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Spectrum of quantum transfer matrices via classical many-body systems. (English) Zbl 1333.81244
Summary: We clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous $${\mathfrak {gl}}_{n}$$-invariant XXX spin chain on $$N$$ sites with twisted boundary conditions can be found in terms of velocities of particles in the rational $$N$$-body Ruijsenaars-Schneider model. The possible values of the velocities are to be found from intersection points of two Lagrangian submanifolds in the phase space of the classical model. One of them is the Lagrangian hyperplane corresponding to fixed coordinates of all $$N$$ particles and the other one is an $$N$$-dimensional Lagrangian submanifold obtained by fixing levels of $$N$$ classical Hamiltonians in involution. The latter are determined by eigenvalues of the twist matrix. To support this picture, we give a direct proof that the eigenvalues of the Lax matrix for the classical Ruijsenaars-Schneider model, where velocities of particles are substituted by eigenvalues of the spin chain Hamiltonians, calculated through the Bethe equations, coincide with eigenvalues of the twist matrix, with certain multiplicities. We also prove a similar statement for the $${\mathfrak {gl}}_{n}$$ Gaudin model with $$N$$ marked points (on the quantum side) and the Calogero-Moser system with $$N$$ particles (on the classical side). The realization of the results obtained in terms of branes and supersymmetric gauge theories is also discussed.

##### MSC:
 81T13 Yang-Mills and other gauge theories in quantum field theory 81R12 Groups and algebras in quantum theory and relations with integrable systems
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##### References:
 [1] 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 [2] Seiberg, N.; Witten, E., Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys., B 431, 484, (1994) · Zbl 1020.81911 [3] 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 [4] Donagi, R.; Witten, E., Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys., B 460, 299, (1996) · Zbl 0996.37507 [5] A. Gorsky and A. Mironov, Integrable many body systems and gauge theories, hep-th/0011197 [INSPIRE]. · Zbl 0979.81096 [6] Adams, M.; Harnad, JP; Hurtubise, J., Dual moment maps into loop algebras, Lett. Math. Phys., 20, 299, (1990) · Zbl 0721.58025 [7] Harnad, JP, Dual isomonodromic deformations and moment maps to loop algebras, Commun. Math. Phys., 166, 337, (1994) · Zbl 0822.58018 [8] E. Mukhin, V. Tarasov and A. Varchenko, Bispectral and (gl_{$$N$$}, gl_{$$M$$} ) dualities, math.QA/0510364. [9] E. Mukhin, V. Tarasov and A. Varchenko, Bispectral and (gl_{$$N$$}, gl_{$$M$$} ) dualities, discrete versus differential, math.QA/0605172. · Zbl 1142.82005 [10] Mironov, A.; Morozov, A.; Runov, B.; Zenkevich, Y.; Zotov, A., Spectral duality between Heisenberg chain and Gaudin model, Lett. Math. Phys., 103, 299, (2013) · Zbl 1262.14040 [11] M. Gaudin, Modèles exacts en mècanique statistique: la mèthode de Bethe et ses gènèralizations, Note CEA 1559 1 and 2 (1972). [12] Gaudin, M., Diagonalisation dùne classe d’hamiltoniens de spin, J. de Physique, 37, 1087, (1976) [13] Calogero, F., Exactly solvable one-dimensional many body problems, Lett. Nuovo Cim., 13, 411, (1975) [14] Calogero, F., On a functional equation connected with integrable many body problems, Lett. Nuovo Cim., 16, 77, (1976) [15] Moser, J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., 16, 1, (1975) [16] Olshanetsky, M.; Perelomov, A., Classical integrable finite dimensional systems related to Lie algebras, Phys. Rept., 71, 313, (1981) [17] Ruijsenaars, SNM; Schneider, H., A new class of integrable systems and its relation to solitons, Ann. Phys., 146, 1, (1986) · Zbl 0554.47003 [18] Ruijsenaars, SNM, Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Commun. Math. Phys., 110, 191, (1987) · Zbl 0673.58024 [19] Gorsky, A.; Gukov, S.; Mironov, A., Multiscale N = 2 SUSY field theories, integrable systems and their stringy/brane origin. 1, Nucl. Phys., B 517, 409, (1998) · Zbl 0920.58068 [20] Braden, H.; Marshakov, A.; Mironov, A.; Morozov, A., On double elliptic integrable systems. 1. A duality argument for the case of SU(2), Nucl. Phys., B 573, 553, (2000) · Zbl 0947.81025 [21] Ruijsenaars, S., Action angle maps and scattering theory for some finite dimensional integrable systems. 1. the pure soliton case, Commun. Math. Phys., 115, 127, (1988) · Zbl 0667.58016 [22] Fock, V.; Gorsky, A.; Nekrasov, N.; Rubtsov, V., Duality in integrable systems and gauge theories, JHEP, 07, 028, (2000) · Zbl 0965.81025 [23] Bulycheva, K.; Chen, H-Y; Gorsky, A.; Koroteev, P., BPS states in omega background and integrability, JHEP, 10, 116, (2012) [24] Mironov, A.; Morozov, A.; Zenkevich, Y.; Zotov, A., Spectral duality in integrable systems from AGT conjecture, JETP Lett., 97, 45, (2013) [25] Mironov, A.; Morozov, A.; Runov, B.; Zenkevich, Y.; Zotov, A., Spectral dualities in XXZ spin chains and five dimensional gauge theories, JHEP, 12, 034, (2013) · Zbl 1342.81310 [26] Mangazeev, VV; Bazhanov, VV; Sergeev, SM, An integrable 3D lattice model with positive Boltzmann weights, J. Phys., A 46, 465206, (2013) · Zbl 1291.82027 [27] Mironov, A.; Morozov, A.; Shakirov, S., Towards a proof of AGT conjecture by methods of matrix models, Int. J. Mod. Phys., A 27, 1230001, (2012) · Zbl 1247.81397 [28] Gaiotto, D., N = 2 dualities, JHEP, 08, 034, (2012) [29] Alday, LF; Gaiotto, D.; Tachikawa, Y., Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys., 91, 167, (2010) · Zbl 1185.81111 [30] Wyllard, N., A(N − 1) conformal Toda field theory correlation functions from conformal N =2 SU(N) quiver gauge theories, JHEP, 11, 002, (2009) [31] Mironov, A.; Morozov, A., On AGT relation in the case of U (3), Nucl. Phys., B 825, 1, (2010) · Zbl 1196.81205 [32] Mironov, A.; Morozov, A., Nekrasov functions and exact Bohr-zommerfeld integrals, JHEP, 04, 040, (2010) · Zbl 1272.81180 [33] Nekrasov, NA; Shatashvili, SL, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl., 192-193, 91, (2009) · Zbl 1180.81125 [34] N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE]. · Zbl 1214.83049 [35] Gaiotto, D.; Koroteev, P., On three dimensional quiver gauge theories and integrability, JHEP, 05, 126, (2013) · Zbl 1342.81284 [36] Givental, A.; Kim, B-s, Quantum cohomology of flag manifolds and Toda lattices, Commun. Math. Phys., 168, 609, (1995) · Zbl 0828.55004 [37] Mukhin, E.; Tarasov, V.; Varchenko, A., Gaudin Hamiltonians generate the Bethe algebra of a tensor power of vector representation of gl_{N}, St. Petersburg Math. J., 22, 463, (2011) · Zbl 1219.82121 [38] Mukhin, E.; Tarasov, V.; Varchenko, A., KZ characteristic variety as the zero set of classical Calogero-Moser Hamiltonians, SIGMA, 8, 72, (2012) · Zbl 1269.82018 [39] Alexandrov, A.; Kazakov, V.; Leurent, S.; Tsuboi, Z.; Zabrodin, A., Classical tau-function for quantum spin chains, JHEP, 09, 064, (2013) · Zbl 1342.81179 [40] Zabrodin, A., The master T-operator for vertex models with trigonometric R-matrices as classical tau-function, Teor. Mat. Fys., 171, 59, (2013) · Zbl 1282.82019 [41] A. Alexandrov, S. Leurent, Z. Tsuboi and A. Zabrodin, The master T-operator for the Gaudin model and the KP hierarchy, arXiv:1306.1111 [INSPIRE]. · Zbl 1323.37037 [42] Krichever, I.; Lipan, O.; Wiegmann, P.; Zabrodin, A., Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations, Commun. Math. Phys., 188, 267, (1997) · Zbl 0896.58035 [43] Faddeev, L.; Sklyanin, E.; Takhtajan, L., The quantum inverse problem method. 1, Theor. Math. Phys., 40, 688, (1980) · Zbl 1138.37331 [44] Takhtajan, L.; Faddeev, L., The quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surveys, 34, 11, (1979) [45] L. Faddeev, How algebraic Bethe ansatz works for integrable model, hep-th/9605187 [INSPIRE]. · Zbl 0934.35170 [46] V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambride University Press, Cambridge U.K. (1997). · Zbl 0787.47006 [47] M. Gaudin, La fonction d’onde de Bethe, Masson, Paris, France (1983). · Zbl 0509.60093 [48] Slavnov, NA, The algebraic Bethe ansatz and quantum integrable systems, Russ. Math. Surv., 62, 727, (2007) · Zbl 1141.81012 [49] Nekrasov, N.; Rosly, A.; Shatashvili, S., Darboux coordinates, Yang-Yang functional and gauge theory, Nucl. Phys. Proc. Suppl., 216, 69, (2011) [50] P.P. Kulish and N.Yu. Reshetikhin, Diagonalisation of gl_{$$N$$}invariant transfer matrices and quantum N-wave system (Lee model), J. Phys.A 16 (1983) L591. [51] Mukhin, E.; Tarasov, V.; Varchenko, A., Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp., 8, 2, (2006) [52] Belliard, S.; Ragoucy, É, Nested Bethe ansatz for ’all’ closed spin chains, J. Phys., A 41, 295202, (2008) · Zbl 1142.82326 [53] Arnaudon, A.; Crampé, N.; Doikou, A.; Frappat, L.; Ragoucy, E., Analytical Bethe ansatz for closed and open gl_{n}-spin chains in any representation, J. Stat. Mech, 0502, p02007, (2005) [54] G. Aminov, S. Arthamonov, A. Smirnov and A. Zotov, Modifications of bundles as generating functions of Lax operators, Preprint ITEP-TH 27/13. [55] Nekrasov, N.; Witten, E., The omega deformation, branes, integrability and Liouville theory, JHEP, 09, 092, (2010) · Zbl 1291.81265 [56] Gaiotto, D.; Witten, E., S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys., 13, 721, (2009) · Zbl 1206.81082 [57] Gaiotto, D.; Witten, E., Supersymmetric boundary conditions in N = 4 super Yang-Mills theory, J. Statist. Phys., 135, 789, (2009) · Zbl 1178.81180 [58] Gorsky, A.; Nekrasov, N., Relativistic Calogero-Moser model as gauged WZW theory, Nucl. Phys., B 436, 582, (1995) · Zbl 1052.81607 [59] K. Bulycheva and A. Gorsky BPS states in Ω-background and torus knots invariants, to appear. · Zbl 1333.81161 [60] G. Aminov, S. Arthamonov, A. Levin, M. Olshanetsky and A. Zotov, Painlevé field theory, arXiv:1306.3265 [INSPIRE]. [61] Levin, AM; Olshanetsky, MA; Zotov, A., Hitchin systems — symplectic Hecke correspondence and two-dimensional version, Commun. Math. Phys., 236, 93, (2003) · Zbl 1088.14008 [62] Zotov, A.; Levin, A., Integrable model of interacting elliptic tops, Theor. Math. Phys., 146, 45, (2006) · Zbl 1177.37061 [63] Levin, A.; Olshanetsky, M.; Smirnov, A.; Zotov, A., Characteristic classes and integrable systems. general construction, Commun. Math. Phys., 316, 1, (2012) · Zbl 1252.30029 [64] Levin, A.; Olshanetsky, M.; Smirnov, A.; Zotov, A., Characteristic classes and integrable systems for simple Lie groups, J. Geom. Phys., 62, 1810, (2012) · Zbl 1261.37031 [65] Levin, A.; Olshanetsky, M.; Smirnov, A.; Zotov, A., Hecke transformations of conformal blocks in WZW theory. I. KZB equations for non-trivial bundles, SIGMA, 8, 095, (2012) · Zbl 1270.14018 [66] Levin, A.; Olshanetsky, M.; Smirnov, A.; Zotov, A., Characteristic classes of SL(N)-bundles and quantum dynamical elliptic R-matrices, J. Phys., A 46, 035201, (2013) · Zbl 1273.81117 [67] Levin, A.; Olshanetsky, M.; Zotov, A., Painleve VI, rigid tops and reflection equation, Commun. Math. Phys., 268, 67, (2006) · Zbl 1116.37038
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