The Hamiltonian structure of discrete KP equations

Abstract

This paper investigates Hamiltonian properties of the algebro-geometric discretization of KP hierarchy introduced in \cite{Gie1}. A Poisson bracket is introduced. The system is related to the periodic band matrix system of \cite{vM-M}. It is shown that the bracket descends to the latter and endows it with bi-Hamiltonian structure together with the first bracket already considered in \cite{vM-M}. On the other hand a bi-Hamiltonian structure for discrete KP seems to be absent for fundamental reasons. It is proven that the conserved quantities of both systems are in involution with respect to the bracket. A construction relating the bracket to a certain intersection pairing of cycles on a discrete torus is shown. This pairing is reminiscent of the intersection pairing in ``string topology'' \cite{C-S}.