1,308 research outputs found
QKZ-Ruijsenaars correspondence revisited
We discuss the Matsuo-Cherednik type correspondence between the quantum
Knizhnik-Zamolodchikov equations associated with and the -particle
quantum Ruijsenaars model, with being not necessarily equal to . The
quasiclassical limit of this construction yields the quantum-classical
correspondence between the quantum spin chains and the classical Ruijsenaars
models.Comment: 14 pages, minor correction
Self-dual form of Ruijsenaars-Schneider models and ILW equation with discrete Laplacian
We discuss a self-dual form or the B\"acklund transformations for the
continuous (in time variable) Ruijsenaars-Schneider model. It is
based on the first order equations in complex variables which include
positions of particles and dual variables. The latter satisfy equations of
motion of the Ruijsenaars-Schneider model. In the elliptic case it
holds while for the rational and trigonometric models is not
necessarily equal to . Our consideration is similar to the previously
obtained results for the Calogero-Moser models which are recovered in the
non-relativistic limit. We also show that the self-dual description of the
Ruijsenaars-Schneider models can be derived from complexified intermediate long
wave equation with discrete Laplacian be means of the simple pole ansatz
likewise the Calogero-Moser models arise from ordinary intermediate long wave
and Benjamin-Ono equations.Comment: 16 pages, references adde
Supersymmetric extension of qKZ-Ruijsenaars correspondence
We describe the correspondence of the Matsuo-Cherednik type between the
quantum -body Ruijsenaars-Schneider model and the quantum
Knizhnik-Zamolodchikov equations related to supergroup . The spectrum
of the Ruijsenaars-Schneider Hamiltonians is shown to be independent of the
-grading for a fixed value of , so that different
qKZ systems of equations lead to the same -body quantum problem. The
obtained results can be viewed as a quantization of the previously described
quantum-classical correspondence between the classical -body
Ruijsenaars-Schneider model and the supersymmetric quantum spin
chains on sites.Comment: 17 page
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 -invariant XXX spin chain on
sites with twisted boundary conditions can be found in terms of velocities of
particles in the rational -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
particles and the other one is an -dimensional Lagrangian submanifold
obtained by fixing levels of 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 Gaudin model with marked points (on the quantum side)
and the Calogero-Moser system with particles (on the classical side). The
realization of the results obtained in terms of branes and supersymmetric gauge
theories is also discussed.Comment: 25 pages, minor correction
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