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

Abstract

We continue the investigation of the connection between the genus expansion of matrix models and the $\hslash$ 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 Brézin-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 $\hslash$, which controls the genus expansion, are solutions of the $\hslash$-$B$KP hierarchy with good quasi-classical behavior. $\hslash$-$B$KP language implies the algorithmic prescription for $\hslash$-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 $\hslash$ 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.

arXiv comment36 pages

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