Geometry of the Pfaff lattices

TL;DR

This paper studies the finite Pfaff lattice as a Hamiltonian system, proving its complete integrability and describing the structure of its isospectral varieties using moment maps and convex polytopes.

Contribution

It establishes the complete integrability of the finite Pfaff lattice and characterizes its real isospectral varieties via moment maps and convex polytopes.

Findings

Proves the complete integrability of the finite Pfaff lattice.

Describes the real isospectral varieties using moment maps.

Identifies the moment map image as a convex polytope with fixed points as vertices.

Abstract

Pfaff lattice was introduced by Adler and van Moerbeke to describe the partition functions for the random matrix models of GOE and GSE type. The partition functions of those matrix models are given by the Pfaffians of certain skew-symmetric matrices called the moment matrices, and they are the $\tau$-functions of the Pfaff lattice. In this paper, we study a finite version of the Pfaff lattice equation as a Hamiltonian system. In particular, we prove the complete integrability in the sense of Arnold-Liouville, and using a moment map, we describe the real isospectral varieties of the Pfaff lattice. The image of the moment map is a convex polytope whose vertices are identified as the fixed points of the flow generated by the Pfaff lattice.