×

Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function. (English) Zbl 1454.81213

Summary: As a development of [the authors, J. High Energy Phys. 2020, No. 4, Paper No. 212, 31 p. (2020; Zbl 1436.81129)], we note that the ordinary Shiraishi functions have an insufficient number of parameters to describe generic eigenfunctions of double elliptic system (Dell). The lacking parameter can be provided by substituting elliptic instead of the ordinary Gamma functions in the coefficients of the series. These new functions (ELS-functions) are conjectured to be functions governed by compactified DIM networks which can simultaneously play the three roles: solutions to non-stationary Dell equations, Dell conformal blocks with the degenerate field (surface operator) insertion, and the corresponding instanton sums in \(6d\) SUSY gauge theories with adjoint matter. We describe the basics of the corresponding construction and make further conjectures about the various limits and dualities which need to be checked to make a precise statement about the Dell description by double-periodic network models with DIM symmetry. We also demonstrate that the ELS-functions provide symmetric polynomials, which are an elliptic generalization of Macdonald ones, and compute the generation function of the elliptic genera of the affine Laumon spaces. In the particular U(1) case, we find an explicit plethystic formula for the \(6d\) partition function, which is a non-trivial elliptic generalization of the \( (q, t) \) Nekrasov-Okounkov formula from \(5d\).

MSC:

81T60 Supersymmetric field theories in quantum mechanics
81R12 Groups and algebras in quantum theory and relations with integrable systems

Citations:

Zbl 1436.81129
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Awata, H.; Kanno, H.; Mironov, A.; Morozov, A., On a complete solution of the quantum Dell system, JHEP, 04, 212 (2020) · Zbl 1436.81129 · doi:10.1007/JHEP04(2020)212
[2] Nekrasov, NA, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys., 7, 831 (2003) · Zbl 1056.81068 · doi:10.4310/ATMP.2003.v7.n5.a4
[3] Nekrasov, NA, Seiberg-Witten prepotential from instanton counting, International Congress of Mathematicians (2003) · Zbl 1056.81068
[4] N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B426 (1994) 19 [Erratum ibid.430 (1994) 485] [hep-th/9407087] [INSPIRE]. · Zbl 0996.81511
[5] Seiberg, N.; Witten, E., Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B, 431, 484 (1994) · Zbl 1020.81911 · doi:10.1016/0550-3213(94)90214-3
[6] Moore, GW; Nekrasov, N.; Shatashvili, S., Integrating over Higgs branches, Commun. Math. Phys., 209, 97 (2000) · Zbl 0981.53082 · doi:10.1007/PL00005525
[7] Losev, A.; Nekrasov, N.; Shatashvili, SL, Issues in topological gauge theory, Nucl. Phys. B, 534, 549 (1998) · Zbl 0954.57013 · doi:10.1016/S0550-3213(98)00628-2
[8] Gorsky, A.; Krichever, I.; Marshakov, A.; Mironov, A.; Morozov, A., Integrability and Seiberg-Witten exact solution, Phys. Lett. B, 355, 466 (1995) · Zbl 0997.81567 · doi:10.1016/0370-2693(95)00723-X
[9] Donagi, R.; Witten, E., Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B, 460, 299 (1996) · Zbl 0996.37507 · doi:10.1016/0550-3213(95)00609-5
[10] N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in 16^thInternational Congress on Mathematical Physics, World Scientific, Singapore (2009), pg. 265 [arXiv:0908.4052] [INSPIRE]. · Zbl 1214.83049
[11] K. Maruyoshi and M. Taki, Deformed prepotential, quantum integrable system and Liouville field theory, Nucl. Phys. B841 (2010) 388 [arXiv:1006.4505] [INSPIRE]. · Zbl 1207.81076
[12] Marshakov, A.; Mironov, A.; Morozov, A., On AGT relations with surface operator insertion and stationary limit of beta-ensembles, J. Geom. Phys., 61, 1203 (2011) · Zbl 1215.81092 · doi:10.1016/j.geomphys.2011.01.012
[13] A. Braverman, Instanton counting via affine Lie algebras. 1. Equivariant J functions of (affine) flag manifolds and Whittaker vectors, in CRM workshop on algebraic structures and moduli spaces, (2004) [math.AG/0401409] [INSPIRE]. · Zbl 1130.14013
[14] A. Braverman and P. Etingof, Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential, math.AG/0409441 [INSPIRE]. · Zbl 1177.14036
[15] Mironov, A.; Morozov, A., Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP, 04, 040 (2010) · Zbl 1272.81180 · doi:10.1007/JHEP04(2010)040
[16] Mironov, A.; Morozov, A., Nekrasov functions from exact BS periods: the case of SU(N ), J. Phys. A, 43, 195401 (2010) · Zbl 1189.81237 · doi:10.1088/1751-8113/43/19/195401
[17] Mironov, A.; Morozov, A.; Shakirov, S., Matrix model conjecture for exact BS periods and Nekrasov functions, JHEP, 02, 030 (2010) · Zbl 1270.81139 · doi:10.1007/JHEP02(2010)030
[18] 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] [INSPIRE]. · Zbl 1185.81111
[19] N. Wyllard, A_N −1conformal Toda field theory correlation functions from conformal N = 2 SU(N ) quiver gauge theories, JHEP11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
[20] Mironov, A.; Morozov, A., On AGT relation in the case of U(3), Nucl. Phys. B, 825, 1 (2010) · Zbl 1196.81205 · doi:10.1016/j.nuclphysb.2009.09.011
[21] Belavin, AA; Polyakov, AM; Zamolodchikov, AB, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B, 241, 333 (1984) · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X
[22] A. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, Phys. Lett. B762 (2016) 196 [arXiv:1603.05467] [INSPIRE]. · Zbl 1390.81216
[23] Awata, H., Explicit examples of DIM constraints for network matrix models, JHEP, 07, 103 (2016) · Zbl 1390.81206 · doi:10.1007/JHEP07(2016)103
[24] Ding, J-T; Iohara, K., Generalization and deformation of Drinfeld quantum affine algebras, Lett. Math. Phys., 41, 181 (1997) · Zbl 0889.17011 · doi:10.1023/A:1007341410987
[25] Miki, K., A (q, γ) analog of the W_1+∞algebra, J. Math. Phys., 48, 123520 (2007) · Zbl 1153.81405 · doi:10.1063/1.2823979
[26] 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. B573 (2000) 553 [hep-th/9906240] [INSPIRE]. · Zbl 0947.81025
[27] Mironov, A.; Morozov, A., Commuting Hamiltonians from Seiberg-Witten theta functions, Phys. Lett. B, 475, 71 (2000) · Zbl 1049.81648 · doi:10.1016/S0370-2693(00)00075-7
[28] A. Mironov and A. Morozov, Double elliptic systems: problems and perspectives, in 3^rdInternational Workshop on Supersymmetries and Quantum Symmetries, (1999) [hep-th/0001168] [INSPIRE].
[29] G. Aminov, A. Mironov and A. Morozov, Modular properties of 6d (DELL) systems, JHEP11 (2017) 023 [arXiv:1709.04897] [INSPIRE]. · Zbl 1383.83146
[30] Itoyama, H.; Morozov, A., Integrability and Seiberg-Witten theory: curves and periods, Nucl. Phys. B, 477, 855 (1996) · Zbl 0925.81362 · doi:10.1016/0550-3213(96)00358-6
[31] Braden, HW; Marshakov, A.; Mironov, A.; Morozov, A., The Ruijsenaars-Schneider model in the context of Seiberg-Witten theory, Nucl. Phys. B, 558, 371 (1999) · Zbl 1068.81601 · doi:10.1016/S0550-3213(99)00414-9
[32] A. Mironov, A. Morozov and Y. Zenkevich, On elementary proof of AGT relations from six dimensions, Phys. Lett. B756 (2016) 208 [arXiv:1512.06701] [INSPIRE]. · Zbl 1400.81168
[33] Mironov, A.; Morozov, A.; Zenkevich, Y., Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings, JHEP, 05, 121 (2016) · Zbl 1388.81588 · doi:10.1007/JHEP05(2016)121
[34] 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]. · Zbl 1388.81623
[35] H. Awata, H. Kanno, A. Mironov and A. Morozov, Shiraishi functor and non-Kerov deformation of Macdonald polynomials, arXiv:2002.12746 [INSPIRE].
[36] 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].
[37] M. Noumi and J. Shiraishi, A direct approach to the bispectral problem for the Ruijsenaars-Macdonald q-difference operators, arXiv:1206.5364.
[38] Awata, H.; Kanno, H., Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP, 05, 039 (2005) · doi:10.1088/1126-6708/2005/05/039
[39] J. Shiraishi, Affine screening operators, affine Laumon spaces and conjectures concerning non-stationary Ruijsenaars functions, J. Integrable Syst.4 (2019) xyz010 [arXiv:1903.07495]. · Zbl 1481.37062
[40] H. Bateman and A. Erdelyi, Higher transcendental functions, volume 2, London, U.K. (1953). · Zbl 0143.29202
[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]. · Zbl 1430.81061
[43] P. Koroteev and S. Shakirov, The quantum DELL system, Lett. Math. Phys.110 (2020) 969 [arXiv:1906.10354] [INSPIRE]. · Zbl 1436.81060
[44] R. Rimányi, V. Tarasov and A. Varchenko, Elliptic and K -theoretic stable envelopes and Newton polytopes, arXiv:1705.09344. · Zbl 1452.55009
[45] 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]. · Zbl 1451.53116
[46] M. Bullimore, M. Fluder, L. Hollands and P. Richmond, The superconformal index and an elliptic algebra of surface defects, JHEP10 (2014) 062 [arXiv:1401.3379] [INSPIRE].
[47] Macdonald, IG, Symmetric functions and Hall polynomials (1995), Oxford, U.K: Oxford University Press, Oxford, U.K · Zbl 0824.05059
[48] P. Koroteev, A-type quiver varieties and ADHM moduli spaces, arXiv:1805.00986 [INSPIRE].
[49] A. Okounkov, Takagi lectures on Donaldson-Thomas theory, arXiv:1802.00779 [INSPIRE]. · Zbl 1411.14066
[50] Hollowood, TJ; Iqbal, A.; Vafa, C., Matrix models, geometric engineering and elliptic genera, JHEP, 03, 069 (2008) · doi:10.1088/1126-6708/2008/03/069
[51] G. Lockhart and C. Vafa, Superconformal partition functions and non-perturbative topological strings, JHEP10 (2018) 051 [arXiv:1210.5909] [INSPIRE]. · Zbl 1402.81248
[52] B. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Triality in little string theories, Phys. Rev. D97 (2018) 046004 [arXiv:1711.07921] [INSPIRE]. · Zbl 1404.83111
[53] F. Nieri and S. Pasquetti, Factorisation and holomorphic blocks in 4d, JHEP11 (2015) 155 [arXiv:1507.00261] [INSPIRE]. · Zbl 1388.81360
[54] Crew, S.; Dorey, N.; Zhang, D., Factorisation of 3d N = 4 twisted indices and the geometry of vortex moduli space, JHEP, 08, 015 (2020) · Zbl 1454.81220 · doi:10.1007/JHEP08(2020)015
[55] 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].
[56] B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-strings, Commun. Math. Phys.334 (2015) 779 [arXiv:1305.6322] [INSPIRE]. · Zbl 1393.81031
[57] B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D89 (2014) 046003 [arXiv:1310.1185] [INSPIRE]. · Zbl 1393.81031
[58] H. Awata and H. Kanno, Refined BPS state counting from Nekrasov’s formula and Macdonald functions, Int. J. Mod. Phys. A24 (2009) 2253 [arXiv:0805.0191] [INSPIRE]. · Zbl 1170.81423
[59] H. Awata, H. Kanno, A. Mironov, A. Morozov and Y. Zenkevich, to appear.
[60] J. Li, K. Liu and J. Zhou, Topological string partition functions as equivariant indices, math.AG/0412089 [INSPIRE]. · Zbl 1129.14024
[61] Awata, H.; Kanno, H., Changing the preferred direction of the refined topological vertex, J. Geom. Phys., 64, 91 (2013) · Zbl 1261.81088 · doi:10.1016/j.geomphys.2012.10.014
[62] A. Iqbal, C. Kozcaz and K. Shabbir, Refined topological vertex, cylindric partitions and the U(1) adjoint theory, Nucl. Phys. B838 (2010) 422 [arXiv:0803.2260] [INSPIRE]. · Zbl 1206.81083
[63] R. Poghossian and M. Samsonyan, Instantons and the 5D U(1) gauge theory with extra adjoint, J. Phys. A42 (2009) 304024 [arXiv:0804.3564] [INSPIRE]. · Zbl 1176.81085
[64] A. Iqbal, B.A. Qureshi and K. Shabbir, (q, t) identities and vertex operators, Mod. Phys. Lett. A31 (2016) 1650065 [INSPIRE]. · Zbl 1339.81066
[65] E. Carlsson, N. Nekrasov and A. Okounkov, Five dimensional gauge theories and vertex operators, Moscow Math. J.14 (2014) 39 [arXiv:1308.2465] [INSPIRE]. · Zbl 1303.14046
[66] Rains, EM; Warnaar, SO, A Nekrasov-Okounkov formula for Macdonald polynomials, J. Algebr. Comb., 48, 1 (2018) · Zbl 1396.05013 · doi:10.1007/s10801-017-0790-2
[67] Dijkgraaf, R.; Moore, GW; Verlinde, EP; Verlinde, HL, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys., 185, 197 (1997) · Zbl 0872.32006 · doi:10.1007/s002200050087
[68] R. Waelder, Equivariant elliptic genera and local McKay correspondences, Asian J. Math.12 (2008) 251 [math.AG/0701336]. · Zbl 1165.14017
[69] Negut, A., Laumon spaces and the Calogero-Sutherland integrable system, Invent. Math., 178, 299 (2009) · Zbl 1185.37140 · doi:10.1007/s00222-009-0198-2
[70] A. Braverman, M. Finkelberg and J. Shiraishi, Macdonald polynomials, Laumon spaces and perverse coherent sheaves, in Perspectives in representation theory, Contemp. Math.610, American Mathematical Society, U.S.A. (2014), pg. 23 [arXiv:1206.3131]. · Zbl 1312.14112
[71] A. Negut, Affine Laumon spaces and integrable systems, arXiv:1112.1756 [INSPIRE]. · Zbl 1185.37140
[72] Schellekens, AN; Warner, NP, Anomalies and modular invariance in string theory, Phys. Lett. B, 177, 317 (1986) · doi:10.1016/0370-2693(86)90760-4
[73] Witten, E., Elliptic genera and quantum field theory, Commun. Math. Phys., 109, 525 (1987) · Zbl 0625.57008 · doi:10.1007/BF01208956
[74] E. Witten, The index of the Dirac operator in loop space, in Elliptic curves and modular forms in algebraic topology, P.S. Landweber ed., Springer, Berlin, Heidelberg, Germany (1988), pg. 161. · Zbl 0679.58045
[75] Dijkgraaf, R.; Moore, GW; Verlinde, EP; Verlinde, HL, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys., 185, 197 (1997) · Zbl 0872.32006 · doi:10.1007/s002200050087
[76] J. Li, K. Liu and J. Zhou, Topological string partition functions as equivariant indices, math.AG/0412089 [INSPIRE]. · Zbl 1129.14024
[77] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lect. Ser.18, American Mathematical Society, U.S.A. (1999). · Zbl 0949.14001
[78] Bruzzo, U.; Fucito, F.; Morales, JF; Tanzini, A., Multiinstanton calculus and equivariant cohomology, JHEP, 05, 054 (2003) · doi:10.1088/1126-6708/2003/05/054
[79] Mehta, VB; Seshadri, CS, Moduli of vector bundles on curves with parabolic structures, Math. Annalen, 248, 205 (1980) · Zbl 0454.14006 · doi:10.1007/BF01420526
[80] Biswas, I., Parabolic bundles as orbifold bundles, Duke Math. J., 88, 305 (1997) · Zbl 0955.14010 · doi:10.1215/S0012-7094-97-08812-8
[81] Feigin, B.; Finkelberg, M.; Negut, A.; Rybnikov, L., Yangians and cohomology rings of Laumon spaces, Selecta Math., 17, 573 (2011) · Zbl 1260.14015 · doi:10.1007/s00029-011-0059-x
[82] A. Braverman, B. Feigin, M. Finkelberg and L. Rybnikov, A finite analog of the AGT relation I: finite W -algebras and quasimaps’ spaces, Commun. Math. Phys.308 (2011) 457 [arXiv:1008.3655] [INSPIRE]. · Zbl 1247.81169
[83] M. Finkelberg and R. Rybnikov, Quantization of Drinfeld Zastava in type A, arXiv:1009.0676. · Zbl 1287.14024
[84] Kanno, H.; Tachikawa, Y., Instanton counting with a surface operator and the chain-saw quiver, JHEP, 06, 119 (2011) · Zbl 1298.81306 · doi:10.1007/JHEP06(2011)119
[85] S. Nawata, Givental J-functions, quantum integrable systems, AGT relation with surface operator, Adv. Theor. Math. Phys.19 (2015) 1277 [arXiv:1408.4132] [INSPIRE]. · Zbl 1342.81313
[86] Bullimore, M.; Kim, H-C; Koroteev, P., Defects and quantum Seiberg-Witten geometry, JHEP, 05, 095 (2015) · Zbl 1388.81788 · doi:10.1007/JHEP05(2015)095
[87] N. Nekrasov, BPS/CFT correspondence IV: σ-models and defects in gauge theory, Lett. Math. Phys.109 (2019) 579 [arXiv:1711.11011] [INSPIRE]. · Zbl 1411.81213
[88] N. Nekrasov, BPS/CFT correspondence V: BPZ and KZ equations from qq-characters, arXiv:1711.11582 [INSPIRE].
[89] Chen, H-Y; Kimura, T.; Lee, N., Quantum elliptic Calogero-Moser systems from gauge origami, JHEP, 02, 108 (2020) · Zbl 1435.81158 · doi:10.1007/JHEP02(2020)108
[90] I.M. Gel’fand, M.I. Graev and I.I. Pyatetskii-Shapiro, Generalized functions, volume 6: representation theory and automorphic functions, AMS Chelsea Publishing 382, (1969). · Zbl 0177.18003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.