Asymptotic Hurwitz numbers

TL;DR

This paper generalizes Hurwitz numbers to all degrees simultaneously, linking them to topological field theory and differential operators, revealing new algebraic relations and connections to string theory models.

Contribution

It extends the construction of Hurwitz numbers to all degrees at once, introducing a new algebraic framework based on Young diagrams and bipartite graphs.

Findings

The algebra of differential operators models open-closed string theory.

Operators from Young diagrams and bipartite graphs produce relations for Hurwitz numbers.

The cut-and-join operator corresponds to the transposition Young diagram.

Abstract

The classical Hurwitz numbers of degree n together with the Hurwitz numbers of the seamed surfaces of degree n give rise to the Klein topological field theory. We extend this construction to the Hurwitz numbers of all degrees at once. The corresponding Cardy-Frobenius algebra is induced by arbitrary Young diagrams and arbitrary bipartite graphs. It turns out to be isomorphic to the algebra of differential operators from arXiv:1210.6955 which serves a model for open-closed string theory. The operator associated with the Young diagram of the transposition of two elements coincides with the cut-and-join operator which gives rise to relations for the classical Hurwitz numbers. We prove that the operators corresponding to arbitrary Young diagrams and bipartite graphs also give rise to relations for the Hurwitz numbers.