Primary invariants of Hurwitz Frobenius manifolds

TL;DR

This paper explores the structure of Hurwitz Frobenius manifolds, linking their primary invariants to topological recursion and periods on Riemann surfaces, providing new insights into their geometric and algebraic properties.

Contribution

It establishes a connection between Hurwitz Frobenius manifolds' primary invariants and topological recursion, offering a method to compute invariants as periods of multidifferentials.

Findings

Primary invariants are obtained as periods of multidifferentials.

Topological recursion computes invariants for a large class of Riemann surfaces.

The correspondence clarifies the geometric meaning of topological recursion.

Abstract

Hurwitz spaces parameterizing covers of the Riemann sphere can be equipped with a Frobenius structure. In this review, we recall the construction of such Hurwitz Frobenius manifolds as well as the correspondence between semisimple Frobenius manifolds and the topological recursion formalism. We then apply this correspondence to Hurwitz Frobenius manifolds by explaining that the corresponding primary invariants can be obtained as periods of multidifferentials globally defined on a compact Riemann surface by topological recursion. Finally, we use this construction to reply to the following question in a large class of cases: given a compact Riemann surface, what does the topological recursion compute?