Genus expansion of matrix models and $\hbar$ expansion of KP hierarchy

TL;DR

This paper explores the genus expansion of matrix model tau-functions within the $ ext{-}KP$ hierarchy, linking it to enumerative geometry and demonstrating the $$-expansion's equivalence to genus expansion in key models.

Contribution

It establishes that several matrix models' $ au$-functions are solutions of the $$-KP hierarchy with genus expansion matching the $$-expansion, connecting recent $$-formulations to Takasaki-Takebe's approach.

Findings

$$-expansion of matrix models matches genus expansion.

$ au$-functions of key models are solutions of $$-KP hierarchy.

Connection between recent $$-formulations and original Takasaki-Takebe approach.

Abstract

We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe arXiv:hep-th/9405096 considering several examples of matrix model $\tau$-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter $\hbar$ are $\tau$-functions of the $\hbar$-KP hierarchy and the expansion in $\hbar$ for the $\hbar$-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the $\hbar$-formulation of the KP hierarchy arXiv:1509.04472, arXiv:1512.07172 with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of $\tau$-functions is straightforward and algorithmic.