Abstract
Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/\( \mathcal{W} \)-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.
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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].
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].
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].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
A. Losev, N. Nekrasov and S.L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B 534 (1998) 549 [hep-th/9711108] [INSPIRE].
A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, hep-th/9801061 [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, D particle bound states and generalized instantons, Commun. Math. Phys. 209 (2000) 77 [hep-th/9803265] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
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].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
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].
A. Zamolodchikov and Al. Zamolodchikov, Conformal field theory and critical phenomena in 2d systems, (2009).
L. Álvarez-Gaumé, Random surfaces, statistical mechanics and string theory, Helv. Phys. Acta 64 (1991) 359 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, Germany (1996).
A. Mironov, S. Mironov, A. Morozov and A. Morozov, CFT exercises for the needs of AGT, Theor. Math. Phys. 165 (2010) 1662 [Teor. Mat. Fiz. 165 (2010) 503] [arXiv:0908.2064] [INSPIRE].
A. Mironov and A. Morozov, On the origin of Virasoro constraints in matrix models: Lagrangian approach, Phys. Lett. B 252 (1990) 47 [INSPIRE].
H. Itoyama and Y. Matsuo, Noncritical Virasoro algebra of d < 1 matrix model and quantized string field, Phys. Lett. B 255 (1991) 202 [INSPIRE].
R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [INSPIRE].
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].
T. Eguchi and K. Maruyoshi, Penner type matrix model and Seiberg-Witten theory, JHEP 02 (2010) 022 [arXiv:0911.4797] [INSPIRE].
T. Eguchi and K. Maruyoshi, Seiberg-Witten theory, matrix model and AGT relation, JHEP 07 (2010) 081 [arXiv:1006.0828] [INSPIRE].
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].
A. Mironov, A. Morozov and S. Shakirov, Matrix model conjecture for exact BS periods and Nekrasov functions, JHEP 02 (2010) 030 [arXiv:0911.5721] [INSPIRE].
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].
A. Mironov, A. Morozov and S. Shakirov, On ‘Dotsenko-Fateev’ representation of the toric conformal blocks, J. Phys. A 44 (2011) 085401 [arXiv:1010.1734] [INSPIRE].
A. Mironov, A. Morozov and S. Shakirov, Towards a proof of AGT conjecture by methods of matrix models, Int. J. Mod. Phys. A 27 (2012) 1230001 [arXiv:1011.5629] [INSPIRE].
P. Sulkowski, Matrix models for beta-ensembles from Nekrasov partition functions, JHEP 04 (2010) 063 [arXiv:0912.5476] [INSPIRE].
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].
A. Mironov, A. Morozov and A. Morozov, Conformal blocks and generalized Selberg integrals, Nucl. Phys. B 843 (2011) 534 [arXiv:1003.5752] [INSPIRE].
Y. Zenkevich, Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions, JHEP 05 (2015) 131 [arXiv:1412.8592] [INSPIRE].
A. Morozov and Y. Zenkevich, Decomposing Nekrasov decomposition, JHEP 02 (2016) 098 [arXiv:1510.01896] [INSPIRE].
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].
A. Mironov, A. Morozov and Y. Zenkevich, Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings, JHEP 05 (2016) 121 [arXiv:1603.00304] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
H. Awata and H. Kanno, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [INSPIRE].
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].
H. Nakajima, Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. Math. 160 (2004) 1057.
H. Nakajima, t-analogs of q-characters of quantum affine algebras of type A n , D n , math/0204184.
H. Nakajima, t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, math/0204185 [INSPIRE].
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].
S. Kanno, Y. Matsuo and S. Shiba, W 1+∞ algebra as a symmetry behind AGT relation, Phys. Rev. D 84 (2011) 026007 [arXiv:1105.1667] [INSPIRE].
S. Kanno, Y. Matsuo and H. Zhang, Virasoro constraint for Nekrasov instanton partition function, JHEP 10 (2012) 097 [arXiv:1207.5658] [INSPIRE].
S. Kanno, Y. Matsuo and H. Zhang, Extended conformal symmetry and recursion formulae for Nekrasov partition function, JHEP 08 (2013) 028 [arXiv:1306.1523] [INSPIRE].
N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, JHEP 03 (2016) 181 [arXiv:1512.05388] [INSPIRE].
T. Kimura and V. Pestun, Quiver W -algebras, arXiv:1512.08533 [INSPIRE].
J.-E. Bourgine, Y. Matsuo and H. Zhang, Holomorphic field realization of SH c and quantum geometry of quiver gauge theories, JHEP 04 (2016) 167 [arXiv:1512.02492] [INSPIRE].
A. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, arXiv:1603.05467 [INSPIRE].
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].
N. Wyllard, A N −1 conformal Toda field theory correlation functions from conformal N = 2 SU(N ) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [INSPIRE].
J.-T. Ding and K. Iohara, Generalization and deformation of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997) 181 [q-alg/9608002] [INSPIRE].
K. Miki, A (q, γ) analog of the W 1+∞ algebra, J. Math. Phys. 48 (2007) 123520.
V. Ginzburg, M. Kapranov and E. Vasserot, Langlands reciprocity for algebraic surfaces, Math. Res. Lett. 2 (1995) 147 [q-alg/9502013].
M. Varagnolo and E. Vasserot, Schur duality in the toroidal setting, Commun. Math. Phys. 182 (1996) 469 [q-alg/9506026].
O. Schiffmann and E. Vasserot, The elliptic Hall algebra, Cherednick Hecke algebras and Macdonald polynomials, Compositio Math. 147 (2011) 188 [arXiv:0802.4001].
O. Schiffmann and E. Vasserot, The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of A 2, Duke Math. J. 162 (2013) 279 [arXiv:0905.2555].
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].
B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi and S. Yanagida, A commutative algebra on degenerate CP 1 and Macdonald polynomials, J. Math. Phys. 50 (2009) 095215 [arXiv:0904.2291].
B. Feigin, A. Hoshino, J. Shibahara, J. Shiraishi and S. Yanagida, Kernel function and quantum algebras, arXiv:1002.2485.
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous \( \mathfrak{g}{\mathfrak{l}}_{\infty } \) : semi-infinite construction of representations, Kyoto J. Math. 51 (2011) 337 [arXiv:1002.3100].
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous \( \mathfrak{g}{\mathfrak{l}}_{\infty } \) : tensor products of Fock modules and W n characters, Kyoto J. Math. 51 (2011) 365 [arXiv:1002.3113] [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum toroidal \( \mathfrak{g}{\mathfrak{l}}_1 \) algebra : plane partitions, Kyoto J. Math. 52 (2012) 621 [arXiv:1110.5310].
H. Awata, B. Feigin and J. Shiraishi, Quantum algebraic approach to refined topological vertex, JHEP 03 (2012) 041 [arXiv:1112.6074] [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum toroidal \( \mathfrak{g}{\mathfrak{l}}_1 \) and Bethe ansatz, J. Phys. A 48 (2015) 244001 [arXiv:1502.07194] [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Finite type modules and Bethe ansatz for quantum toroidal \( \mathfrak{g}{\mathfrak{l}}_1 \), arXiv:1603.02765.
A. Okounkov and A. Smirnov, Quantum difference equation for Nakajima varieties, arXiv:1602.09007 [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, Generalized matrix models as conformal field theories: discrete case, Phys. Lett. B 265 (1991) 99 [INSPIRE].
A. Mironov and S. Pakulyak, On the continuum limit of the conformal matrix models, Theor. Math. Phys. 95 (1993) 604 [Teor. Mat. Fiz. 95 (1993) 317] [Int. J. Mod. Phys. A 8 (1993) 3107] [hep-th/9209100] [INSPIRE].
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].
A. Morozov, String theory: what is it?, Phys. Usp. 35 (1992) 671 [Erratum ibid. 35 (1992) 1003].
A. Morozov, Integrability and matrix models, Phys. Usp. 37 (1994) 1.
A. Morozov, Matrix models as integrable systems, hep-th/9502091 [INSPIRE].
A. Morozov, Challenges of matrix models, hep-th/0502010 [INSPIRE].
A. Mironov, 2D gravity and matrix models. 1. 2D gravity, Int. J. Mod. Phys. A 9 (1994) 4355 [hep-th/9312212] [INSPIRE].
A. Mironov, Matrix models of two-dimensional gravity, Phys. Part. Nucl. 33 (2002) 537 [Fiz. Elem. Chast. Atom. Yadra 33 (2002) 1051] [INSPIRE].
A. Mironov, Quantum deformations of tau functions, bilinear identities and representation theory, hep-th/9409190 [INSPIRE].
M. Aganagic, N. Haouzi, C. Kozcaz and S. Shakirov, Gauge/Liouville triality, arXiv:1309.1687 [INSPIRE].
M. Aganagic, N. Haouzi and S. Shakirov, A n -triality, arXiv:1403.3657 [INSPIRE].
M. Aganagic and N. Haouzi, ADE little string theory on a Riemann surface (and triality), arXiv:1506.04183 [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Representations of quantum toroidal \( \mathfrak{g}{\mathfrak{l}}_n \), arXiv:1204.5378.
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Branching rules for quantum toroidal \( \mathfrak{g}{\mathfrak{l}}_n \), arXiv:1309.2147 [INSPIRE].
A. Tsymbaliuk, Several realizations of Fock modules for quantum toroidal algebras of sl(n), arXiv:1603.08915.
P. Goddard, A. Kent and D.I. Olive, Virasoro algebras and coset space models, Phys. Lett. B 152 (1985) 88 [INSPIRE].
A. Tsuchiya and Y. Kanie, Vertex operators in the conformal field theory on P 1 and monodromy representations of the braid group, Lett. Math. Phys. 13 (1987) 303 [INSPIRE].
A. Tsuchiya and Y. Kanie, Vertex operators in conformal field theory on P 1 and monodromy representations of braid group, Adv. Stud. Pure Math. 16 (1988) 297 [Erratum ibid. 19 (1989) 675] [INSPIRE].
E. Mukhin, V. Tarasov and A. Varchenko, Bispectral and \( \left(\mathfrak{g}{\mathfrak{l}}_N,\;\mathfrak{g}{\mathfrak{l}}_M\right) \) dualities, math/0510364.
E. Mukhin, V. Tarasov and A. Varchenko, Bispectral and \( \left(\mathfrak{g}{\mathfrak{l}}_N,\;\mathfrak{g}{\mathfrak{l}}_M\right) \) dualities, discrete versus differential, Adv. Math. 218 (2008) 216 [math/0605172].
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] [Pisma Zh. Eksp. Teor. Fiz. 97 (2013) 49] [INSPIRE].
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].
A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral dualities in XXZ spin chains and five dimensional gauge theories, JHEP 12 (2013) 034 [arXiv:1307.1502] [INSPIRE].
L. Bao, E. Pomoni, M. Taki and F. Yagi, M 5-branes, toric diagrams and gauge theory duality, JHEP 04 (2012) 105 [arXiv:1112.5228] [INSPIRE].
M. Wakimoto, Fock representations of the affine lie algebra A (1)1 , Commun. Math. Phys. 104 (1986) 605 [INSPIRE].
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].
B. Feigin and E. Frenkel, Quantization of the Drinfeld-Sokolov reduction, Phys. Lett. B 246 (1990) 75 [INSPIRE].
A. Tsuchiya and Y. Kanie, Fock space representations of the Virasoro algebra — intertwining operators, Publ. R.I.M.S. Kyoto Univ. 22 (1986) 259.
M. Kato and S. Matsuda, Construction of singular vertex operators as degenerate primary conformal fields, Phys. Lett. B 172 (1986) 216 [INSPIRE].
G. Felder, BRST approach to minimal models, Nucl. Phys. B 317 (1989) 215 [Erratum ibid. B 324 (1989) 548] [INSPIRE].
K. Mimachi and Y. Yamada, Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials, Commun. Math. Phys. 174 (1995) 447.
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].
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].
H. Awata, Y. Matsuo, S. Odake and J. Shiraishi, A note on Calogero-Sutherland model, W N singular vectors and generalized matrix models, Soryushiron Kenkyu 91 (1995) A69 [hep-th/9503028] [INSPIRE].
S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE].
S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4D N = 2 gauge theories: 1, Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110] [INSPIRE].
B. Kol, 5D field theories and M-theory, JHEP 11 (1999) 026 [hep-th/9705031] [INSPIRE].
O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].
A. Gorsky, S. Gukov and A. Mironov, SUSY field theories, integrable systems and their stringy/brane origin. 2, Nucl. Phys. B 518 (1998) 689 [hep-th/9710239] [INSPIRE].
B. Kol and J. Rahmfeld, BPS spectrum of five-dimensional field theories, (p, q) webs and curve counting, JHEP 08 (1998) 006 [hep-th/9801067] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].
C.P. Herzog, I.R. Klebanov, S.S. Pufu and T. Tesileanu, Multi-matrix models and tri-Sasaki Einstein spaces, Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [INSPIRE].
M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 03 (2012) P03001 [arXiv:1110.4066] [INSPIRE].
H. Awata, S. Hirano and M. Shigemori, The partition function of ABJ theory, Prog. Theor. Exp. Phys. 2013 (2013) 053B04 [arXiv:1212.2966] [INSPIRE].
R. Lawrence and L. Rozansky, Witten-Reshetikhin-Turaev invariants of Seifert manifolds, Commun. Math. Phys. 205 (1999) 287.
M. Mariño, Chern-Simons theory, matrix integrals and perturbative three manifold invariants, Commun. Math. Phys. 253 (2004) 25 [hep-th/0207096] [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, Matrix model as a mirror of Chern-Simons theory, JHEP 02 (2004) 010 [hep-th/0211098] [INSPIRE].
C. Beasley and E. Witten, Non-Abelian localization for Chern-Simons theory, J. Diff . Geom. 70 (2005) 183 [hep-th/0503126] [INSPIRE].
C. Beasley, Localization for Wilson loops in Chern-Simons theory, Adv. Theor. Math. Phys. 17 (2013) 1 [arXiv:0911.2687] [INSPIRE].
M. Tierz, Soft matrix models and Chern-Simons partition functions, Mod. Phys. Lett. A 19 (2004) 1365 [hep-th/0212128] [INSPIRE].
A. Brini, B. Eynard and M. Mariño, Torus knots and mirror symmetry, Annales Henri Poincaré 13 (2012) 1873 [arXiv:1105.2012] [INSPIRE].
A. Alexandrov, A. Mironov, A. Morozov and a. Morozov, Towards matrix model representation of HOMFLY polynomials, JETP Lett. 100 (2014) 271 [arXiv:1407.3754] [INSPIRE].
A. Mironov, A. Morozov and A. Sleptsov, Genus expansion of HOMFLY polynomials, Theor. Math. Phys. 177 (2013) 1435 [Teor. Mat. Fiz. 177 (2013) 179] [arXiv:1303.1015] [INSPIRE].
A. Mironov, A. Morozov and A. Sleptsov, On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions, Eur. Phys. J. C 73 (2013) 2492 [arXiv:1304.7499] [INSPIRE].
O. Dubinkin, On the Virasoro constraints for torus knots, J. Phys. A 47 (2014) 485203 [arXiv:1307.7909] [INSPIRE].
G.W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].
N. Guay, Affine Yangians and deformed double current algebras in type A, Adv. Math. 211 (2007) 436.
D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287 [INSPIRE].
N. Arbesfeld and O. Schiffmann, A presentation of the deformed W 1+∞ algebra, arXiv:1209.0429.
O. Schiffmann and E. Vasserot, Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A 2, Publ. Math. Inst. Hautes Etudes Sci. 118 (2013) 213 [arXiv:1202.2756].
A. Smirnov, On the instanton R-matrix, arXiv:1302.0799 [INSPIRE].
A. Smirnov, Polynomials associated with fixed points on the instanton moduli space, arXiv:1404.5304 [INSPIRE].
A. Tsymbaliuk, The affine Yangian of \( \mathfrak{g}{\mathfrak{l}}_1 \) revisited, arXiv:1404.5240.
R.-D. Zhu and Y. Matsuo, Yangian associated with 2D N = 1 SCFT, Prog. Theor. Exp. Phys. 2015 (2015) 093A01 [arXiv:1504.04150] [INSPIRE].
M. Fukuda, S. Nakamura, Y. Matsuo and R.-D. Zhu, SH c realization of minimal model CFT: triality, poset and Burge condition, JHEP 11 (2015) 168 [arXiv:1509.01000] [INSPIRE].
M. Bernshtein and A. Tsymbaliuk, Homomorphisms between different quantum toroidal and affine Yangian algebras, arXiv:1512.09109.
T. Procházka, W -symmetry, topological vertex and affine Yangian, arXiv:1512.07178 [INSPIRE].
Y. Saito, Elliptic Ding-Iohara algebra and the free field realization of the elliptic Macdonald operator, arXiv:1301.4912.
Y. Saito, Elliptic Ding-Iohara algebra and commutative families of the elliptic Macdonald operator, arXiv:1309.7094.
A. Iqbal, C. Kozcaz and S.-T. Yau, Elliptic Virasoro conformal blocks, arXiv:1511.00458 [INSPIRE].
F. Nieri, An elliptic Virasoro symmetry in 6d, arXiv:1511.00574 [INSPIRE].
A. Iqbal, All genus topological string amplitudes and five-brane webs as Feynman diagrams, hep-th/0207114 [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, Prog. Math. 244 (2006) 597 [hep-th/0309208] [INSPIRE].
A. Iqbal, N. Nekrasov, A. Okounkov and C. Vafa, Quantum foam and topological strings, JHEP 04 (2008) 011 [hep-th/0312022] [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, The vertex on a strip, Adv. Theor. Math. Phys. 10 (2006) 317 [hep-th/0410174] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. I, Invent. Math. 162 (2005) 313 [math/0306198] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. II. K-theoretic partition function, math/0505553 [INSPIRE].
H. Nakajima and K. Yoshioka, Lectures on instanton counting, math/0311058 [INSPIRE].
E. Carlsson, N. Nekrasov and A. Okounkov, Five dimensional gauge theories and vertex operators, arXiv:1308.2465 [INSPIRE].
M. Taki, Refined topological vertex and instanton counting, JHEP 03 (2008) 048 [arXiv:0710.1776] [INSPIRE].
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].
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].
A. Mironov and A. Morozov, Commuting Hamiltonians from Seiberg-Witten theta functions, Phys. Lett. B 475 (2000) 71 [hep-th/9912088] [INSPIRE].
A. Mironov and A. Morozov, Double elliptic systems: problems and perspectives, hep-th/0001168 [INSPIRE].
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].
G. Aminov, H.W. Braden, A. Mironov, A. Morozov and A. Zotov, Seiberg-Witten curves and double-elliptic integrable systems, JHEP 01 (2015) 033 [arXiv:1410.0698] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, del Pezzo surfaces and type-I’ theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].
D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
M. Taki, Notes on enhancement of flavor symmetry and 5d superconformal index, arXiv:1310.7509 [INSPIRE].
M. Taki, Seiberg duality, 5d SCFTs and Nekrasov partition functions, arXiv:1401.7200 [INSPIRE].
V. Mitev, E. Pomoni, M. Taki and F. Yagi, Fiber-base duality and global symmetry enhancement, JHEP 04 (2015) 052 [arXiv:1411.2450] [INSPIRE].
S.-S. Kim, M. Taki and F. Yagi, Tao probing the end of the world, Prog. Theor. Exp. Phys. 2015 (2015) 083B02 [arXiv:1504.03672] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, A new 5d description of 6d D-type minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [INSPIRE].
P. Di Francesco, M. Gaudin, C. Itzykson and F. Lesage, Laughlin’s wave functions, Coulomb gases and expansions of the discriminant, Int. J. Mod. Phys. A 9 (1994) 4257 [hep-th/9401163] [INSPIRE].
A. Zabrodin, Random matrices and Laplacian growth, arXiv:0907.4929.
A. Morozov and S. Shakirov, The matrix model version of AGT conjecture and CIV-DV prepotential, JHEP 08 (2010) 066 [arXiv:1004.2917] [INSPIRE].
L. Chekhov, Logarithmic potential β-ensembles and Feynman graphs, arXiv:1009.5940 [INSPIRE].
A. Mironov, A. Morozov, A. Popolitov and S. Shakirov, Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble, Theor. Math. Phys. 171 (2012) 505 [Teor. Mat. Fiz. 171 (2012) 96] [arXiv:1103.5470] [INSPIRE].
H. Awata and Y. Yamada, Five-dimensional AGT conjecture and the deformed Virasoro algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [INSPIRE].
H. Awata and Y. Yamada, Five-dimensional AGT relation and the deformed beta-ensemble, Prog. Theor. Phys. 124 (2010) 227 [arXiv:1004.5122] [INSPIRE].
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].
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].
H. Itoyama, T. Oota and R. Yoshioka, q-vertex operator from 5D Nekrasov function, arXiv:1602.01209 [INSPIRE].
A. Nedelin and M. Zabzine, q-Virasoro constraints in matrix models, arXiv:1511.03471 [INSPIRE].
R. Yoshioka, The integral representation of solutions of KZ equation and a modification by \( \mathcal{K} \) operator insertion, arXiv:1512.01084 [INSPIRE].
Y. Zenkevich, Quantum spectral curve for (q, t)-matrix model, arXiv:1507.00519 [INSPIRE].
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].
F. Nieri, S. Pasquetti, F. Passerini and A. Torrielli, 5D partition functions, q-Virasoro systems and integrable spin-chains, JHEP 12 (2014) 040 [arXiv:1312.1294] [INSPIRE].
Y. Ohkubo, H. Awata and H. Fujino, Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence, arXiv:1512.08016 [INSPIRE].
V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE].
A. Belavin and V. Belavin, AGT conjecture and integrable structure of conformal field theory for c = 1, Nucl. Phys. B 850 (2011) 199 [arXiv:1102.0343] [INSPIRE].
Y. Matsuo, C. Rim and H. Zhang, Construction of Gaiotto states with fundamental multiplets through degenerate DAHA, JHEP 09 (2014) 028 [arXiv:1405.3141] [INSPIRE].
E. Carlsson and A. Okounkov, Exts and vertex operators, arXiv:0801.2565.
A. Negut, Exts and the AGT relations, arXiv:1510.05482 [INSPIRE].
A. Mironov, A. Morozov and S. Shakirov, Brezin-Gross-Witten model as ‘pure gauge’ limit of Selberg integrals, JHEP 03 (2011) 102 [arXiv:1011.3481] [INSPIRE].
A. Morozov and A. Smirnov, Towards the proof of AGT relations with the help of the generalized Jack polynomials, Lett. Math. Phys. 104 (2014) 585 [arXiv:1307.2576] [INSPIRE].
S. Mironov, A. Morozov and Y. Zenkevich, Generalized Jack polynomials and the AGT relations for the SU(3) group, JETP Lett. 99 (2014) 109 [arXiv:1312.5732] [INSPIRE].
Y. Ohkubo, Existence and orthogonality of generalized Jack polynomials and its q-deformation, arXiv:1404.5401 [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum toroidal \( \mathfrak{g}{\mathfrak{l}}_1 \) and Bethe ansatz, J. Phys. A 48 (2015) 244001 [arXiv:1502.07194] [INSPIRE].
H. Awata, M. Fukuma, Y. Matsuo and S. Odake, Representation theory of the W 1+∞ algebra, Prog. Theor. Phys. Suppl. 118 (1995) 343 [hep-th/9408158] [INSPIRE].
A.V. Litvinov, On spectrum of ILW hierarchy in conformal field theory, JHEP 11 (2013) 155 [arXiv:1307.8094] [INSPIRE].
M.N. Alfimov and A.V. Litvinov, On spectrum of ILW hierarchy in conformal field theory II: coset CFT’s, JHEP 02 (2015) 150 [arXiv:1411.3313] [INSPIRE].
G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N ) quantum intermediate long wave hydrodynamics, JHEP 07 (2014) 141 [arXiv:1403.6454] [INSPIRE].
G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Quantum cohomology and quantum hydrodynamics from supersymmetric quiver gauge theories, arXiv:1505.07116 [INSPIRE].
P. Koroteev and A. Sciarappa, Quantum hydrodynamics from large-N supersymmetric gauge theories, arXiv:1510.00972 [INSPIRE].
P. Koroteev and A. Sciarappa, On elliptic algebras and large-N supersymmetric gauge theories, arXiv:1601.08238 [INSPIRE].
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Awata, H., Kanno, H., Matsumoto, T. et al. Explicit examples of DIM constraints for network matrix models. J. High Energ. Phys. 2016, 103 (2016). https://doi.org/10.1007/JHEP07(2016)103
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DOI: https://doi.org/10.1007/JHEP07(2016)103