Abstract
We argue that MacMahon representation of Ding-Iohara-Miki (DIM) algebra spanned by plane partitions is closely related to the Hilbert space of a 3d field theory. Using affine matrix model we propose a generalization of Bethe equations associated to DIM algebra with solutions also labelled by plane partitions. In a certain limit we identify the eigenstates of the Bethe system as new triple Macdonald polynomials depending on an infinite number of families of time variables. We interpret these results as first hints of the existence of an integrable 3d field theory, in which DIM algebra plays the same role as affine algebras in 2d WZNW models.
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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. 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, NATO Sci. Ser. C 520 (1999) 359 [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].
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].
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].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE].
D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287 [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].
P. Koroteev and A. Sciarappa, Quantum hydrodynamics from large-N supersymmetric gauge theories, Lett. Math. Phys. 108 (2018) 45 [arXiv:1510.00972] [INSPIRE].
P. Koroteev and A. Sciarappa, On elliptic algebras and large-N supersymmetric gauge theories, J. Math. Phys. 57 (2016) 112302 [arXiv:1601.08238] [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 \( \mathfrak{g}{\mathfrak{l}}_1 \), Commun. Math. Phys. 356 (2017) 285 [arXiv:1603.02765] [INSPIRE].
B. Feigin, M. Jimbo and E. Mukhin, Integrals of motion from quantum toroidal algebras, J. Phys. A 50 (2017) 464001 [arXiv:1705.07984] [INSPIRE].
M. Aganagic, K. Costello, J. McNamara and C. Vafa, Topological Chern-Simons/matter theories, arXiv:1706.09977 [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].
V. Kac and A. Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Commun. Math. Phys. 157 (1993) 429 [hep-th/9308153] [INSPIRE].
G. de B. Robinson, On the representations of the symmetric group, Amer. J. Math. 60 (1938) 745.
C. Schensted, Longest increasing and decreasing subsequences, Canadian J. Math. 13 (1961) 179.
D.E. Knuth, Permutations, matrices and generalised Young tableaux, Pacific J. Math. 34 (1970) 709.
E.A. Bender and D.E. Knuth, Enumeration of plane partitions, J. Comb. Theory A 13 (1972) 40.
A.A. Jucys, The bijection between plane partitions and nonnegative integer matrices, Lithuanian Math. J. 35 (1995) 163.
R.P. Stanley, Enumerative Combinatorics. 2, Cambridge University Press, Cambridge U.K. (1999).
J.-t. Ding and K. Iohara, Generalization and deformation of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997) 181 [INSPIRE].
K. Miki, A (q, γ) analog of the W(1 + ∞) algebra, J. Math. Phys. 48 (2007) 123520.
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. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, Phys. Lett. B 762 (2016) 196 [arXiv:1603.05467] [INSPIRE].
H. Awata et al., Explicit examples of DIM constraints for network matrix models, JHEP 07 (2016) 103 [arXiv:1604.08366] [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, 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].
M. Taki, Refined topological vertex and instanton counting, JHEP 03 (2008) 048 [arXiv:0710.1776] [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].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [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].
Y. Zenkevich, Quantum spectral curve for (q, t)-matrix model, Lett. Math. Phys. 108 (2018) 413 [arXiv:1507.00519] [INSPIRE].
Ya. Kononov and A. Morozov, On factorization of generalized Macdonald polynomials, Eur. Phys. J. C 76 (2016) 424 [arXiv:1607.00615] [INSPIRE].
Y. Zenkevich, Refined toric branes, surface operators and factorization of generalized Macdonald polynomials, JHEP 09 (2017) 070 [arXiv:1612.09570] [INSPIRE].
M. Bershtein and O. Foda, AGT, Burge pairs and minimal models, JHEP 06 (2014) 177 [arXiv:1404.7075] [INSPIRE].
V. Belavin, O. Foda and R. Santachiara, AGT, N-Burge partitions and \( \mathcal{W} \) N minimal models, JHEP 10 (2015) 073 [arXiv:1507.03540] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, Conformal blocks of W N minimal models and AGT correspondence, JHEP 07 (2014) 024 [arXiv:1404.7094] [INSPIRE].
H. Awata et al., Toric Calabi-Yau threefolds as quantum integrable systems. \( \mathrm{\mathcal{R}} \) -matrix and RTT relations, JHEP 10 (2016) 047 [arXiv:1608.05351] [INSPIRE].
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].
H. Awata et al., Generalized Knizhnik-Zamolodchikov equation for Ding-Iohara-Miki algebra, Phys. Rev. D 96 (2017) 026021 [arXiv:1703.06084] [INSPIRE].
H. Awata et al., (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces, JHEP 03 (2018) 192 [arXiv:1712.08016] [INSPIRE].
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 [Funkt. Anal. Pril. 16 (1982) 47] [INSPIRE].
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].
T. Kimura and V. Pestun, Quiver W-algebras, Lett. Math. Phys. 108 (2018) 1351 [arXiv:1512.08533] [INSPIRE].
T. Kimura and V. Pestun, Quiver elliptic W-algebras, Lett. Math. Phys. 108 (2018) 1383 [arXiv:1608.04651] [INSPIRE].
T. Kimura and V. Pestun, Fractional quiver W-algebras, Lett. Math. Phys. 108 (2018) 2425 [arXiv:1705.04410] [INSPIRE].
R. Dijkgraaf, D. Orlando and S. Reffert, Quantum crystals and spin chains, Nucl. Phys. B 811 (2009) 463 [arXiv:0803.1927] [INSPIRE].
H. Awata, B. Feigin and J. Shiraishi, Quantum algebraic approach to refined topological vertex, JHEP 03 (2012) 041 [arXiv:1112.6074] [INSPIRE].
N. Nekrasov, Magnificent four, arXiv:1712.08128 [INSPIRE].
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Zenkevich, Y. 3d field theory, plane partitions and triple Macdonald polynomials. J. High Energ. Phys. 2019, 12 (2019). https://doi.org/10.1007/JHEP06(2019)012
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DOI: https://doi.org/10.1007/JHEP06(2019)012