Enumeration of hypermaps and Hirota equations for extended rationally constrained KP

TL;DR

This paper proves that the all genera partition function for a specific Hurwitz Frobenius manifold is a tau function of a rational KP hierarchy reduction, linking it to hypermap enumeration.

Contribution

It establishes the connection between the partition function of a Hurwitz Frobenius manifold and a rational KP hierarchy tau function, confirming a conjecture and providing enumerative interpretation.

Findings

Partition function is a tau function of a rational KP hierarchy reduction.

Partial enumerative meaning for the partition function related to hypermaps.

Confirmed conjecture by Liu, Zhang, and Zhou.

Abstract

We consider the Hurwitz Dubrovin--Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also known as the total descendant potential) associated with this Dubrovin--Frobenius manifold is a tau function of a rational reduction of the Kadomtsev--Petviashvili hierarchy. This statement was conjectured by Liu, Zhang, and Zhou. We also provide a partial enumerative meaning for this partition function associating one particular set of times with enumeration of rooted hypermaps.