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

Abstract

AbstractWe continue the investigation of the connection between the genus expansion of matrix models and the $$\hbar $$ ħ expansion of integrable hierarchies. In this paper, we focus on the BKP 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 $$\hbar $$ ħ , which controls the genus expansion, are solutions of the $$\hbar $$ ħ -BKP hierarchy with good quasi-classical behavior. $$\hbar $$ ħ -BKP language implies the algorithmic prescription for $$\hbar $$ ħ -deformation of the mentioned models in terms of hypergeometric BKP $$\tau $$ τ -functions and gives insight into the similarities and differences between the models. Firstly, the insertion of $$\hbar $$ ħ 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 $$\tau $$ τ -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 BKP: 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.