Advertisement

Communications in Mathematical Physics

, Volume 358, Issue 3, pp 1041–1064 | Cite as

Quantum Hitchin Systems via \({\beta}\)-Deformed Matrix Models

  • Giulio Bonelli
  • Kazunobu Maruyoshi
  • Alessandro Tanzini
Article
  • 66 Downloads

Abstract

We study the quantization of Hitchin systems in terms of \({\beta}\)-deformations of generalized matrix models related to conformal blocks of Liouville theory on punctured Riemann surfaces. We show that in a suitable limit, corresponding to the Nekrasov–Shatashvili one, the loop equations of the matrix model reproduce the Hamiltonians of the quantum Hitchin system on the sphere and the torus with marked points. The eigenvalues of these Hamiltonians are shown to be the \({\epsilon_1}\)-deformation of the chiral observables of the corresponding \({{\mathcal{N}=2}}\) four dimensional gauge theory. Moreover, we find the exact wave-functions in terms of the matrix model representation of the conformal blocks with degenerate field insertions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in \({N = 2}\) supersymmetric Yang–Mills theory. Nucl. Phys. B 426, 19 (1994) [Erratum-ibid. B 430, 485 (1994)] arXiv:hep-th/9407087
  2. 2.
    Seiberg N., Witten E.: Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nucl. Phys. B 431, 484 (1994) arXiv:hep-th/9408099 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gorsky A., Krichever I., Marshakov A., Mironov A., Morozov A.: Integrability and Seiberg–Witten exact solution. Phys. Lett. B 355, 466 (1995) arXiv:hep-th/9505035 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Martinec E.J., Warner N.P.: Integrable systems and supersymmetric gauge theory. Nucl. Phys. B 459, 97 (1996) arXiv:hep-th/9509161 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Nakatsu T., Takasaki K.: Whitham–Toda hierarchy and N = 2 supersymmetric Yang–Mills theory. Mod. Phys. Lett. A 11, 157 (1996) arXiv:hep-th/9509162 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Itoyama H., Morozov A.: Integrability and Seiberg–Witten theory: curves and periods. Nucl. Phys. B 477, 855 (1996) arXiv:hep-th/9511126 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Itoyama H., Morozov A.: Prepotential and the Seiberg–Witten theory. Nucl. Phys. B 491, 529 (1997) arXiv:hep-th/9512161 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gorsky A., Marshakov A., Mironov A., Morozov A.: N = 2 supersymmetric QCD and integrable spin chains: rational case \({N_f < 2N_c}\). Phys. Lett. B 380, 75 (1996) arXiv:hep-th/9603140 ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Donagi R., Witten E.: Supersymmetric Yang–Mills theory and integrable systems. Nucl. Phys. B 460, 299 (1996) arXiv:hep-th/9510101 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gaiotto D.: N N = 2 dualities. JHEP i208, 034 (2012) arXiv:0904.2715 [hep-th]ADSCrossRefGoogle Scholar
  11. 11.
    Gaiotto D., Moore G.W., Neitzke A.: Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys. 299, 163 (2010) arXiv:0807.4723 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nekrasov N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7(5), 831–864 (2003) arXiv:hep-th/0306211 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nekrasov, N., Okounkov, A.: Seiberg–Witten theory and random partitions, In: Etingof, P. et al. (eds.) The Unity Mathematics. Progress in Mathematics 244. Birkäuser, Boston (2006) arXiv:hep-th/0306238
  14. 14.
    Flume R., Poghossian R.: An Algorithm for the microscopic evaluation of the coefficients of the Seiberg–Witten prepotential. Int. J. Mod. Phys. A 18, 2541 (2003) arXiv:hep-th/0208176 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bruzzo U., Fucito F., Morales J.F., Tanzini A.: Multiinstanton calculus and equivariant cohomology. JHEP 0305, 054 (2003) arXiv:hep-th/0211108 ADSCrossRefGoogle Scholar
  16. 16.
    Fucito F., Morales J.F., Poghossian R.: Instantons on quivers and orientifolds. JHEP 0410, 037 (2004) arXiv:hep-th/0408090 ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Alday L.F., Gaiotto D., Tachikawa Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167 (2010) arXiv:0906.3219 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bonelli G., Tanzini A.: Hitchin systems, N = 2 gauge theories and W-gravity. Phys. Lett. B 691, 111 (2010) arXiv:0909.4031 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Teschner J.: Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I. Adv. Theor. Math. Phys. 15(2), 471–564 (2011) arXiv:1005.2846 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. arXiv:0908.4052 [hep-th]
  21. 21.
    Mironov A., Morozov A.: Nekrasov functions and exact Bohr–Sommerfeld integrals. JHEP 1004, 040 (2010) arXiv:0910.5670 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mironov A., Morozov A.: Nekrasov functions from exact BS periods: the case of SU(N). J. Phys. A 43, 195401 (2010) arXiv:0911.2396 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Popolitov, A.: On relation between Nekrasov functions and BS periods in pure SU(N) case. arXiv:1001.1407 [hep-th]
  24. 24.
    Nekrasov N., Witten E.: The omega deformation, branes, integrability, and Liouville theory. JHEP 1009, 092 (2010) arXiv:1002.0888 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Orlando D., Reffert S.: Relating gauge theories via Gauge/Bethe correspondence. JHEP 1010, 071 (2010) arXiv:1005.4445 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    He W., Miao Y.G.: On the magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory. Phys. Rev. D 82, 025020 (2010) arXiv:1006.1214 [hep-th]ADSCrossRefGoogle Scholar
  27. 27.
    Kozlowski, K.K., Teschner, J.: TBA for the Toda chain, In: New trends in quantum integrable systems. In: Feigih, B. et al. (eds.) Proceedings of the Infinite Analysis 09, pp. 195–219 (2010) arXiv:1006.2906 [math-ph]
  28. 28.
    Poghossian R.: Deforming SW curve. JHEP 1104, 033 (2011) arXiv:1006.4822 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Marshakov, A., Mironov, A., Morozov, A.: On AGT relations with surface operator insertion and stationary limit of beta-ensembles. Teor. Mat. Fiz. 164:1:3 (2010). arXiv:1011.4491 [hep-th]
  30. 30.
    Yamada Y.: A quantum isomonodromy equation and its application to N = 2 SU(N) gauge theories. J. Phys. A 44, 055403 (2011) arXiv:1011.0292 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Piatek M.: Classical conformal blocks from TBA for the elliptic Calogero–Moser system. JHEP 1106, 050 (2011) arXiv:1102.5403 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nekrasov N., Rosly A., Shatashvili S.: Darboux coordinates, Yang–Yang functional, and gauge theory. Nucl. Phys. B Proc. Suppl. 216(1), 69–93 (2011) arXiv:1103.3919 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Fucito, F., Morales, J.F., Poghossian, R., Pacifici, D.R.: gauge theories on \({\Omega}\)-backgrounds from non commutative Seiberg–Witten curves. JHEP 1105:098 (2011). arXiv:1103.4495 [hep-th]
  34. 34.
    Zenkevich Y.: Nekrasov prepotential with fundamental matter from the quantum spin chain. Phys. Lett. B 701(5), 630–639 (2011) arXiv:1103.4843 [math-ph]ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Dorey N., Hollowood T.J., Lee S.: Quantization of integrable systems and a 2d/4d duality. JHEP 1110, 077 (2011) arXiv:1103.5726 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Chen, H.-Y., Dorey, N., Hollowood, T.J., Lee, S.: A new 2d/4d duality via integrability. JHEP 1109:040 (2011). arXiv:1104.3021 [hep-th]
  37. 37.
    Alday L.F., Tachikawa Y.: Affine SL(2) conformal blocks from 4d gauge theories. Lett. Math. Phys. 94, 87–114 (2010) arXiv:1005.4469 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Maruyoshi K., Taki M.: Deformed prepotential, quantum integrable system and Liouville field theory. Nucl. Phys. B 841(3), 388–425 (2010) arXiv:1006.4505 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Alday L.F., Gaiotto D., Gukov S., Tachikawa Y., Verlinde H.: Loop and surface operators in N = 2 gauge theory and Liouville modular geometry. JHEP 1001, 113 (2010) arXiv:0909.0945 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Dimofte T., Gukov S., Hollands L.: Vortex counting and Lagrangian 3-manifolds. Lett. Math. Phys. 98(3), 225–287 (2011) arXiv:1006.0977 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kozcaz C., Pasquetti S., Wyllard N.: A&B model approaches to surface operators and Toda theories. JHEP 1008, 042 (2010) arXiv:1004.2025 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Taki M.: Surface operator, bubbling Calabi–Yau and AGT relation. JHEP 1107, 047 (2011) arXiv:1007.2524 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Awata H., Fuji H., Kanno H., Manabe M., Yamada Y.: Localization with a surface operator, irregular conformal blocks and open topological string. Adv. Theor. Math. Phys. 16(3), 725–804 (2012) arXiv:1008.0574 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Kozcaz C., Pasquetti S., Passerini F., Wyllard N.: Affine sl(N) conformal blocks from N = 2 SU(N) gauge theories. JHEP 1101, 045 (2011) arXiv:1008.1412 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Bonelli G., Tanzini A., Zhao J.: Vertices, vortices and interacting surface operators. JHEP 1206, 178 (2012) arXiv:1102.0184 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Dijkgraaf, R., Vafa, C.: Toda theories, matrix models, topological strings, and N = 2 Gauge systems. arXiv:0909.2453 [hep-th]
  47. 47.
    Itoyama H., Maruyoshi K., Oota T.: The Quiver matrix model and 2d–4d conformal connection. Prog. Theor. Phys. 123, 957 (2010) arXiv:0911.4244 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  48. 48.
    Eguchi T., Maruyoshi K.: Penner type matrix model and Seiberg–Witten theory. JHEP 1002, 022 (2010) arXiv:0911.4797 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Schiappa R., Wyllard N.: An A r threesome: matrix models, 2d CFTs and 4d N = 2 gauge theories. J. Math. Phys. 51, 082304 (2010) arXiv:0911.5337 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Mironov A., Morozov A., Shakirov S.: Matrix model conjecture for exact BS periods and Nekrasov functions. JHEP 1002, 030 (2010) arXiv:0911.5721 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Fujita M., Hatsuda Y., Tai T.-S.: Genus-one correction to asymptotically free Seiberg–Witten prepotential from Dijkgraaf–Vafa matrix model. JHEP 1003, 046 (2010) arXiv:0912.2988 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Sulkowski P.: Matrix models for beta-ensembles from Nekrasov partition functions. JHEP 1004, 063 (2010) arXiv:0912.5476 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Mironov A., Morozov A., Shakirov S.: Conformal blocks as Dotsenko–Fateev integral discriminants. Int. J. Mod. Phys. A 25, 3173 (2010) arXiv:1001.0563 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Itoyama H., Oota T.: Method of generating q-expansion coefficients for conformal block and N = 2 Nekrasov function by beta-deformed matrix model. Nucl. Phys. B 838, 298 (2010) arXiv:1003.2929 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  55. 55.
    Mironov A., Morozov A., Morozov A.: Matrix model version of AGT conjecture and generalized Selberg integrals. Nucl. Phys. B 843(2), 534–557 (2011) arXiv:1003.5752 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  56. 56.
    Eguchi T., Maruyoshi K.: Seiberg–Witten theory, matrix model and AGT relation. JHEP 1007, 081 (2010) arXiv:1006.0828 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Itoyama H., Oota T., Yonezawa N.: Massive scaling limit of beta-deformed matrix model of Selberg type. Phys. Rev. D. 82, 085031 (2010) arXiv:1008.1861 [hep-th]ADSCrossRefGoogle Scholar
  58. 58.
    Brini A., Marino M., Stevan S.: The uses of the refined matrix model recursion. J. Math. Phys. 32, 052305 (2011) arXiv:1010.1210 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Cheng M.C.N., Dijkgraaf R., Vafa C.: Non-perturbative topological strings and conformal blocks. JHEP 1109, 022 (2011) arXiv:1010.4573 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Santillan, O.P.: Geometric transitions, double scaling limits and gauge theories. arXiv:1103.1422 [hep-th]
  61. 61.
    Mironov A., Morozov A., Popolitov A., Shakirov S.: Resolvents and Seiberg–Witten representation for Gaussian beta-ensemble. Theor. Math. Phys. 171(1), 505–522 (2012) arXiv:1103.5470 [hep-th]CrossRefzbMATHGoogle Scholar
  62. 62.
    Itoyama H., Yonezawa N.: \({\epsilon}\)-Corrected Seiberg–Witten prepotential obtained from half genus expansion in beta-deformed matrix model. Int. J. Mod. Phys. A 26, 3439–3467 (2011) arXiv:1104.2738 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  63. 63.
    Maruyoshi K., Yagi F.: Seiberg–Witten curve via generalized matrix model. JHEP 1101, 042 (2011) arXiv:1009.5553 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Mironov A., Morozov A., Shakirov S.: On ‘Dotsenko–Fateev’ representation of the toric conformal blocks. J. Phys. A 44, 085401 (2011) arXiv:1010.1734 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Bonelli G., Maruyoshi K., Tanzini A., Yagi F.: Generalized matrix models and AGT correspondence at all genera. JHEP 1107, 055 (2011) arXiv:1011.5417 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Eynard B., Marchal O.: Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry. JHEP 0903, 094 (2009) arXiv:0809.3367 [math-ph]ADSMathSciNetCrossRefGoogle Scholar
  67. 67.
    Chekhov, L., Eynard, B., Marchal, O.: Topological expansion of the Bethe ansatz, and quantum algebraic geometry. arXiv:0911.1664 [math-ph]
  68. 68.
    Chekhov L.O., Eynard B., Marchal O.: Topological expansion of \({\beta}\)-ensemble model and quantum algebraic geometry in the sectorwise approach. Theor. Math. Phys. 166, 141–185 (2011) arXiv:1009.6007 [math-ph]MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Frenkel, E.: Lectures on the Langlands program and conformal field theory, In: Cartier, P. et al. (eds.) Frontiers in Number Theory, Physics, Geometry II, vol. 88, pp. 387–533. Springer, Berlin (2007). arXiv:hep-th/0512172
  70. 70.
    Witten E.: Solutions of four-dimensional field theories via M theory. Nucl. Phys. B500, 3–42 (1997) arXiv:hep-th/9703166 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Gawedzki K., Tran-Ngoc-Bich P.: Hitchin systems at low genera. J. Math. Phys. 41, 4695–4712 (2000) arXiv:hep-th/9803101 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Dotsenko V.S., Fateev V.A.: Conformal algebra and multipoint correlation functions in 2D statistical models. Nucl. Phys. B 240, 312 (1984)ADSCrossRefGoogle Scholar
  73. 73.
    Dotsenko V.S., Fateev V.A.: Four point correlation functions and the operator algebra in the two-dimensional conformal invariant theories with the central charge C \({<}\) 1. Nucl. Phys. B 251, 691 (1985)ADSCrossRefGoogle Scholar
  74. 74.
    Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Flume R., Fucito F., Morales J.F., Poghossian R.: Matone’s relation in the presence of gravitational couplings. JHEP 0404, 008 (2004) arXiv:hep-th/0403057 ADSMathSciNetCrossRefGoogle Scholar
  76. 76.
    Fucito F., Morales J.F., Poghossian R., Tanzini A.: N = 1 superpotentials from multi-instanton calculus. JHEP 0601, 031 (2006) arXiv:hep-th/0510173 ADSMathSciNetCrossRefGoogle Scholar
  77. 77.
    Matone M.: Instantons and recursion relations in N = 2 SUSY gauge theory. Phys. Lett. B 357, 342 (1995) arXiv:hep-th/9506102 ADSMathSciNetCrossRefGoogle Scholar
  78. 78.
    Sonnenschein J., Theisen S., Yankielowicz S.: On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories. Phys. Lett. B 367, 145 (1996) arXiv:hep-th/9510129 ADSMathSciNetCrossRefGoogle Scholar
  79. 79.
    Eguchi T., Yang S.K.: Prepotentials of N = 2 supersymmetric gauge theories and soliton equations. Mod. Phys. Lett. A 11, 131 (1996) arXiv:hep-th/9510183 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Knizhnik V.G., Zamolodchikov A.B.: Current algebra and Wess–Zumino model in two dimensions. Nucl. Phys. B 247, 83 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Goulian M., Li M.: Correlation functions in Liouville theory. Phys. Rev. Lett. 66, 2051 (1991)ADSCrossRefGoogle Scholar
  82. 82.
    Bernard D.: On the Wess–Zumino–Witten models on the torus. Nucl. Phys. B 303, 77 (1988)ADSMathSciNetCrossRefGoogle Scholar
  83. 83.
    Etingof, P.I., Kirillov, A.A.: Representation of affine Lie algebras, parabolic differential equations and Lame functions. arXiv:hep-th/9310083
  84. 84.
    Felder G., Weiczerkowski C.: Conformal blocks on elliptic curves and the Knizhnik–Zamolodchikov–Bernard equations. Commun. Math. Phys. 176, 133–162 (1996) arXiv:hep-th/9411004 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    Eguchi T., Ooguri H.: Conformal and current algebras on general Riemann surface. Nucl. Phys. B 282, 308 (1987)ADSMathSciNetCrossRefGoogle Scholar
  86. 86.
    Dorey N., Khoze V.V., Mattis M.P.: On mass-deformed N = 4 supersymmetric Yang–Mills theory. Phys. Lett. B 396, 141 (1997) arXiv:hep-th/9612231 ADSMathSciNetCrossRefGoogle Scholar
  87. 87.
    Fateev V.A., Litvinov A.V.: On AGT conjecture. JHEP 1002, 014 (2010) arXiv:0912.0504 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  88. 88.
    Awata H., Yamada Y.: Five-dimensional AGT relation and the deformed beta-ensemble. Prog. Theor. Phys. 124, 227 (2010) arXiv:1004.5122 [hep-th]ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.International School of Advanced Studies (SISSA)TriesteItaly
  2. 2.INFN, Sezione di TriesteTriesteItaly
  3. 3.Faculty of Science and Technology Seikei UniversityTokyoJapan

Personalised recommendations