Elliptic solutions to matrix KP hierarchy and spin generalization of elliptic Calogero–Moser model

Abstract

We consider solutions of the matrix Kadomtsev–Petviashvili (KP) hierarchy that are elliptic functions of the first hierarchical time t₁ = x. It is known that poles x_i and matrix residues at the poles ${\rho }_i^{\alpha \beta }=a_i^{\alpha }{b}_i^{\beta }$ of such solutions as functions of the time t₂ move as particles of spin generalization of the elliptic Calogero–Moser model (elliptic Gibbons–Hermsen model). In this paper, we establish the correspondence with the spin elliptic Calogero–Moser model for the whole matrix KP hierarchy. Namely, we show that the dynamics of poles and matrix residues of the solutions with respect to the kth hierarchical time of the matrix KP hierarchy is Hamiltonian with the Hamiltonian H_k obtained via an expansion of the spectral curve near the marked points. The Hamiltonians are identified with the Hamiltonians of the elliptic spin Calogero–Moser system with coordinates x_i and spin degrees of freedom $a_i^{\alpha },b_i^{\beta }$ .