Open Hurwitz numbers and the mKP hierarchy

Abstract

We give a natural definition of open Hurwitz numbers, where the weight of each ramified covering includes an integer parameter $N$ taken to the power that is equal to the number of boundary components of a Riemann surface with boundary mapping to $\mathbb{CP}^1$. We prove that the resulting sequence of partition functions, depending on $N\in\mathbb{Z}$, is a tau-sequence of the mKP hierarchy, or in other words it is a sequence of tau-functions of the KP hierarchy where each tau-function is obtained from the previous one by a B\"acklund-Darboux transformation. Our result is motivated by a previous observation of Alexandrov and the first two authors that the refined intersection numbers on the moduli spaces of Riemann surfaces with boundary give a tau-sequence of the mKP hierarchy.