Emergent Geometry of KP Hierarchy

TL;DR

This paper develops a method to construct quantum deformations of spectral curves as tau-functions of the KP hierarchy, providing new insights into their geometric and intersection number applications.

Contribution

It introduces a novel quantum deformation framework for spectral curves that explains previous results and extends to higher Weil-Petersson volumes and Witten's r-spin intersection numbers.

Findings

Quantum deformation of spectral curves as KP tau-functions

Application to Witten-Kontsevich tau-function

Extension to higher Weil-Petersson volumes and r-spin numbers

Abstract

We explain how to construct a quantum deformation of a spectral curve to a tau-function of the KP hierarchy. This construction is applied to Witten-Kontsevich tau-function to give a natural explanation of some earlier work. We also apply it to higher Weil-Petersson volumes and Witten's r-spin intersection numbers.