Spectrum of quantum transfer matrices via classical many-body systems

In this paper 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.

Funding Agency	Grant Number
Russian Foundation for Basic Research	12-02-00284 12-02-91052 11-02-01220 12-02-91052 12-02-92108 14-01-00860
Ministry of Education and Science of the Russian Federation	8207 NSh-3349.2012.2 NSh-4724.2014.2
Dynasty Foundation
The work of A. G. was supported in part by grants RFBR-12-02-00284 and PICS-12-02-91052. A. G. thanks the organizers of Simons Summer School at Simons Center for Geometry and Physics where the part of this work has been done for the hospitality and support. The work of A.Zabrodin was supported in part by RFBR grant 11-02-01220, by joint RFBR grants 12-02-91052-CNRS, 12-02-92108-JSPS and by Ministry of Science and Education of Russian Federation under contract 8207 and by grant NSh-3349.2012.2 for support of leading scientific schools. The work of A.Zotov was supported in part by RFBR grants 14-01-00860, by grant NSh-4724.2014.2 for support of leading scientific schools and by the D. Zimin's fund "Dynasty".