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

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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.

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References

  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. 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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Martinec E.J., Warner N.P.: Integrable systems and supersymmetric gauge theory. Nucl. Phys. B 459, 97 (1996) arXiv:hep-th/9509161

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Itoyama H., Morozov A.: Integrability and Seiberg–Witten theory: curves and periods. Nucl. Phys. B 477, 855 (1996) arXiv:hep-th/9511126

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Itoyama H., Morozov A.: Prepotential and the Seiberg–Witten theory. Nucl. Phys. B 491, 529 (1997) arXiv:hep-th/9512161

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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

    Article  ADS  MathSciNet  Google Scholar 

  9. Donagi R., Witten E.: Supersymmetric Yang–Mills theory and integrable systems. Nucl. Phys. B 460, 299 (1996) arXiv:hep-th/9510101

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Gaiotto D.: N N = 2 dualities. JHEP i208, 034 (2012) arXiv:0904.2715 [hep-th]

    Article  ADS  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Nekrasov N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7(5), 831–864 (2003) arXiv:hep-th/0306211

    Article  MathSciNet  MATH  Google Scholar 

  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. 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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Bruzzo U., Fucito F., Morales J.F., Tanzini A.: Multiinstanton calculus and equivariant cohomology. JHEP 0305, 054 (2003) arXiv:hep-th/0211108

    Article  ADS  Google Scholar 

  16. Fucito F., Morales J.F., Poghossian R.: Instantons on quivers and orientifolds. JHEP 0410, 037 (2004) arXiv:hep-th/0408090

    Article  ADS  MathSciNet  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  20. Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. arXiv:0908.4052 [hep-th]

  21. Mironov A., Morozov A.: Nekrasov functions and exact Bohr–Sommerfeld integrals. JHEP 1004, 040 (2010) arXiv:0910.5670 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Popolitov, A.: On relation between Nekrasov functions and BS periods in pure SU(N) case. arXiv:1001.1407 [hep-th]

  24. Nekrasov N., Witten E.: The omega deformation, branes, integrability, and Liouville theory. JHEP 1009, 092 (2010) arXiv:1002.0888 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Orlando D., Reffert S.: Relating gauge theories via Gauge/Bethe correspondence. JHEP 1010, 071 (2010) arXiv:1005.4445 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  Google Scholar 

  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. Poghossian R.: Deforming SW curve. JHEP 1104, 033 (2011) arXiv:1006.4822 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. 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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Piatek M.: Classical conformal blocks from TBA for the elliptic Calogero–Moser system. JHEP 1106, 050 (2011) arXiv:1102.5403 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. 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]

    Article  ADS  MathSciNet  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. 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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Taki M.: Surface operator, bubbling Calabi–Yau and AGT relation. JHEP 1107, 047 (2011) arXiv:1007.2524 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Bonelli G., Tanzini A., Zhao J.: Vertices, vortices and interacting surface operators. JHEP 1206, 178 (2012) arXiv:1102.0184 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  46. Dijkgraaf, R., Vafa, C.: Toda theories, matrix models, topological strings, and N = 2 Gauge systems. arXiv:0909.2453 [hep-th]

  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]

    Article  ADS  MATH  Google Scholar 

  48. Eguchi T., Maruyoshi K.: Penner type matrix model and Seiberg–Witten theory. JHEP 1002, 022 (2010) arXiv:0911.4797 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Sulkowski P.: Matrix models for beta-ensembles from Nekrasov partition functions. JHEP 1004, 063 (2010) arXiv:0912.5476 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MATH  Google Scholar 

  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]

    Article  ADS  MATH  Google Scholar 

  56. Eguchi T., Maruyoshi K.: Seiberg–Witten theory, matrix model and AGT relation. JHEP 1007, 081 (2010) arXiv:1006.0828 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. Santillan, O.P.: Geometric transitions, double scaling limits and gauge theories. arXiv:1103.1422 [hep-th]

  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]

    Article  MATH  Google Scholar 

  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]

    Article  ADS  MATH  Google Scholar 

  63. Maruyoshi K., Yagi F.: Seiberg–Witten curve via generalized matrix model. JHEP 1101, 042 (2011) arXiv:1009.5553 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  Google Scholar 

  67. Chekhov, L., Eynard, B., Marchal, O.: Topological expansion of the Bethe ansatz, and quantum algebraic geometry. arXiv:0911.1664 [math-ph]

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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. Witten E.: Solutions of four-dimensional field theories via M theory. Nucl. Phys. B500, 3–42 (1997) arXiv:hep-th/9703166 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  72. Dotsenko V.S., Fateev V.A.: Conformal algebra and multipoint correlation functions in 2D statistical models. Nucl. Phys. B 240, 312 (1984)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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

    Article  ADS  MathSciNet  Google Scholar 

  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

    Article  ADS  MathSciNet  Google Scholar 

  77. Matone M.: Instantons and recursion relations in N = 2 SUSY gauge theory. Phys. Lett. B 357, 342 (1995) arXiv:hep-th/9506102

    Article  ADS  MathSciNet  Google Scholar 

  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

    Article  ADS  MathSciNet  Google Scholar 

  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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  80. Knizhnik V.G., Zamolodchikov A.B.: Current algebra and Wess–Zumino model in two dimensions. Nucl. Phys. B 247, 83 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. Goulian M., Li M.: Correlation functions in Liouville theory. Phys. Rev. Lett. 66, 2051 (1991)

    Article  ADS  Google Scholar 

  82. Bernard D.: On the Wess–Zumino–Witten models on the torus. Nucl. Phys. B 303, 77 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  83. Etingof, P.I., Kirillov, A.A.: Representation of affine Lie algebras, parabolic differential equations and Lame functions. arXiv:hep-th/9310083

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  85. Eguchi T., Ooguri H.: Conformal and current algebras on general Riemann surface. Nucl. Phys. B 282, 308 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  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

    Article  ADS  MathSciNet  Google Scholar 

  87. Fateev V.A., Litvinov A.V.: On AGT conjecture. JHEP 1002, 014 (2010) arXiv:0912.0504 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MATH  Google Scholar 

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  1. Kazunobu Maruyoshi

    Present address: Faculty of Science and Technology Seikei University, 3-3-1 Kichijoji-Kitamachi, Musashino, Tokyo, 180-8633, Japan

Authors and Affiliations

  1. International School of Advanced Studies (SISSA), via Bonomea 265, 34136, Trieste, Italy

    Giulio Bonelli, Kazunobu Maruyoshi & Alessandro Tanzini

  2. INFN, Sezione di Trieste, Trieste, Italy

    Giulio Bonelli, Kazunobu Maruyoshi & Alessandro Tanzini

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  1. Giulio Bonelli
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Bonelli, G., Maruyoshi, K. & Tanzini, A. Quantum Hitchin Systems via \({\beta}\)-Deformed Matrix Models. Commun. Math. Phys. 358, 1041–1064 (2018). https://doi.org/10.1007/s00220-017-3053-0

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