di-Langlands correspondence and extended observables. (English) Zbl 07899764
Summary: We explore the difference Langlands correspondence using the four dimensional \(\mathcal{N} = 2\) super-QCD. Surface defects and surface observables play the crucial role. As an application, we give the first construction of the full set of quantum integrals, i.e. commuting differential operators, such that the partition function of the so-called regular monodromy surface defect is their joint eigenvectors in an evaluation module over the Yangian \(Y(\mathfrak{gl}(2))\), making it the wavefunction of a \(N\)-site \(\mathfrak{gl}(2)\) spin chain with bi-infinite spin modules. We construct the \(\mathbf{Q}\)- and \(\tilde{\mathbf{Q}}\)-surface observables which are believed to be the \(Q\)-operators on the bi-infinite module over the Yangian \(Y(\mathfrak{gl}(2))\), and compute their eigenvalues, the \(Q\)-functions, as vevs of the surface observables.
MSC:
| 81Txx | Quantum field theory; related classical field theories |
| 17Bxx | Lie algebras and Lie superalgebras |
| 81Rxx | Groups and algebras in quantum theory |
Keywords:
lattice integrable models; supersymmetric gauge theory; quantum groups; duality in gauge field theoriesReferences:
| [1] | A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B241 (1984) 333 [INSPIRE]. · Zbl 0661.17013 |
| [2] | Seiberg, N.; Witten, E., Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B, 426, 19, 1994 · Zbl 0996.81510 · doi:10.1016/0550-3213(94)90124-4 |
| [3] | 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 |
| [4] | Nekrasov, N.; Tsymbaliuk, A., Surface defects in gauge theory and KZ equation, Lett. Math. Phys., 112, 28, 2022 · Zbl 1490.81107 · doi:10.1007/s11005-022-01511-8 |
| [5] | Aganagic, M.; Okounkov, A., Quasimap counts and Bethe eigenfunctions, Moscow Math. J., 17, 565, 2017 · Zbl 1426.82018 · doi:10.17323/1609-4514-2017-17-4-565-600 |
| [6] | Dedushenko, M.; Nekrasov, N., Interfaces and quantum algebras, I: Stable envelopes, J. Geom. Phys., 194, 104991, 2023 · Zbl 1534.17015 · doi:10.1016/j.geomphys.2023.104991 |
| [7] | L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in the proceedings of the Les Houches School of Physics: Astrophysical Sources of Gravitational Radiation, Les Houches, France, September 26 - October 06 (1995) [hep-th/9605187] [INSPIRE]. |
| [8] | Nekrasov, N.; Okounkov, A., Seiberg-Witten theory and random partitions, Prog. Math., 244, 525, 2006 · Zbl 1233.14029 · doi:10.1007/0-8176-4467-9_15 |
| [9] | Jeong, S.; Nekrasov, N., Opers, surface defects, and Yang-Yang functional, Adv. Theor. Math. Phys., 24, 1789, 2020 · Zbl 1524.81101 · doi:10.4310/ATMP.2020.v24.n7.a4 |
| [10] | A. Beilinson and Y. Drinfeld, Quantization of Hitchin’s Integrable System and Hecke Eigensheaves [INSPIRE]. · Zbl 0864.14007 |
| [11] | A. Beilinson and V. Drinfeld, Opers, math/0501398. |
| [12] | Gaiotto, D., N = 2 dualities, JHEP, 08, 034, 2012 · Zbl 1397.81362 · doi:10.1007/JHEP08(2012)034 |
| [13] | Gaiotto, D.; Moore, GW; Neitzke, A., Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math., 234, 239, 2013 · Zbl 1358.81150 · doi:10.1016/j.aim.2012.09.027 |
| [14] | A. Grekov, I. Krichever and N. Nekrasov, Difference opers from linear quiver theories, to appear. |
| [15] | E. Witten, Topological Quantum Field Theory, Commun. Math. Phys.117 (1988) 353 [INSPIRE]. · Zbl 0656.53078 |
| [16] | Bershadsky, M.; Vafa, C.; Sadov, V., D-branes and topological field theories, Nucl. Phys. B, 463, 420, 1996 · Zbl 1004.81560 · doi:10.1016/0550-3213(96)00026-0 |
| [17] | Harvey, JA; Moore, GW; Strominger, A., Reducing S duality to T duality, Phys. Rev. D, 52, 7161, 1995 · doi:10.1103/PhysRevD.52.7161 |
| [18] | Nekrasov, N.; Witten, E., The Omega Deformation, Branes, Integrability, and Liouville Theory, JHEP, 09, 092, 2010 · Zbl 1291.81265 · doi:10.1007/JHEP09(2010)092 |
| [19] | S. Jeong, N. Lee and N. Nekrasov, Parallel surface defects, Hecke operators, and quantum Hitchin system, arXiv:2304.04656 [INSPIRE]. |
| [20] | E. Frenkel, D. Gaitsgory and K. Vilonen, On the geometric Langlands conjecture, math/0012255. · Zbl 1071.11039 |
| [21] | E. Frenkel, Ramifications of the geometric Langlands Program, math/0611294 [INSPIRE]. · Zbl 1194.22022 |
| [22] | Kapustin, A.; Witten, E., Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys., 1, 1, 2007 · Zbl 1128.22013 · doi:10.4310/CNTP.2007.v1.n1.a1 |
| [23] | S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE]. · Zbl 1237.14024 |
| [24] | Gaiotto, D., S-duality of boundary conditions and the Geometric Langlands program, Proc. Symp. Pure Math., 98, 139, 2018 · Zbl 1452.81163 · doi:10.1090/pspum/098/01721 |
| [25] | Frenkel, E.; Gaiotto, D., Quantum Langlands dualities of boundary conditions, D-modules, and conformal blocks, Commun. Num. Theor. Phys., 14, 199, 2020 · Zbl 1445.14024 · doi:10.4310/CNTP.2020.v14.n2.a1 |
| [26] | E. Frenkel, Lectures on the Langlands program and conformal field theory, in the proceedings of the Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry, Les Houches, France, March 09-21 (2003) [doi:10.1007/978-3-540-30308-4_11] [hep-th/0512172] [INSPIRE]. |
| [27] | Teschner, J., Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I, Adv. Theor. Math. Phys., 15, 471, 2011 · Zbl 1442.81059 · doi:10.4310/ATMP.2011.v15.n2.a6 |
| [28] | Balasubramanian, A.; Teschner, J., Supersymmetric field theories and geometric Langlands: The other side of the coin, Proc. Symp. Pure Math., 98, 79, 2018 · Zbl 1452.81160 · doi:10.1090/pspum/098/01723 |
| [29] | J. Teschner, Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence, arXiv:1707.07873 [INSPIRE]. · Zbl 1439.14051 |
| [30] | Cherkis, SA; Kapustin, A., Nahm transform for periodic monopoles and N = 2 superYang-Mills theory, Commun. Math. Phys., 218, 333, 2001 · doi:10.1007/PL00005558 |
| [31] | Cherkis, SA; Kapustin, A., Periodic monopoles with singularities and N = 2 super QCD, Commun. Math. Phys., 234, 1, 2003 · doi:10.1007/s00220-002-0786-0 |
| [32] | N. Nekrasov and V. Pestun, Seiberg-Witten Geometry of Four-Dimensional \(\mathcal{N} = 2\) Quiver Gauge Theories, SIGMA19 (2023) 047 [arXiv:1211.2240] [INSPIRE]. · Zbl 07727614 |
| [33] | Nekrasov, N.; Pestun, V.; Shatashvili, S., Quantum geometry and quiver gauge theories, Commun. Math. Phys., 357, 519, 2018 · Zbl 1393.81033 · doi:10.1007/s00220-017-3071-y |
| [34] | C. Elliott and V. Pestun, Multiplicative Hitchin systems and supersymmetric gauge theory, Selecta Math. (2019) 1 [arXiv:1812.05516] [INSPIRE]. · Zbl 1423.14081 |
| [35] | B. Charbonneau and J. Hurtubise, Singular Hermitian-Einstein Monopoles on the Product of a Circle and a Riemann Surface, Int. Math. Res. Not.2011 (2010) 175. · Zbl 1252.53036 |
| [36] | Jeong, S.; Zhang, X., A note on chiral trace relations from qq-characters, JHEP, 04, 026, 2020 · doi:10.1007/JHEP04(2020)026 |
| [37] | Jeong, S., Splitting of surface defect partition functions and integrable systems, Nucl. Phys. B, 938, 775, 2019 · Zbl 1405.81076 · doi:10.1016/j.nuclphysb.2018.12.007 |
| [38] | Nekrasov, N., BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, JHEP, 03, 181, 2016 · Zbl 1388.81872 · doi:10.1007/JHEP03(2016)181 |
| [39] | Nekrasov, N., BPS/CFT correspondence II: Instantons at crossroads, Moduli and Compactness Theorem, Adv. Theor. Math. Phys., 21, 503, 2017 · Zbl 1508.81938 · doi:10.4310/ATMP.2017.v21.n2.a4 |
| [40] | Nekrasov, N., BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters, Commun. Math. Phys., 358, 863, 2018 · Zbl 1386.81125 · doi:10.1007/s00220-017-3057-9 |
| [41] | Kimura, T.; Pestun, V., Quiver W-algebras, Lett. Math. Phys., 108, 1351, 2018 · Zbl 1388.81850 · doi:10.1007/s11005-018-1072-1 |
| [42] | J.-E. Bourgine et al., (p, q)-webs of DIM representations, 5d \(\mathcal{N} = 1\) instanton partition functions and qq-characters, JHEP11 (2017) 034 [arXiv:1703.10759] [INSPIRE]. · Zbl 1383.83156 |
| [43] | Awata, H., (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces, JHEP, 03, 192, 2018 · Zbl 1388.81623 · doi:10.1007/JHEP03(2018)192 |
| [44] | J.-E. Bourgine and S. Jeong, New quantum toroidal algebras from 5D \(\mathcal{N} = 1\) instantons on orbifolds, JHEP05 (2020) 127 [arXiv:1906.01625] [INSPIRE]. · Zbl 1437.81055 |
| [45] | N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in the proceedings of the 16th International Congress on Mathematical Physics, Prague, Czechia, August 03-08 (2009) [doi:10.1142/9789814304634_0015] [arXiv:0908.4052] [INSPIRE]. · Zbl 1214.83049 |
| [46] | Poghossian, R., Deforming SW curve, JHEP, 04, 033, 2011 · Zbl 1250.81111 · doi:10.1007/JHEP04(2011)033 |
| [47] | N. Nekrasov, 2d CFT-type equations from 4d gauge theory, Lecture at the Langlands Program and Physics, conference at IAS, Princeton, U.S.A., March 8-10 (2004). |
| [48] | Gukov, S.; Witten, E., Rigid Surface Operators, Adv. Theor. Math. Phys., 14, 87, 2010 · Zbl 1203.81114 · doi:10.4310/ATMP.2010.v14.n1.a3 |
| [49] | Nekrasov, N., BPS/CFT correspondence IV: sigma models and defects in gauge theory, Lett. Math. Phys., 109, 579, 2019 · Zbl 1411.81213 · doi:10.1007/s11005-018-1115-7 |
| [50] | 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 |
| [51] | Jeong, S.; Nekrasov, N., Riemann-Hilbert correspondence and blown up surface defects, JHEP, 12, 006, 2020 · Zbl 1457.81057 · doi:10.1007/JHEP12(2020)006 |
| [52] | Jeong, S.; Lee, N.; Nekrasov, N., Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations, JHEP, 10, 120, 2021 · doi:10.1007/JHEP10(2021)120 |
| [53] | Koroteev, P.; Sage, DS; Zeitlin, AM, ( SL (N ), q)-Opers, the q-Langlands Correspondence, and Quantum/Classical Duality, Commun. Math. Phys., 381, 641, 2021 · Zbl 1462.82013 · doi:10.1007/s00220-020-03891-1 |
| [54] | Frenkel, E.; Koroteev, P.; Sage, DS; Zeitlin, AM, q-opers, QQ-systems, and Bethe Ansatz, J. Eur. Math. Soc., 26, 355, 2023 · Zbl 1534.82005 · doi:10.4171/jems/1268 |
| [55] | P. Koroteev and A.M. Zeitlin, The Zoo of Opers and Dualities, Int. Math. Res. Not.2024 (2024) 6850 [arXiv:2208.08031] [INSPIRE]. |
| [56] | Frassek, R.; Pestun, V., A Family of GL_rMultiplicative Higgs Bundles on Rational Base, SIGMA, 15, 031, 2019 · Zbl 1464.16029 |
| [57] | E. Mukhin and A. Varchenko, Discrete Miura Opers and Solutions of the Bethe Ansatz Equations, Commun. Math. Phys.256 (2005) 565. · Zbl 1127.82020 |
| [58] | Hollands, L.; Neitzke, A., Spectral Networks and Fenchel-Nielsen Coordinates, Lett. Math. Phys., 106, 811, 2016 · Zbl 1345.32020 · doi:10.1007/s11005-016-0842-x |
| [59] | Hollands, L.; Kidwai, O., Higher length-twist coordinates, generalized Heun’s opers, and twisted superpotentials, Adv. Theor. Math. Phys., 22, 1713, 2018 · Zbl 07430956 · doi:10.4310/ATMP.2018.v22.n7.a2 |
| [60] | Nekrasov, N.; Rosly, A.; Shatashvili, S., Darboux coordinates, Yang-Yang functional, and gauge theory, Nucl. Phys. B Proc. Suppl., 216, 69, 2011 · doi:10.1016/j.nuclphysbps.2011.04.150 |
| [61] | S. Jeong and N. Lee, Bispectral duality and separation of variables from surface defect transition, arXiv:2402.13889 [INSPIRE]. |
| [62] | S. Jeong and X. Zhang, BPZ equations for higher degenerate fields and non-perturbative Dyson-Schwinger equations, arXiv:1710.06970 [INSPIRE]. |
| [63] | Bazhanov, VV; Lukowski, T.; Meneghelli, C.; Staudacher, M., A Shortcut to the Q-Operator, J. Stat. Mech., 1011, 2010 · doi:10.1088/1742-5468/2010/11/P11002 |
| [64] | D. Talalaev, Quantization of the Gaudin system, hep-th/0404153 [INSPIRE]. · Zbl 1111.82015 |
| [65] | Chervov, A.; Falqui, G., Manin matrices and Talalaev’s formula, J. Phys. A, 41, 194006, 2008 · Zbl 1151.81022 · doi:10.1088/1751-8113/41/19/194006 |
| [66] | Y.I. Manin, Quantum Groups and Noncommutative Geometry, Springer International Publishing (2018) [doi:10.1007/978-3-319-97987-8]. · Zbl 1430.16001 |
| [67] | M. Nazarov and G. Olshanski, Bethe Subalgebras in Twisted Yangians, q-alg/9507003 [doi:10.1007/BF02099459]. · Zbl 0876.17015 |
| [68] | E. Mukhin, V. Tarasov and A. Varchenko, Bethe eigenvectors of higher transfer matrices, J. Stat. Mech.2006 (2006) P08002 [math/0605015]. · Zbl 1456.82301 |
| [69] | Chervov, A.; Talalaev, D., KZ equation, G-opers and quantum Drinfeld-Sokolov reduction, J. Math. Sci., 158, 904, 2009 · Zbl 1179.82052 · doi:10.1007/s10958-009-9415-1 |
| [70] | A. Chervov and D. Talalaev, Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, hep-th/0604128 [INSPIRE]. · Zbl 1179.82052 |
| [71] | A. Beilinson and J. Bernstein, Localisation de g-modules, C. R. Acad. Sci. Paris Sér. I Math.292(1) (1981) 15. · Zbl 0476.14019 |
| [72] | Dorey, N.; Lee, S.; Hollowood, TJ, Quantization of Integrable Systems and a 2d/4d Duality, JHEP, 10, 077, 2011 · Zbl 1303.81114 · doi:10.1007/JHEP10(2011)077 |
| [73] | N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. B Proc. Suppl.192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE]. · Zbl 1180.81125 |
| [74] | Nekrasov, NA; Shatashvili, SL, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl., 177, 105, 2009 · Zbl 1173.81325 · doi:10.1143/PTPS.177.105 |
| [75] | N. Lee and N. Nekrasov, Quantum spin systems and supersymmetric gauge theories. Part I, JHEP03 (2021) 093 [arXiv:2009.11199] [INSPIRE]. |
| [76] | E. Frenkel and N. Reshetikhin, Quantum affine algebras and deformations of the Virasoro and 237-1237-1237-1, Commun. Math. Phys.178 (1996) 237. · Zbl 0869.17014 |
| [77] | Aganagic, M.; Frenkel, E.; Okounkov, A., Quantum q-Langlands Correspondence, Trans. Moscow Math. Soc., 79, 1, 2018 · Zbl 1422.22021 · doi:10.1090/mosc/278 |
| [78] | E.E. Mukhin, V.O. Tarasov and A.N. Varchenko, Bispectral and \(\left({\mathfrak{gl}}_N,{\mathfrak{gl}}_M\right)\) dualities, Functional Analysis and Other Mathematics1 (2007) 47. |
| [79] | Mironov, A.; Morozov, A.; Zenkevich, Y.; Zotov, A., Spectral Duality in Integrable Systems from AGT Conjecture, JETP Lett., 97, 45, 2013 · doi:10.1134/S0021364013010062 |
| [80] | Bulycheva, K.; Chen, H-Y; Gorsky, A.; Koroteev, P., BPS States in Omega Background and Integrability, JHEP, 10, 116, 2012 · Zbl 1397.81344 · doi:10.1007/JHEP10(2012)116 |
| [81] | Gaiotto, D.; Koroteev, P., On Three Dimensional Quiver Gauge Theories and Integrability, JHEP, 05, 126, 2013 · Zbl 1342.81284 · doi:10.1007/JHEP05(2013)126 |
| [82] | N. Haouzi, A new realization of quantum algebras in gauge theory and Ramification in the Langlands program, arXiv:2311.04367 [INSPIRE]. |
| [83] | 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 |
| [84] | A. Grekov and N. Nekrasov, Elliptic Calogero-Moser system, crossed and folded instantons, and bilinear identities, arXiv:2310.04571 [INSPIRE]. |
| [85] | B. Feigin and E. Frenkel, Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras, Int. J. Mod. Phys. A7S1A (1992) 197 [INSPIRE]. · Zbl 0925.17022 |
| [86] | K. Iohara, Bosonic representations of Yangian double with, J. Phys. A29 (1996) 4593. · Zbl 0899.17009 |
| [87] | N. Jing, S. Kožić, A. Molev and F. Yang, Center of the quantum affine vertex algebra in type A, J. Algebra496 (2018) 138 [INSPIRE]. · Zbl 1432.17014 |
| [88] | E. Frenkel and N. Reshetikhin, Towards deformed chiral algebras, in the proceedings of the 21st International Colloquium on Group Theoretical Methods in Physics, Goslar, Germany, July 16-20 (1996) [INSPIRE]. · Zbl 0869.17014 |
| [89] | P. Etingof and D. Kazhdan, Quantization of Lie bialgebras, Part V: Quantum vertex operator algebras, Selecta Math.6 (2000) 105. · Zbl 0948.17008 |
| [90] | Y. Fan and N. Jing, Center of the Yangian double in type A, Sci. China Math. (2024) [arXiv:2207.01712]. |
| [91] | B.-Y. Hou and W.-L. Yang, A h-bar deformed Virasoro algebra as hidden symmetry of the restricted sine-Gordon model, hep-th/9612235 [INSPIRE]. |
| [92] | X.-M. Ding, B.-Y. Hou and L. Zhao, ħ (Yangian) deformation of the miura map and Virasoro algebra, Int. J. Mod. Phys. A13 (1998) 1129. · Zbl 0965.81036 |
| [93] | P. Etingof, E. Frenkel and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves, arXiv:1908.09677 [INSPIRE]. · Zbl 1475.14021 |
| [94] | Etingof, P.; Frenkel, E.; Kazhdan, D., Hecke operators and analytic Langlands correspondence for curves over local fields, Duke Math. J., 172, 2015, 2023 · Zbl 1539.11146 · doi:10.1215/00127094-2022-0068 |
| [95] | P. Etingof, E. Frenkel and D. Kazhdan, Analytic Langlands correspondence for PGL_2on ℙ^1with parabolic structures over local fields, Geom. Funct. Anal.32 (2022) 725 [arXiv:2106.05243] [INSPIRE]. · Zbl 1495.14021 |
| [96] | Etingof, P.; Frenkel, E.; Kazhdan, D., A general framework and examples of the analytic Langlands correspondence, Pure Appl. Math. Quart., 20, 307, 2024 · Zbl 07826260 · doi:10.4310/PAMQ.2024.v20.n1.a8 |
| [97] | Gaiotto, D.; Witten, E., Gauge Theory and the Analytic Form of the Geometric Langlands Program, Annales Henri Poincare, 25, 557, 2024 · Zbl 07802655 · doi:10.1007/s00023-022-01225-6 |
| [98] | D. Gaiotto and J. Teschner, Quantum Analytic Langlands Correspondence, arXiv:2402.00494 [INSPIRE]. |
| [99] | D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, arXiv:1211.1287 [INSPIRE]. · Zbl 1422.14002 |
| [100] | Aganagic, M.; Okounkov, A., Elliptic stable envelopes, J. Am. Math. Soc., 34, 79, 2021 · Zbl 1524.14098 · doi:10.1090/jams/954 |
| [101] | Bullimore, M.; Kim, H-C; Lukowski, T., Expanding the Bethe/Gauge Dictionary, JHEP, 11, 055, 2017 · Zbl 1383.81282 · doi:10.1007/JHEP11(2017)055 |
| [102] | M. Bullimore and D. Zhang, 3d \(\mathcal{N} = 4\) Gauge Theories on an Elliptic Curve, SciPost Phys.13 (2022) 005 [arXiv:2109.10907] [INSPIRE]. |
| [103] | N. Ishtiaque, S.F. Moosavian and Y. Zhou, Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical R-Matrices for Superspin Chains from The Bethe/Gauge Correspondence, arXiv:2308.12333 [INSPIRE]. |
| [104] | A.I. Molev, Yangians and their applications, Handbook of Algebra3 (2003) 907 [math/0211288] [INSPIRE]. · Zbl 1086.17008 |
| [105] | Loebbert, F., Lectures on Yangian Symmetry, J. Phys. A, 49, 323002, 2016 · Zbl 1356.81173 · doi:10.1088/1751-8113/49/32/323002 |
| [106] | V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl.32 (1985) 254 [INSPIRE]. · Zbl 0588.17015 |
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.
