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

TL;DR

This paper explores the connection between genus expansion in matrix models and the ar expansion in the ar-BKP hierarchy, focusing on models like BGW, Kontsevich, and spin Hurwitz numbers, revealing new deformation insights.

Contribution

It introduces an ar-arBKP framework for these models, providing an algorithmic approach to ar-deformations and highlighting differences in genus expansion prescriptions.

Findings

Partition functions are solutions of ar-BKP hierarchy with good quasi-classical behavior.

Kontsevich model is a specific hypergeometric ar-BKP ar-tau function.

Spin Hurwitz numbers have a different genus expansion prescription.

Abstract

We continue the investigation of the connection between the genus expansion of matrix models and the $\hbar$ expansion of integrable hierarchies started in arXiv:2008.06416. In this paper, we focus on the $B$KP hierarchy, which corresponds to the infinite-dimensional Lie algebra of type $B$. We consider the genus expansion of such important solutions as Br\'{e}zin-Gross-Witten (BGW) model, Kontsevich model, and generating functions for spin Hurwitz numbers with completed cycles. We show that these partition functions with inserted parameter $\hbar$, which controls the genus expansion, are solutions of the $\hbar$-$B$KP hierarchy with good quasi-classical behavior. $\hbar$-$B$KP language implies the algorithmic prescription for $\hbar$-deformation of the mentioned models in terms of hypergeometric $B$KP $\tau$-functions and gives insight into the similarities and differences between the models. Firstly, the insertion of $\hbar$ into the Kontsevich model is similar to the one in the BGW model, though the Kontsevich model seems to be a very specific example of hypergeometric $\tau$-function. Secondly, generating functions for spin Hurwitz numbers appear to possess a different prescription for genus expansion. This property of spin Hurwitz numbers is not the unique feature of $B$KP: already in the KP hierarchy, one can observe that generating functions for ordinary Hurwitz numbers with completed cycles are deformed differently from the standard matrix model examples.