The Mathematical Beauty of Physics (Adv. Ser. Math. Phys., Vol. 24, J. M. Drouffe and J. B. Zuber, eds.), World Scientific, Singapore (1997), pp. 333–366; arXiv:hep-th/9607163v1 (1996).Google Scholar
  44. 44.
    N. J. Hitchin, Proc. London Math. Soc. (3), 55, 59–126 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    A. Kapustin and E. Witten, Commun. Number Theory Phys., 1, 1–236 (2007); arXiv:hep-th/0604151v3 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    N. Hitchin, Duke Math. J., 54, 91–114 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    R. Donagi, “Spectral covers,” in: Current Topics in Complex Algebraic Geometry (MSRI Publ., Vol. 28, H. Clemens and J. Kollár, eds.), Cambridge Univ. Press, Cambridge (1995), pp. 65–86; arXiv:alg-geom/9505009v1 (1995).Google Scholar
  48. 48.
    N. Nekrasov and E. Witten, JHEP, 1009, 092 (2010); arXiv:1002.0888v2 [hep-th] (2010).CrossRefADSMathSciNetGoogle Scholar
  49. 49.
    D. Gaiotto and E. Witten, Adv. Theoret. Math. Phys., 13, 721–896; arXiv:0807.3720v1 [hep-th] (2008).CrossRefADSMathSciNetGoogle Scholar
  50. 50.
    D. Gaiotto and E. Witten, “Supersymmetric boundary conditions in N=4 super Yang-Mills theory,” arXiv:0804.2902v2 [hep-th] (2008).Google Scholar
  51. 51.
    A. Beilinson and V. Drinfeld, “Quantization of Hitchin’s integrable system and Hecke eigensheaves,” unpublished.Google Scholar
  52. 52.
    S. Gukov and E. Witten, “Branes and quantization,” arXiv:0809.0305v2 [hep-th] (2008).Google Scholar
  53. 53.
    L. F. Alday, D. Gaiotto, and Y. Tachikawa, Lett. Math. Phys., 91, 167–197 (2010); arXiv:0906.3219v2 [hep-th] (2009).CrossRefADSzbMATHMathSciNetGoogle Scholar
  54. 54.
    J. Teschner, Adv. Theor. Math. Phys., 15, 471–564 (2011); arXiv:1005.2846v2 [hep-th] (2010).CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    P. G. Zograf and L. A. Takhtadzhyan, Math. USSR-Sb., 60, 143–161 (1988).CrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    V. Fock and A. Goncharov, Publ. Math. Inst. Hautes Études Sci., 103, 1–211 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    N. Nekrasov, Commun. Math. Phys., 180, 587–603 (1996); arXiv:hep-th/9503157v4 (1995).CrossRefADSzbMATHMathSciNetGoogle Scholar
  58. 58.
    R. Donagi and E. Markman, “Spectral covers, algebraically completely integrable hamiltonian systems, and moduli of bundles,” in: Integrable Systems and Quantum Groups (Lect. Notes Math., Vol. 1620, M. Francaviglia and S. Greco, eds.) (1996), pp. 1–119; arXiv:alg-geom/9507017v2 (1995).CrossRefGoogle Scholar
  59. 59.
    A. Kapustin and S. Sethi, Adv. Theor. Math. Phys., 2, 571–591 (1998); arXiv:hep-th/9804027v2 (1998).zbMATHMathSciNetGoogle Scholar
  60. 60.
    E. Markman, Compositio Math., 93, 255–290 (1994).zbMATHMathSciNetGoogle Scholar
  61. 61.
    A. Gorsky, N. Nekrasov, and V. Rubtsov, Commun. Math. Phys., 222, 299–318 (2001); arXiv:hep-th/9901089v3 (1999).CrossRefADSzbMATHMathSciNetGoogle Scholar
  62. 62.
    V. V. Fock and A. A. Rosly, “Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix [in Russian],” Preprint ITEP-72-92, Inst. Theor. Exp. Phys., Moscow (1992); English transl., AMS Transl. Ser. 2, 191, 67–86 (1999); arXiv:math/9802054v2 (1998).Google Scholar
  63. 63.
    V. V. Fock and A. A. Roslyi, Theor. Math. Phys., 95, 526–534 (1993).CrossRefzbMATHMathSciNetGoogle Scholar
  64. 64.
    V. V. Fock and A. A. Rosly, Internat. J. Mod. Phys. B, 11, 3195–3206 (1997).CrossRefADSzbMATHMathSciNetGoogle Scholar
  65. 65.
    W. M. Goldman, Adv. Math., 54, 200–225 (1984).CrossRefzbMATHGoogle Scholar
  66. 66.
    W. M. Goldman, Invent. Math., 85, 263–302 (1986).CrossRefADSzbMATHMathSciNetGoogle Scholar
  67. 67.
    V. G. Turaev, Ann. Sci. École Norm. Sup. (4), 24, 635–704 (1991).zbMATHMathSciNetGoogle Scholar
  68. 68.
    A. N. Tyurin, Quantization, Classical and Quantum Field Theory, and Theta Functions (CRM Monogr. Ser., Vol. 21), Amer. Math. Soc., Providence, R. I. (2003).zbMATHGoogle Scholar
  69. 69.
    D. A. Derevnin and A. D. Mednykh, Russ. Math. Surveys, 60, 346–348 (2005).CrossRefADSzbMATHMathSciNetGoogle Scholar
  70. 70.
    L. Schläfli, Theorie der vielfachen Kontinuität (Gesammelte Mathematishe Abhandlungen, Vol. 1), Birkhäuser, Basel (1950).Google Scholar
  71. 71.
    N. I. Lobatschefskij, “Imaginäre Geometrie und ihre Anwendung auf einige Integrale,” in: Deutsche Übersetzung von H. Liebmann, Teubner, Leipzig (1904).Google Scholar
  72. 72.
    Yu. Cho and H. Kim, Discrete Comput. Geom., 22, 347–366 (1999).CrossRefzbMATHMathSciNetGoogle Scholar
  73. 73.
    J. Milnor, “The Schläfli differential equality,” in: Collected Papers, Vol. 1, Geometry, Publish or Perish, Houston, Tex. (1994), pp. 281–295.Google Scholar
  74. 74.
    M. Kapovich, J. J. Millson, and T. Treloar, “The symplectic geometry of polygons in hyperbolic 3-space,” arXiv:math/9907143v2 (1999).Google Scholar
  75. 75.
    A. A. Klyachko, “Spatial polygons and stable configurations of points in the projective line,” in: Algebraic Geometry and Its Applications (Asp. Math., Vol. 25), Vieweg, Braunschweig (1994), pp. 67–84.CrossRefGoogle Scholar
  76. 76.
    P. Foth, J. Geom. Phys., 58, 825–832 (2007); arXiv:math/0703525v2 (2007).CrossRefADSMathSciNetGoogle Scholar
  77. 77.
    A. Beilinson and V. Drinfeld, “Opers,” arXiv:math.AG/0501398v1 (2005).Google Scholar
  78. 78.
    A. Bilal, I. Kogan, and V. Fock, “On the origin of W-algebras,” Preprint CERN-TH.5965/90, CERN, Geneva (1990).Google Scholar
  79. 79.
    A. Gerasimov, A. Levin, and A. Marshakov, Nucl. Phys. B, 360, 537–558 (1991).CrossRefADSMathSciNetGoogle Scholar
  80. 80.
    N. Nekrasov and V. Pestun, “Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories,” arXiv:1211.2240v1 [hep-th] (2012).Google Scholar
  81. 81.
    E. Sklyanin, J. Sov. Math., 47, 2473–2488 (1989).CrossRefzbMATHMathSciNetGoogle Scholar
  82. 82.
    E. Frenkel, “Affine algebras, langlands duality, and Bethe ansatz,” in: XIth International Congress on Mathematical Physics (D. Iagolnitzer, ed.), International Press, Cambridge, Mass. (1995), pp. 606–642; arXiv:q-alg/9506003v3 (1995).Google Scholar
  83. 83.
    A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov, and S. L. Shatashvili, Internat. J. Mod. Phys. A, 5, 2495–2589 (1990).CrossRefADSMathSciNetGoogle Scholar
  84. 84.
    V. Schechtman and A. Varchenko, “Integral representations of N-point conformal correlators in the WZW model,” Preprint MPI/89-51, Max-Planck Institut, Bonn (1989).Google Scholar
  85. 85.
    B. Feigin, E. Frenkel, and N. Reshetikhin, Commun. Math. Phys., 166, 27–62 (1994); arXiv:hep-th/9402022v2 (1994).CrossRefADSzbMATHMathSciNetGoogle Scholar
  86. 86.
    N. Reshetikhin and A. Varchenko, “Quasiclassical asymptotics of solutions to the KZ equations,” in: Geometry, Topology, and Physics for Raoul Bott (Conf. Proc. Lect. Notes Geom. Topol., Vol. 4, S.-T. Yau, ed.), International Press, Cambridge, Mass. (1995), pp. 293–322; arXiv:hep-th/9402126v3 (1994).Google Scholar
  87. 87.
    G. Felder and A. Varchenko, Compositio Math., 107, 143–175 (1997); arXiv:hep-th/9511120v1 (1995).CrossRefzbMATHMathSciNetGoogle Scholar
  88. 88.
    A. B. Zamolodchikov and Al. B. Zamolodchikov, Nucl. Phys. B, 477, No. 2, 577–605 (1996); arXiv:hep-th/9506136v2 (1995).CrossRefADSzbMATHMathSciNetGoogle Scholar
  89. 89.
    E. Aldrovandi and L. A. Takhtajan, Commun. Math. Phys., 227, 303–348 (2002); arXiv:math/0006147v1 (2000).CrossRefADSzbMATHMathSciNetGoogle Scholar
  90. 90.
    L. F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa, and H. Verlinde, JHEP, 1001, 113 (2010); arXiv:0909.0945v3 [hep-th] (2009).CrossRefADSMathSciNetGoogle Scholar
  91. 91.
    D. Gaiotto, “Asymptotically free N=2 theories and irregular conformal blocks,” arXiv:0908.0307v1 [hep-th] (2009).Google Scholar
  92. 92.
    E. Witten, Anal. Appl. (Singapore), 6, 429–501 (2008); arXiv:0710.0631v1 [hep-th] (2007).CrossRefzbMATHMathSciNetGoogle Scholar
  93. 93.
    B. Feigin, E. Frenkel, and V. Toledano-Laredo, Adv. Math., 223, 873–948 (2010); arXiv:math/0612798v3 (2006).CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • N. A. Nekrasov
    • 1
    • 2
    • 3
    • 4
    Email author
  • A. A. Rosly
    • 4
    • 3
  • S. L. Shatashvili
    • 3
    • 5
    • 6
    • 7
    • 8
  1. 1.Simons Center for Geometry and PhysicsStony Brook UniversityStony BrookUSA
  2. 2.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  3. 3.Institute for Information Transmission ProblemsMoscowRussia
  4. 4.Institute for Theoretical and Experimental PhysicsMoscowRussia
  5. 5.Hamilton Mathematics InstituteTrinity CollegeDublinIreland
  6. 6.School of MathematicsTrinity CollegeDublinIreland
  7. 7.Louis Michel ChairInstitut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  8. 8.Euler International Mathematical InstituteSt. PetersburgRussia

Personalised recommendations