Tau functions, infinite Grassmannians and lattice recurrences

TL;DR

This paper explores how evaluating KP and CKP tau functions at lattice points in infinite Grassmannians yields solutions to various integrable lattice recurrences, connecting algebraic geometry with discrete integrable systems.

Contribution

It introduces a geometric framework linking tau functions on Grassmannians to solutions of lattice recurrences like the Kashaev and hexahedron relations.

Findings

Solutions to hyperdeterminantal relations derived from KP tau functions.

Discretizations of KP and CKP hierarchies produce integrable lattice recurrences.

Connections established between infinite Grassmannians and discrete integrable systems.

Abstract

The addition formulae for KP $\tau$-functions, when evaluated at lattice points in the KP flow group orbits in the infinite dimensional Sato-Segal-Wilson Grassmannian, give infinite parametric families of solutions to discretizations of the KP hierarchy. The CKP hierarchy may similarly be viewed as commuting flows on the Lagrangian sub-Grassmannian of maximal isotropic subspaces with respect to a suitably defined symplectic form. Evaluating the $\tau$-functions at a sublattice of points within the KP orbit, the resulting discretization gives solutions both to the hyperdeterminantal relations (or Kashaev recurrence) and the hexahedron (or Kenyon-Pemantle) recurrence.