Geometry of Spectral Curves and All Order Dispersive Integrable System

TL;DR

This paper introduces a formal asymptotic framework for Tau functions and spinor kernels parametrized by algebraic curve deformations, connecting integrable systems, topological recursion, and geometry.

Contribution

It defines a new asymptotic series for Tau functions involving theta functions, linking large N limits of algebraic curve deformations to integrable hierarchies and geometric structures.

Findings

Tau function satisfies Hirota equations to first two orders

Conjecture that Hirota equations hold to all orders

Reconstruction of isomonodromic problems from the spinor kernel

Abstract

We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between "correlators", the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.