Communications in Number Theory and Physics

Volume 17 (2023)

Number 3

Enumeration of hypermaps and Hirota equations for extended rationally constrained KP

Pages: 643 – 708



G. Carlet (Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université Bourgogne Franche-Comté, Dijon, France)

J. van de Leur (Mathematical Institute, Utrecht University, Utrecht, The Netherlands)

H. Posthuma (Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands)

S. Shadrin (Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands)


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.


Frobenius manifolds, Kadomtsev–Petviashvili hierarchy, Hirota equations, Lax equations, enumeration of hypermaps

2010 Mathematics Subject Classification

Primary 14H81, 37K10, 53D45. Secondary 05A15, 05C30, 14H70, 37K20.

This work is supported by the EIPHI Graduate School (contract ANR-17-EURE-0002). H. P. and S. S. were supported by the Netherlands Organization for Scientific Research.

Received 29 November 2022

Accepted 30 June 2023

Published 7 November 2023