Awata, Hidetoshi; Kanno, Hiroaki; Mironov, Andrei; Morozov, Alexei; Suetake, Kazuma; Zenkevich, Yegor \((q, t)\)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces. (English) Zbl 1388.81623 J. High Energy Phys. 2018, No. 3, Paper No. 192, 70 p. (2018). Summary: We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody \( {U}_q{\left(\widehat{\mathfrak{g}}\right)}_k \) to generic quantum toroidal algebras. A nearly exhaustive presentation is given for both \( {U}_{q,t}\left({\widehat{\widehat{\mathfrak{gl}}}}_1\right) \) and \( {U}_{q,t}\left({\widehat{\widehat{\mathfrak{gl}}}}_n\right) \) when the screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations. Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of the Fock representations of the quantum toroidal algebra of type \(A_n\). The correlation functions of these operators satisfy the \((q, t)\)-Knizhnik-Zamolodchikov (KZ) equation, which features the \(\mathcal{R}\)-matrix. The matching with the Nekrasov function for the instanton counting on the ALE space is worked out explicitly. We also present an important application of the DIM formalism to the study of \(6d\) gauge theories described by the double elliptic integrable systems. We show that the modular and periodicity properties of the gauge theories are neatly explained by the network matrix models providing solutions to the elliptic \((q, t)\)-KZ equations. Cited in 31 Documents MSC: 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 83E30 String and superstring theories in gravitational theory 81T45 Topological field theories in quantum mechanics Keywords:Knizhnik-Zamolodchikov (KZ) equation; Nekrasov partition function; conformal and W symmetry; conformal field theory; supersymmetric gauge theory; topological strings; Ding-Iohara-Miki (DIM) algebra PDFBibTeX XMLCite \textit{H. Awata} et al., J. High Energy Phys. 2018, No. 3, Paper No. 192, 70 p. (2018; Zbl 1388.81623) Full Text: DOI arXiv References: [1] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys.B 241 (1984) 333 [INSPIRE]. · Zbl 0661.17013 [2] A. Zamolodchikov and Al. Zamolodchikov, Conformal field theory and critical phenomena in 2d systems, MCCME, Moscow, Russia, (2009), ISBN 978-5-94057-520-7. · Zbl 0712.17028 [3] L. Álvarez-Gaumé, Random surfaces, statistical mechanics and string theory, Helv. Phys. Acta64 (1991) 359 [INSPIRE]. [4] P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer, (1996). · Zbl 0869.53052 [5] 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 [6] N. Wyllard, AN −1conformal Toda field theory correlation functions from conformal N = 2 SU(N ) quiver gauge theories, JHEP11 (2009) 002 [arXiv:0907.2189] [INSPIRE]. [7] A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys.B 825 (2010) 1 [arXiv:0908.2569] [INSPIRE]. · Zbl 1196.81205 [8] H. Awata and H. Kanno, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP05 (2005) 039 [hep-th/0502061] [INSPIRE]. [9] H. Awata and Y. Yamada, Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra, JHEP01 (2010) 125 [arXiv:0910.4431] [INSPIRE]. · Zbl 1269.81157 [10] H. Awata and Y. Yamada, Five-dimensional AGT Relation and the Deformed beta-ensemble, Prog. Theor. Phys.124 (2010) 227 [arXiv:1004.5122] [INSPIRE]. · Zbl 1201.81074 [11] S. Yanagida, Five-dimensional SU(2) AGT conjecture and recursive formula of deformed Gaiotto state, J. Math. Phys.51 (2010) 123506 [arXiv:1005.0216] [INSPIRE]. · Zbl 1314.81147 [12] 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]. · Zbl 1273.81178 [13] H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP06 (2011) 119 [arXiv:1105.0357] [INSPIRE]. · Zbl 1298.81306 [14] A. Mironov, A. Morozov, S. Shakirov and A. Smirnov, Proving AGT conjecture as HS duality: extension to five dimensions, Nucl. Phys.B 855 (2012) 128 [arXiv:1105.0948] [INSPIRE]. · Zbl 1229.81184 [15] H. Kanno and M. Taki, Generalized Whittaker states for instanton counting with fundamental hypermultiplets, JHEP05 (2012) 052 [arXiv:1203.1427] [INSPIRE]. · Zbl 1348.81423 [16] F. Nieri, S. Pasquetti and F. Passerini, 3d and 5d Gauge Theory Partition Functions as q-deformed CFT Correlators, Lett. Math. Phys.105 (2015) 109 [arXiv:1303.2626] [INSPIRE]. · Zbl 1305.81112 [17] F. Nieri, S. Pasquetti, F. Passerini and A. Torrielli, 5D partition functions, q-Virasoro systems and integrable spin-chains, JHEP12 (2014) 040 [arXiv:1312.1294] [INSPIRE]. [18] M.-C. Tan, An M-Theoretic Derivation of a 5d and 6d AGT Correspondence and Relativistic and Elliptized Integrable Systems, JHEP12 (2013) 031 [arXiv:1309.4775] [INSPIRE]. [19] M.-C. Tan, Higher AGT Correspondences, W-algebras and Higher Quantum Geometric Langlands Duality from M-theory, arXiv:1607.08330 [INSPIRE]. · Zbl 1401.83026 [20] H. Itoyama, T. Oota and R. Yoshioka, q-Vertex Operator from 5D Nekrasov Function, J. Phys.A 49 (2016) 345201 [arXiv:1602.01209] [INSPIRE]. · Zbl 1362.82022 [21] A. Nedelin and M. Zabzine, q-Virasoro constraints in matrix models, JHEP03 (2017) 098 [arXiv:1511.03471] [INSPIRE]. · Zbl 1377.81180 [22] R. Yoshioka, The integral representation of solutions of KZ equation and a modification byK \[\mathcal{K}\] operator insertion, arXiv:1512.01084 [INSPIRE]. [23] H. Awata, H. Fujino and Y. Ohkubo, Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence, J. Math. Phys.58 (2017) 071704 [arXiv:1512.08016] [INSPIRE]. · Zbl 1422.17024 [24] S. Pasquetti, Holomorphic blocks and the 5d AGT correspondence, J. Phys.A 50 (2017) 443016 [arXiv:1608.02968] [INSPIRE]. · Zbl 1377.81098 [25] N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys.B 426 (1994) 19 [Erratum ibid.B 430 (1994) 485] [hep-th/9407087] [INSPIRE]. · Zbl 0996.81511 [26] N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys.B 431 (1994) 484 [hep-th/9408099] [INSPIRE]. · Zbl 1020.81911 [27] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett.B 355 (1995) 466 [hep-th/9505035] [INSPIRE]. · Zbl 0997.81567 [28] R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys.B 460 (1996) 299 [hep-th/9510101] [INSPIRE]. · Zbl 0996.37507 [29] E.J. Martinec, Integrable structures in supersymmetric gauge and string theory, Phys. Lett.B 367 (1996) 91 [hep-th/9510204] [INSPIRE]. [30] E.J. Martinec and N.P. Warner, Integrability in N = 2 gauge theory: A proof, hep-th/9511052 [INSPIRE]. [31] A. Gorsky and A. Mironov, Integrable many body systems and gauge theories, hep-th/0011197 [INSPIRE]. · Zbl 0979.81096 [32] N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th/0206161] [INSPIRE]. · Zbl 1056.81068 [33] R. Flume and R. Poghossian, An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int. J. Mod. Phys.A 18 (2003) 2541 [hep-th/0208176] [INSPIRE]. · Zbl 1069.81569 [34] N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math.244 (2006) 525 [hep-th/0306238] [INSPIRE]. · Zbl 1233.14029 [35] J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett.B 37 (1971) 95 [INSPIRE]. [36] S.P. Novikov, The Hamiltonian formalism and a many valued analog of Morse theory, Usp. Mat. Nauk37N5 (1982) 3 [INSPIRE]. · Zbl 0571.58011 [37] E. Witten, Nonabelian Bosonization in Two-Dimensions, Commun. Math. Phys.92 (1984) 455 [INSPIRE]. · Zbl 0536.58012 [38] A.M. Polyakov and P.B. Wiegmann, Theory of Nonabelian Goldstone Bosons, Phys. Lett.B 131 (1983) 121 [INSPIRE]. [39] A.M. Polyakov and P.B. Wiegmann, Goldstone Fields in Two-Dimensions with Multivalued Actions, Phys. Lett.B 141 (1984) 223 [INSPIRE]. [40] B.L. Feigin and D.B. Fuks, Invariant skew symmetric differential operators on the line and verma modules over the Virasoro algebra, Funct. Anal. Appl.16 (1982) 114 [INSPIRE]. · Zbl 0505.58031 [41] V.S. Dotsenko and V.A. Fateev, Conformal Algebra and Multipoint Correlation Functions in Two-Dimensional Statistical Models, Nucl. Phys.B 240 (1984) 312 [INSPIRE]. [42] M. Wakimoto, Fock representations of the affine lie algebra A1(1), Commun. Math. Phys.104 (1986) 605 [INSPIRE]. · Zbl 0587.17009 [43] A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S.L. Shatashvili, Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys.A 5 (1990) 2495 [INSPIRE]. [44] V.S. Dotsenko, The Free Field Representation of the SU(2) Conformal Field Theory, Nucl. Phys.B 338 (1990) 747 [INSPIRE]. [45] V.S. Dotsenko, Solving the SU(2) conformal field theory with the Wakimoto free field representation, Nucl. Phys.B 358 (1991) 547 [INSPIRE]. [46] B. Feigin and E. Frenkel, Quantization of the Drinfeld-Sokolov reduction, Phys. Lett.B 246 (1990) 75 [INSPIRE]. · Zbl 1242.17023 [47] A. Mironov, A. Morozov and S. Shakirov, Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions, JHEP02 (2010) 030 [arXiv:0911.5721] [INSPIRE]. · Zbl 1270.81139 [48] A. Mironov, A. Morozov and S. Shakirov, Conformal blocks as Dotsenko-Fateev Integral Discriminants, Int. J. Mod. Phys.A 25 (2010) 3173 [arXiv:1001.0563] [INSPIRE]. · Zbl 1193.81091 [49] A. Mironov, A. Morozov and S. Shakirov, Brezin-Gross-Witten model as ‘pure gauge’ limit of Selberg integrals, JHEP03 (2011) 102 [arXiv:1011.3481] [INSPIRE]. · Zbl 1301.81151 [50] A. Morozov and Y. Zenkevich, Decomposing Nekrasov Decomposition, JHEP02 (2016) 098 [arXiv:1510.01896] [INSPIRE]. · Zbl 1388.81686 [51] 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]. · Zbl 1390.81216 [52] 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]. · Zbl 1388.81588 [53] 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]. · Zbl 1390.81216 [54] H. Awata et al., Explicit examples of DIM constraints for network matrix models, JHEP07 (2016) 103 [arXiv:1604.08366] [INSPIRE]. · Zbl 1390.81206 [55] H. Awata et al., Toric Calabi-Yau threefolds as quantum integrable systems. ℛ-matrix andℛTT \[\text{\mathcal{R}}\mathcal{T}\mathcal{T}\] relations, JHEP10 (2016) 047 [arXiv:1608.05351] [INSPIRE]. · Zbl 1390.81221 [56] H. Awata et al., Anomaly in RTT relation for DIM algebra and network matrix models, Nucl. Phys.B 918 (2017) 358 [arXiv:1611.07304] [INSPIRE]. · Zbl 1360.81261 [57] H. Awata et al., Generalized Knizhnik-Zamolodchikov equation for Ding-Iohara-Miki algebra, Phys. Rev.D 96 (2017) 026021 [arXiv:1703.06084] [INSPIRE]. [58] J.-E. Bourgine, Y. Matsuo and H. Zhang, Holomorphic field realization of SHcand quantum geometry of quiver gauge theories, JHEP04 (2016) 167 [arXiv:1512.02492] [INSPIRE]. [59] J.-E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang and R.-D. Zhu, Coherent states in quantumW1+∞\[ {\mathcal{W}}_{1+\infty }\] algebra and qq-character for 5d Super Yang-Mills, PTEP2016 (2016) 123B05 [arXiv:1606.08020] [INSPIRE]. · Zbl 1361.81064 [60] J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo and R.-D. Zhu, (p,q)-webs of DIM representations, 5d N = 1 instanton partition functions and qq-characters, JHEP11 (2017) 034 [arXiv:1703.10759] [INSPIRE]. · Zbl 1383.83156 [61] M. Fukuda, K. Harada, Y. Matsuo and R.-D. Zhu, The Maulik-Okounkov R-matrix from the Ding-Iohara-Miki algebra, PTEP2017 (2017) 093A01 [arXiv:1705.02941] [INSPIRE]. · Zbl 1524.81082 [62] J.-E. Bourgine, M. Fukuda, Y. Matsuo and R.-D. Zhu, Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver, JHEP12 (2017) 015 [arXiv:1709.01954] [INSPIRE]. · Zbl 1383.81265 [63] T. Kimura and V. Pestun, Quiver W-algebras, arXiv:1512.08533 [INSPIRE]. · Zbl 1388.81850 [64] T. Kimura and V. Pestun, Quiver elliptic W-algebras, arXiv:1608.04651 [INSPIRE]. · Zbl 1388.81851 [65] T. Kimura and V. Pestun, Fractional quiver W-algebras, arXiv:1705.04410 [INSPIRE]. · Zbl 1388.81850 [66] A. Okounkov, Lectures on K-theoretic computations in enumerative geometry, arXiv:1512.07363 [INSPIRE]. · Zbl 1402.19001 [67] A. Okounkov and A. Smirnov, Quantum difference equation for Nakajima varieties, arXiv:1602.09007 [INSPIRE]. · Zbl 1504.14096 [68] A. Smirnov, Rationality of capped descendent vertex in K-theory, arXiv:1612.01048 [INSPIRE]. [69] J. Ding and K. Iohara, Generalization of Drinfeld quantum affine algebras, Lett. Math. Phys.41 (1997) 181 [q-alg/9608002]. · Zbl 0889.17011 [70] K. Miki, A (q, γ) analog of the W1+∞algebra, J. Math. Phys.48 (2007) 123520. · Zbl 1153.81405 [71] B. Feigin and A. Tsymbaliuk, Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra, Kyoto J. Math.51 (2011) 831 [arXiv:0904.1679]. · Zbl 1242.14006 [72] B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuousgl∞\[ \mathfrak{g}{\mathfrak{l}}_{\infty } \]: Semi-infinite construction of representations, Kyoto J. Math.51 (2011) 337 [arXiv:1002.3100]. · Zbl 1278.17012 [73] B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi and S. Yanagida, A commutative algebra on degenerate CP1and Macdonald polynomials, J. Math. Phys.50 (2009) 095215 [arXiv:0904.2291]. · Zbl 1248.33034 [74] B. Feigin, A. Hoshino, J. Shibahara, J. Shiraishi and S. Yanagida, Kernel function and quantum algebras, arXiv:1002.2485. [75] B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuousgl∞\[ \mathfrak{g}{\mathfrak{l}}_{\infty } \]: Tensor products of Fock modules and Wncharacters, arXiv:1002.3113 [INSPIRE]. · Zbl 1278.17013 [76] H. Awata, B. Feigin, A. Hoshino, M. Kanai, J. Shiraishi and S. Yanagida, Notes on Ding-Iohara algebra and AGT conjecture, RIMS Kōkyūroku1765 (2011) 12. [77] H. Awata, B. Feigin, A. Hoshino, M. Kanai, J. Shiraishi and S. Yanagida, Notes on Ding-Iohara algebra and AGT conjecture, arXiv:1106.4088 [INSPIRE]. [78] B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum toroidalgl \[1 \mathfrak{g}{\mathfrak{l}}_1\] algebra: plane partitions, Kyoto J. Math.52 (2012) 621 [arXiv:1110.5310]. · Zbl 1315.17010 [79] B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum toroidalgl \[1 \mathfrak{g}{\mathfrak{l}}_1\] and Bethe ansatz, J. Phys.A 48 (2015) 244001 [arXiv:1502.07194] [INSPIRE]. · Zbl 1318.81039 [80] H. Awata, B. Feigin and J. Shiraishi, Quantum Algebraic Approach to Refined Topological Vertex, JHEP03 (2012) 041 [arXiv:1112.6074] [INSPIRE]. · Zbl 1309.81112 [81] B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Representations of quantum toroidalgln \[\mathfrak{g}{\mathfrak{l}}_n\] J. Algebra380 (2013) 78 [arXiv:1204.5378]. · Zbl 1293.17017 [82] B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Branching rules for quantum toroidalgln \[\mathfrak{g}{\mathfrak{l}}_n \], Adv. Math.300 (2016) 229 [arXiv:1309.2147] [INSPIRE]. · Zbl 1402.17031 [83] B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Finite Type Modules and Bethe Ansatz for Quantum Toroidalgl \[1 \mathfrak{g}{\mathfrak{l}}_1 \], Commun. Math. Phys.356 (2017) 285 [arXiv:1603.02765] [INSPIRE]. · Zbl 1425.17020 [84] B. Feigin, M. Jimbo and E. Mukhin, Integrals of motion from quantum toroidal algebras, J. Phys.A 50 (2017) 464001 [arXiv:1705.07984] [INSPIRE]. · Zbl 1386.82037 [85] G. Aminov, A. Mironov and A. Morozov, Modular properties of 6d (DELL) systems, JHEP11 (2017) 023 [arXiv:1709.04897] [INSPIRE]. · Zbl 1383.83146 [86] V.G. Knizhnik and A.B. Zamolodchikov, Current Algebra and Wess-Zumino Model in Two-Dimensions, Nucl. Phys.B 247 (1984) 83 [INSPIRE]. · Zbl 0661.17020 [87] P.I. Etingof, I.B. Frenkel and A.A. Kirillov Jr., Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, AMS (1998). · Zbl 0903.17006 [88] I.B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations, Commun. Math. Phys.146 (1992) 1 [INSPIRE]. · Zbl 0760.17006 [89] D. Bernard, On the Wess-Zumino-Witten Models on the Torus, Nucl. Phys.B 303 (1988) 77 [INSPIRE]. [90] D. Bernard, On the Wess-Zumino-Witten Models on Riemann Surfaces, Nucl. Phys.B 309 (1988) 145 [INSPIRE]. [91] G. Felder and A. Varchenko, Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations, Int. Math. Res. Not. (1995) 221 [hep-th/9502165] [INSPIRE]. · Zbl 0840.17020 [92] P. Etingof and A. Varchenko, Traces of intertwiners for quantum groups and difference equations, I, math/9907181. · Zbl 1004.17006 [93] P. Etingof and A. Varchenko, The orthogonality and qKZB-heat equation for traces of Uq (g)-intertwiners, math/0302071. · Zbl 1160.17305 [94] P. Etingof, O. Schiffmann and A. Varchenko, Traces of intertwiners for quantum groups and difference equations, II, math/0207157. · Zbl 1018.17013 [95] Y. Sun, Traces of intertwiners for quantum affine algebras and difference equations (after Etingof-Schiffmann-Varchenko), arXiv:1609.09038. · Zbl 1423.81109 [96] V.V. Schechtman and A.N. Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys.20 (1990) 279 [INSPIRE]. · Zbl 0719.35079 [97] V. Schechtman and A. Varchenko, Arrangements of Hyperplanes and Lie Algebra Homology, Invent. Math.106 (1991) 139. · Zbl 0754.17024 [98] H. Awata, A. Tsuchiya and Y. Yamada, Integral formulas for the WZNW correlation functions, Nucl. Phys.B 365 (1991) 680 [INSPIRE]. [99] H. Awata, Screening currents ward identity and integral formulas for the WZNW correlation functions, Prog. Theor. Phys. Suppl.110 (1992) 303 [hep-th/9202032] [INSPIRE]. · Zbl 0832.17030 [100] H. Awata, S. Odake and J. Shiraishi, Nagoya Repository, http://hdl.handle.net/2237/25736, (1993). [101] G. Felder, V. Tarasov and A. Varchenko, Solutions of the elliptic qKZB equations and Bethe ansatz, Am. Math. Soc. Transl.180 (1997) 45. · Zbl 0884.65127 [102] G. Felder, V. Tarasov and A. Varchenko, Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations, Int. J. Math.10 (1999) 943 [q-alg/9705017]. · Zbl 1038.32500 [103] G. Felder and A. Varchenko, The q-deformed Knizhnik-Zamolodchikov-Bernard heat equation, Comm. Math. Phys.221 (2001) 549 [math/9809139]. · Zbl 1057.32006 [104] 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]. · Zbl 0947.81025 [105] A. Mironov and A. Morozov, Commuting Hamiltonians from Seiberg-Witten theta functions, Phys. Lett.B 475 (2000) 71 [hep-th/9912088] [INSPIRE]. · Zbl 1049.81648 [106] A. Mironov and A. Morozov, Double elliptic systems: Problems and perspectives, in Proceedings, International Workshop on Supersymmetries and Quantum Symmetries (SQS’99): Moscow, Russia, July 27-31, 1999, hep-th/0001168 [INSPIRE]. [107] 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]. · Zbl 1331.81135 [108] 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]. · Zbl 1388.81194 [109] 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]. [110] H. Nakajima, Instantons on ALE spaces, quiver varieties and Kac-Moody algebras, Duke Math. J.76 (1994) 365 [INSPIRE]. · Zbl 0826.17026 [111] H. Nakajima, Quiver varieties and finite dimensional representations of quantum affine algebras, J. Am. Math. Soc.14 (2001) 145 [math/9912158]. · Zbl 0981.17016 [112] V. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR283 (1985) 1060. [113] M. Jimbo, A q difference analog of U(g) and the Yang-Baxter equation, Lett. Math. Phys.10 (1985) 63 [INSPIRE]. · Zbl 0587.17004 [114] M. Jimbo, A q Analog ofUglN+\[1 \text{U}\left(\mathfrak{g}{\mathfrak{l}}_{N+1}\right) \], Hecke Algebra and the Yang-Baxter Equation, Lett. Math. Phys.11 (1986) 247 [INSPIRE]. · Zbl 0602.17005 [115] M. Jimbo, Quantum R Matrix for the Generalized Toda System, Commun. Math. Phys.102 (1986) 537 [INSPIRE]. · Zbl 0604.58013 [116] V.G. Drinfeld, A new realization of Yangians and quantum affine algebras, Sov. Math. Dokl.36 (1988) 212. [117] M. Varagnolo and E. Vasserot, Double-loop algebras and the Fock space, Invent. Math.133 (1998) 133 [q-alg/9612035]. · Zbl 0904.17014 [118] M. Varagnolo and E. Vasserot, On the K-theory of the cyclic quiver variety, Math. Res. Lett.18 (1999) 1005 [math/9902091]. · Zbl 1014.17017 [119] K. Nagao, Quiver varieties and Frenkel-Kac construction, J. Algebra321 (2009) 3764 [math/0703107]. · Zbl 1196.17021 [120] K. Nagao, K-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials, Osaka J. Math.46 (2009) 877 [arXiv:0709.1767]. · Zbl 1246.17020 [121] A. Negut, Quantum Algebras and Cyclic Quiver Varieties, arXiv:1504.06525. · Zbl 1407.16029 [122] Y. Saito, Quantum toroidal algebras and their vertex representations, Publ. Res. Inst. Math. Sci. Kyoto34 (1998) 155 [q-alg/9611030]. · Zbl 0982.17008 [123] Y. Saito, K. Takemura and D. Uglov, Toroidal actions on level 1 modules ofUqsl^n \[{U}_q\left(s{\widehat{l}}_n\right) \], Transform. Groups3 (1998) 75 [q-alg/9702024]. · Zbl 0915.17012 [124] M. Taki, Refined Topological Vertex and Instanton Counting, JHEP03 (2008) 048 [arXiv:0710.1776] [INSPIRE]. [125] 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]. · Zbl 1170.81423 [126] A. Marshakov, A. Mironov and A. Morozov, Generalized matrix models as conformal field theories: Discrete case, Phys. Lett.B 265 (1991) 99 [INSPIRE]. [127] A. Mironov and S. Pakulyak, On the continuum limit of the conformal matrix models, Theor. Math. Phys.95 (1993) 604 [hep-th/9209100] [INSPIRE]. · Zbl 0852.35135 [128] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and S. Pakuliak, Conformal matrix models as an alternative to conventional multimatrix models, Nucl. Phys.B 404 (1993) 717 [hep-th/9208044] [INSPIRE]. · Zbl 1009.81550 [129] H. Awata, Y. Matsuo, S. Odake and J. Shiraishi, Collective field theory, Calogero-Sutherland model and generalized matrix models, Phys. Lett.B 347 (1995) 49 [hep-th/9411053] [INSPIRE]. · Zbl 0894.17027 [130] H. Awata, Y. Matsuo, S. Odake and J. Shiraishi, Excited states of Calogero-Sutherland model and singular vectors of the W(N) algebra, Nucl. Phys.B 449 (1995) 347 [hep-th/9503043] [INSPIRE]. · Zbl 0894.17024 [131] H. Awata, Y. Matsuo, S. Odake and J. Shiraishi, A note on Calogero-Sutherland model, W(n) singular vectors and generalized matrix models, Soryushiron Kenkyu91 (1995) A69 [hep-th/9503028] [INSPIRE]. · Zbl 0894.17027 [132] R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings and N = 2 Gauge Systems, arXiv:0909.2453 [INSPIRE]. · Zbl 0999.81068 [133] H. Itoyama, K. Maruyoshi and T. Oota, The Quiver Matrix Model and 2d-4d Conformal Connection, Prog. Theor. Phys.123 (2010) 957 [arXiv:0911.4244] [INSPIRE]. · Zbl 1195.81103 [134] T. Eguchi and K. Maruyoshi, Penner Type Matrix Model and Seiberg-Witten Theory, JHEP02 (2010) 022 [arXiv:0911.4797] [INSPIRE]. · Zbl 1270.81165 [135] T. Eguchi and K. Maruyoshi, Seiberg-Witten theory, matrix model and AGT relation, JHEP07 (2010) 081 [arXiv:1006.0828] [INSPIRE]. · Zbl 1290.81063 [136] R. Schiappa and N. Wyllard, An A(r) threesome: Matrix models, 2d CFTs and 4d N = 2 gauge theories, J. Math. Phys.51 (2010) 082304 [arXiv:0911.5337] [INSPIRE]. · Zbl 1312.81108 [137] P. Sulkowski, Matrix models for beta-ensembles from Nekrasov partition functions, JHEP04 (2010) 063 [arXiv:0912.5476] [INSPIRE]. · Zbl 1272.81172 [138] H. Itoyama and T. Oota, Method of Generating q-Expansion Coefficients for Conformal Block and N = 2 Nekrasov Function by beta-Deformed Matrix Model, Nucl. Phys.B 838 (2010) 298 [arXiv:1003.2929] [INSPIRE]. · Zbl 1206.81102 [139] A. Mironov, A. Morozov and A. Morozov, Conformal blocks and generalized Selberg integrals, Nucl. Phys.B 843 (2011) 534 [arXiv:1003.5752] [INSPIRE]. · Zbl 1207.81146 [140] F. Fucito, J.F. Morales and R. Poghossian, Multi instanton calculus on ALE spaces, Nucl. Phys.B 703 (2004) 518 [hep-th/0406243] [INSPIRE]. · Zbl 1198.53024 [141] S. Fujii and S. Minabe, A combinatorial study on quiver varieties, SIGMA13 (2017) 052 [math/0510455] [INSPIRE]. · Zbl 1376.14015 [142] K. Takemura and D. Uglov, The orthogonal eigenbasis and norms of eigenvectors in the Spin Calogero-Sutherland Model, J. Phys.A 30 (1997) 3685 [solv-int/9611006]. · Zbl 0964.33009 [143] D. Uglov, Yangian Gelfand-Zetlin bases,glN \[\mathfrak{g}{\mathfrak{l}}_N\] Jack polynomials and computation of dynamical correlation functions in the spin Calogero-Sutherland model, Commun. Math. Phys.193 (1998) 663 [hep-th/9702020] [INSPIRE]. · Zbl 0962.17012 [144] V. Belavin and B. Feigin, Super Liouville conformal blocks from N = 2 SU(2) quiver gauge theories, JHEP07 (2011) 079 [arXiv:1105.5800] [INSPIRE]. · Zbl 1298.81154 [145] T. Nishioka and Y. Tachikawa, Central charges of para-Liouville and Toda theories from M-5-branes, Phys. Rev.D 84 (2011) 046009 [arXiv:1106.1172] [INSPIRE]. [146] A.A. Belavin, M.A. Bershtein, B.L. Feigin, A.V. Litvinov and G.M. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory, Commun. Math. Phys.319 (2013) 269 [arXiv:1111.2803] [INSPIRE]. · Zbl 1263.81252 [147] A.A. Belavin, M.A. Bershtein and G.M. Tarnopolsky, Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity, JHEP03 (2013) 019 [arXiv:1211.2788] [INSPIRE]. · Zbl 1342.81173 [148] H. Itoyama, T. Oota and R. Yoshioka, 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space, Nucl. Phys.B 877 (2013) 506 [arXiv:1308.2068] [INSPIRE]. · Zbl 1284.81172 [149] H. Itoyama, T. Oota and R. Yoshioka, q-Virasoro/W Algebra at Root of Unity and Parafermions, Nucl. Phys.B 889 (2014) 25 [arXiv:1408.4216] [INSPIRE]. · Zbl 1326.81179 [150] M. Jimbo, Topics from Representations of Uq (g). An Introductory Guide to Physicists, Nankai Lectures on Mathematical Physics, World Scientific, Singapore, (1992), pp. 1-61. [151] E. Mukhin, V. Tarasov and A. Varchenko, Bispectral andglN,glM \[\left(\mathfrak{g}{\mathfrak{l}}_N,\mathfrak{g}{\mathfrak{l}}_M\right)\] Dualities, math/0510364. [152] E. Mukhin, V. Tarasov and A. Varchenko, Bispectral and (glN,glM \[\mathfrak{g}{\mathfrak{l}}_N,\mathfrak{g}{\mathfrak{l}}_M )\] Dualities, Discrete Versus Differential, Adv. Math.218 (2008) 216 [math/0605172]. · Zbl 1142.82005 [153] 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]. [154] 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]. · Zbl 1262.14040 [155] 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]. · Zbl 1342.81310 [156] L. Bao, E. Pomoni, M. Taki and F. Yagi, M5-Branes, Toric Diagrams and Gauge Theory Duality, JHEP04 (2012) 105 [arXiv:1112.5228] [INSPIRE]. · Zbl 1348.81397 [157] M. Aganagic, N. Haouzi, C. Kozcaz and S. Shakirov, Gauge/Liouville Triality, arXiv:1309.1687 [INSPIRE]. [158] M. Aganagic, N. Haouzi and S. Shakirov, An-Triality, arXiv:1403.3657 [INSPIRE]. [159] M. Aganagic and N. Haouzi, ADE Little String Theory on a Riemann Surface (and Triality), arXiv:1506.04183 [INSPIRE]. [160] 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, August 3-8, 2009, pp. 265-289, arXiv:0908.4052 [INSPIRE]. · Zbl 1214.83049 [161] A. Mironov and A. Morozov, Nekrasov Functions and Exact Bohr-Zommerfeld Integrals, JHEP04 (2010) 040 [arXiv:0910.5670] [INSPIRE]. · Zbl 1272.81180 [162] 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]. · Zbl 1189.81237 [163] 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]. · Zbl 1215.81092 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.