On a complete solution of the quantum Dell system

1. H.W. Braden, A. Marshakov, A. Mironov and A. Morozov, On double elliptic integrable systems. 1. A Duality argument for the case of SU(2), Nucl. Phys.B 573 (2000) 553 [hep-th/9906240] [INSPIRE].

2. S.N.M. Ruijsenaars, Action Angle Maps and Scattering Theory for Some Finite Dimensional Integrable Systems. 1. The Pure Soliton Case, Commun. Math. Phys.115 (1988) 127 [INSPIRE].

3. V. Fock, Three Remarks on Group Invariants Related to Flat Connections, in Geometry and Integrable Models, P. Pyatov and S. Solodukhin eds., World Scientific (1995), p. 20.

4. V.V. Fock and A.A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and r matrix, Am. Math. Soc. Transl.191 (1999) 67 [math/9802054] [INSPIRE].

5. V. Fock, A. Gorsky, N. Nekrasov and V. Rubtsov, Duality in integrable systems and gauge theories, JHEP07 (2000) 028 [hep-th/9906235] [INSPIRE].

   Article  ADS  MathSciNet  MATH  Google Scholar 

6. A. Gorsky and V. Rubtsov, Dualities in integrable systems: Geometrical aspects, in Integrable structures of exactly solvable two-dimensional models of quantum field theory. Proceedings, NATO Advanced Research Workshop on dynamical symmetries of integrable quantum field theories and lattice models, Kiev, Ukraine, 25–30 September 2000, pp. 173–198 (2000) [hep-th/0103004] [INSPIRE].

7. A. Mironov, Seiberg-Witten theory and duality in integrable systems, in 34th Annual Winter School on Nuclear and Particle Physics (PNPI 2000), Gatchina, Russia, 14–20 February 2000 (2000) [hep-th/0011093] [INSPIRE].

8. A. Gorsky and A. Mironov, Integrable many body systems and gauge theories, hep-th/0011197 [INSPIRE].

9. A. Mironov and A. Morozov, Commuting Hamiltonians from Seiberg-Witten theta functions, Phys. Lett.B 475 (2000) 71 [hep-th/9912088] [INSPIRE].

   Article  ADS  MATH  Google Scholar 

10. A. Mironov and A. Morozov, Double elliptic systems: Problems and perspectives, in Proceedings, International Workshop on Supersymmetries and Quantum Symmetries (SQS’99), Moscow, Russia, 27–31 July 1999 (1999) [hep-th/0001168] [INSPIRE].

11. A. Mironov, Selfdual Hamiltonians as deformations of free systems, Theor. Math. Phys.129 (2001) 1581 [hep-th/0104253] [INSPIRE].

    Article  MATH  Google Scholar 

12. A. Mironov, Integrability in string/field theories and Hamiltonian flows in the space of physical systems, Theor. Math. Phys.135 (2003) 814 [hep-th/0205202] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

13. A. Mironov and A. Morozov, p,q duality and Hamiltonian flows in the space of integrable systems or integrable systems as canonical transforms of the free ones, Phys. Lett.B 524 (2002) 217 [hep-th/0107114] [INSPIRE].

14. E. Mukhin, V. Tarasov, A. Varchenko, Bispectral and (\( \mathfrak{gl} \)_N, \( \mathfrak{gl} \)_M ) Dualities, math/0510364.

15. E.Mukhin, V.Tarasov, A.Varchenko, Bispectral and (\( \mathfrak{gl} \)_N, \( \mathfrak{gl} \)_M ) dualities, discrete versus differential, Adv. Math.218 (2008) 216 [math/0605172].

16. A. Mironov, A. Morozov, Y. Zenkevich and A. Zotov, Spectral Duality in Integrable Systems from AGT Conjecture, JETP Lett.97 (2013) 45 [arXiv:1204.0913] [INSPIRE].

    Article  ADS  Google Scholar 

17. A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral Duality Between Heisenberg Chain and Gaudin Model, Lett. Math. Phys.103 (2013) 299 [arXiv:1206.6349] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

18. A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral dualities in XXZ spin chains and five dimensional gauge theories, JHEP12 (2013) 034 [arXiv:1307.1502] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

19. L. Bao, E. Pomoni, M. Taki and F. Yagi, M5-Branes, Toric Diagrams and Gauge Theory Duality, JHEP04 (2012) 105 [arXiv:1112.5228] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

20. A. Mironov, A. Morozov and Y. Zenkevich, On elementary proof of AGT relations from six dimensions, Phys. Lett.B 756 (2016) 208 [arXiv:1512.06701] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

21. A. Mironov, A. Morozov and Y. Zenkevich, Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings, JHEP05 (2016) 121 [arXiv:1603.00304] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

22. H. Awata, H. Kanno, A. Mironov, A. Morozov, K. Suetake and Y. Zenkevich, (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces, JHEP03 (2018) 192 [arXiv:1712.08016] [INSPIRE].

23. G. Aminov, A. Mironov and A. Morozov, New non-linear equations and modular form expansion for double-elliptic Seiberg-Witten prepotential, Eur. Phys. J.C 76 (2016) 433 [arXiv:1606.05274] [INSPIRE].

    Article  ADS  Google Scholar 

24. G. Aminov, A. Mironov and A. Morozov, Modular properties of 6d (DELL) systems, JHEP11 (2017) 023 [arXiv:1709.04897] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

25. H.W. Braden and T.J. Hollowood, The Curve of compactified 6-D gauge theories and integrable systems, JHEP12 (2003) 023 [hep-th/0311024] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

26. G. Aminov, A. Mironov, A. Morozov and A. Zotov, Three-particle Integrable Systems with Elliptic Dependence on Momenta and Theta Function Identities, Phys. Lett.B 726 (2013) 802 [arXiv:1307.1465] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

27. G. Aminov, H.W. Braden, A. Mironov, A. Morozov and A. Zotov, Seiberg-Witten curves and double-elliptic integrable systems, JHEP01 (2015) 033 [arXiv:1410.0698] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

28. N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09), Prague, Czech Republic, 3–8 August 2009, pp. 265–289 (2009) [DOI] [arXiv:0908.4052] [INSPIRE].

29. N. Nekrasov, A. Rosly and S. Shatashvili, Darboux coordinates, Yang-Yang functional and gauge theory, Nucl. Phys. Proc. Suppl.216 (2011) 69 [arXiv:1103.3919] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

30. A. Mironov and A. Morozov, Nekrasov Functions and Exact Bohr-Zommerfeld Integrals, JHEP04 (2010) 040 [arXiv:0910.5670] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

31. A. Mironov and A. Morozov, Nekrasov Functions from Exact BS Periods: The Case of SU(N ), J. Phys.A 43 (2010) 195401 [arXiv:0911.2396] [INSPIRE].

    ADS  MATH  Google Scholar 

32. P. Koroteev and S. Shakirov, The Quantum DELL System, Lett. Math. Phys.110 (2020) 969 [arXiv:1906.10354] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

33. M. Noumi and J. Shiraishi, A direct approach to the bispectral problem for the Ruijsenaars-Macdonald q-difference operators, arXiv:1206.5364.

34. H. Awata et al., Explicit examples of DIM constraints for network matrix models, JHEP07 (2016) 103 [arXiv:1604.08366] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

35. R. Rimányi, A. Smirnov, A. Varchenko and Z. Zhou, 3d Mirror Symmetry and Elliptic Stable Envelopes, arXiv:1902.03677 [INSPIRE].

36. R. Rimányi, A. Smirnov, A. Varchenko and Z. Zhou, Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety, SIGMA15 (2019) 093 [arXiv:1906.00134] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

37. A. Okounkov and A. Smirnov, Quantum difference equation for Nakajima varieties, arXiv:1602.09007 [INSPIRE].

38. M. Aganagic and A. Okounkov, Elliptic stable envelopes, arXiv:1604.00423 [INSPIRE].

39. Y. Saito, Elliptic Ding-Iohara Algebra and the Free Field Realization of the Elliptic Macdonald Operator, arXiv:1301.4912.

40. Y. Saito, Elliptic Ding-Iohara Algebra and Commutative Families of the Elliptic Macdonald Operator, SIGMA10 (2014) 021 [arXiv:1309.7094].

    MATH  Google Scholar 

41. A. Iqbal, C. Kozcaz and S.-T. Yau, Elliptic Virasoro Conformal Blocks, arXiv:1511.00458 [INSPIRE].

42. F. Nieri, An elliptic Virasoro symmetry in 6d, Lett. Math. Phys.107 (2017) 2147 [arXiv:1511.00574] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

43. A. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, Phys. Lett.B 762 (2016) 196 [arXiv:1603.05467] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

44. R. Lodin, F. Nieri and M. Zabzine, Elliptic modular double and 4d partition functions, J. Phys.A 51 (2018) 045402 [arXiv:1703.04614] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

45. F. Nieri, Y. Pan and M. Zabzine, q-Virasoro modular triple, Commun. Math. Phys.366 (2019) 397 [arXiv:1710.07170] [INSPIRE].

46. S.V. Kerov, Hall-Littlewood functions and orthogonal polynomials Funct. Anal. App.25 (1991) 65.

    MATH  Google Scholar 

47. A. Mironov and A. Morozov, Kerov functions revisited, arXiv:1811.01184 [INSPIRE].

48. A. Mironov and A. Morozov, Kerov functions for composite representations and Macdonald ideal, Nucl. Phys.B 944 (2019) 114641 [arXiv:1903.00773] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

49. A. Mironov and A. Morozov, On Hamiltonians for Kerov functions, Eur. Phys. J.C 80 (2020) 277 [arXiv:1908.05176] [INSPIRE].

    Article  ADS  Google Scholar 

50. J. Shiraishi, Affine screening operators, affine Laumon spaces, and conjectures concerning non-stationary Ruijsenaars functions, J. Integrable Syst.4 (2019) xyz010 [arXiv:1903.07495].

51. P. Etingof and A. Varchenko, Traces of intertwiners for quantum groups and difference equations, I, math/9907181.

52. O. Babelon and D. Bernard, The sine-Gordon solitons as a N body problem, Phys. Lett.B 317 (1993) 363 [hep-th/9309154] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

53. H. Airault, H. McKean and J. Moser, Rational and elliptic solutions of the Korteweg-De Vries equation and a related many-body problem, Commun. Pure Appl. Math.30 (1977) 95.

    Article  ADS  MathSciNet  MATH  Google Scholar 

54. I. Krichever, Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. Anal. Appl.14 (1980) 282.

    Article  MATH  Google Scholar 

55. T. Shiota, Calogero-Moser hierarchy and KP hierarchy, J. Math. Phys.35 (1994) 5844 [hep-th/9402021] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

56. I. Krichever, O. Babelon, E. Billey and M. Talon, Spin generalization of the Calogero-Moser system and the matrix KP equation, hep-th/9411160 [INSPIRE].

57. I. Krichever and A. Zabrodin, Spin generalization of the Ruijsenaars-Schneider model, nonAbelian 2-D Toda chain and representations of Sklyanin algebra, Russ. Math. Surveys50 (1995) 1101 [hep-th/9505039] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

58. H. Bateman and A. Erdelyi, Higher transcendental functions, vol. 3, McGraw-Hill Book Company, London (1955).

59. D. Mumford, Tata Lectures on Theta I, Birkhäuser, Boston (1983).

60. D. Mumford, Tata Lectures on Theta II, Birkhäuser, Boston (1984).

61. A. Erdélyi, Integraldarstellungen hyper-geometrischer Funktionen, Quart. J. Math.8 (1937) 267.

    Article  ADS  MATH  Google Scholar 

62. A. Morozov and L. Vinet, q hypergeometric functions in the formalism of free fields, Mod. Phys. Lett.A 8 (1993) 2891 [hep-th/9309026] [INSPIRE].

63. H. Awata, H. Kanno, A. Mironov and A. Morozov, Elliptic lift of the Shiraishi function as a candidate for the non-stationary double-elliptic function, to appear.

64. D. Gaiotto, Surface Operators in N = 2 4d Gauge Theories, JHEP11 (2012) 090 [arXiv:0911.1316] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

65. D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys.13 (2009) 721 [arXiv:0807.3720] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

66. A. Nedelin, S. Pasquetti and Y. Zenkevich, T [SU(N )] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences, JHEP02 (2019) 176 [arXiv:1712.08140] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

67. Y. Yoshida and K. Sugiyama, Localization of 3d \( \mathcal{N} \) = 2 Supersymmetric Theories on S¹× D² , arXiv:1409.6713 [INSPIRE].

68. M. Bullimore, H.-C. Kim and P. Koroteev, Defects and Quantum Seiberg-Witten Geometry, JHEP05 (2015) 095 [arXiv:1412.6081] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

69. Y. Zenkevich, Higgsed network calculus, arXiv:1812.11961 [INSPIRE].

70. M. Fukuda, Y. Ohkubo and J. Shiraishi, Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction, arXiv:1903.05905 [INSPIRE].

71. M. Fukuda, Y. Ohkubo and J. Shiraishi, Non-stationary Ruijsenaars functions for κ = t^−1/Nand intertwining operators of Ding-Iohara-Miki algebra, arXiv:2002.00243 [INSPIRE].

72. M. Fukuda, private communication.

73. H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP06 (2011) 119 [arXiv:1105.0357] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

74. B. Feigin, M. Finkelberg, A. Negut and R. Rybnikov, Yangians and cohomology rings of Laumon spaces, Sel. Math. New Ser.17 (2011) 573 [arXiv:0812.4656].

    Article  MathSciNet  MATH  Google Scholar 

75. A. Braverman, B. Feigin, M. Finkelberg and L. Rybnikov, A Finite analog of the AGT relation I: F inite W -algebras and quasimaps’ spaces, Commun. Math. Phys.308 (2011) 457 [arXiv:1008.3655] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

76. M. Finkelberg and R. Rybnikov, Quantization of Drinfeld Zastava, arXiv:1009.0676.

77. S. Nawata, Givental J-functions, Quantum integrable systems, AGT relation with surface operator, Adv. Theor. Math. Phys.19 (2015) 1277 [arXiv:1408.4132] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

78. N. Nekrasov, BPS/CFT correspondence IV: σ-models and defects in gauge theory, Lett. Math. Phys.109 (2019) 579 [arXiv:1711.11011] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

79. H. Awata and H. Kanno, Refined BPS state counting from Nekrasov’s formula and Macdonald functions, Int. J. Mod. Phys.A 24 (2009) 2253 [arXiv:0805.0191] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

80. H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society (1999).

81. U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP05 (2003) 054 [hep-th/0211108] [INSPIRE].

    Article  ADS  Google Scholar 

82. L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys.94 (2010) 87 [arXiv:1005.4469] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

83. C. Kozcaz, S. Pasquetti, F. Passerini and N. Wyllard, Affine sl(N ) conformal blocks from N = 2 SU(N ) gauge theories, JHEP01 (2011) 045 [arXiv:1008.1412] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

84. A. Braverman, Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors, math/0401409.

85. A. Braverman and P. Etingof, Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential, math/0409441.

86. S. Jeong, Splitting of surface defect partition functions and integrable systems, Nucl. Phys.B 938 (2019) 775 [arXiv:1709.04926] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

87. S. Jeong and N. Nekrasov, Opers, surface defects and Yang-Yang functional, arXiv:1806.08270 [INSPIRE].

88. H. Awata and Y. Yamada, Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra, JHEP01 (2010) 125 [arXiv:0910.4431] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

89. K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys.B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

90. A. Marshakov, A. Mironov and A. Morozov, On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles, J. Geom. Phys.61 (2011) 1203 [arXiv:1011.4491] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

91. H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String, Adv. Theor. Math. Phys.16 (2012) 725 [arXiv:1008.0574] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

92. N. Wyllard, Instanton partition functions in N = 2 SU(N ) gauge theories with a general surface operator and their W-algebra duals, JHEP02 (2011) 114 [arXiv:1012.1355] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

93. E. Frenkel, S. Gukov and J. Teschner, Surface Operators and Separation of Variables, JHEP01 (2016) 179 [arXiv:1506.07508] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references