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

TL;DR

This paper establishes a connection between solutions of the matrix KP hierarchy that are elliptic functions and the spin elliptic Calogero-Moser model, showing that pole dynamics are governed by Hamiltonian systems derived from spectral curve expansions.

Contribution

It extends the correspondence between elliptic solutions of the matrix KP hierarchy and the spin elliptic Calogero-Moser model to the entire hierarchy, identifying Hamiltonians with spectral curve expansions.

Findings

Pole dynamics are Hamiltonian systems with specific Hamiltonians.

Hamiltonians correspond to spectral curve expansions near marked points.

The work links matrix KP solutions to spin Calogero-Moser models across the hierarchy.

Abstract

We consider solutions of the matrix KP hierarchy that are elliptic functions of the first hierarchical time $t_1=x$. 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 $t_2$ 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 $k$-th 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 $a_i^{\alpha}, \, b_i^{\beta}$.