The structure of Hurwitz numbers with fixed ramification profile and varying genus

Abstract

In 1891, Hurwitz introduced the enumeration of genus $g$, degree $d$, branched covers of the Riemann sphere with simple ramification over prescribed points and no branching elsewhere. He showed that for fixed degree $d$, the enumeration possesses a remarkable structure. More precisely, it can be expressed as a linear combination of exponentials $m^{2g-2+2d}$, where $m$ ranges over the integers from $1$ to $\binom{d}{2}$. In this paper, we generalise this structural result to Hurwitz numbers that enumerate branched covers which also have a prescribed ramification profile over one point. Our proof fundamentally uses the infinite wedge space, in particular the connected correlators of products of $\mathcal{E}$-operators. The recent study of Hurwitz numbers has often focussed on their structure with fixed genus and varying ramification profile. Our main result is orthogonal to this, allowing for the explicit calculation and the asymptotic analysis of Hurwitz numbers in large genus. We pose the broad question of which other enumerative problems exhibit analogous structure. We prove that orbifold Hurwitz numbers can also be expressed as a linear combination of exponentials and conjecture that monotone Hurwitz numbers share a similar structure, but with the inclusion of an additional linear term.