Tropical Real Hurwitz numbers

TL;DR

This paper introduces tropical analogues of real Hurwitz numbers, proves their equivalence to classical ones through a correspondence theorem, and applies this to double Hurwitz numbers, expanding the understanding of real algebraic covers.

Contribution

It defines tropical real Hurwitz numbers, establishes a correspondence theorem with classical numbers, and extends results to double Hurwitz numbers.

Findings

Tropical real Hurwitz numbers are equivalent to classical real Hurwitz numbers.

The correspondence theorem bridges tropical and classical enumerations.

Application to double Hurwitz numbers generalizes previous results.

Abstract

In this paper, we define tropical analogues of real Hurwitz numbers, i.e. numbers of covers of surfaces with compatible involutions satisfying prescribed ramification properties. We prove a correspondence theorem stating the equality of the tropical numbers with their real counterparts. We apply this theorem to the case of double Hurwitz numbers (which generalizes our result from arXiv:1409.8095).