Geometry of the Pfaff lattices

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.

💡 Research Summary

The paper revisits the Pfaff lattice, originally introduced by Adler and van Moerbeke as a tool for describing the partition functions of the Gaussian Orthogonal Ensemble (GOE) and Gaussian Symplectic Ensemble (GSE) random‑matrix models. In those models the partition functions are expressed as Pfaffians of skew‑symmetric “moment” matrices, and these Pfaffians serve as τ‑functions of the Pfaff lattice. The authors focus on a finite‑dimensional version of the lattice, treating it as a Hamiltonian system, and they establish several fundamental results.

First, they formulate the finite Pfaff lattice as a Lax equation
  L̇ =