Primitive forms and Frobenius structures on the Hurwitz spaces

TL;DR

This paper introduces primitive forms for Hurwitz covers, establishing a correspondence with semi-simple Frobenius structures, and explores their relation to Frobenius manifolds and the Eynard--Orantin recursion.

Contribution

It defines primitive forms for Hurwitz spaces, links them to Frobenius structures, and connects these concepts to Frobenius manifolds and recursion methods.

Findings

Primitive forms correspond to semi-simple Frobenius structures.

Polynomial primitive forms relate to Dubrovin's Hurwitz Frobenius manifolds.

The theory links Eynard--Orantin recursion with Frobenius manifolds.

Abstract

The main goal of this paper is to introduce the notion of a primitive form for a generic family of Hurwitz covers of $\mathbb{P}^1$ with a fixed ramification profile over infinity. We prove that primitive forms are in one-to-one correspondence with semi-simple Frobenius structures on the base of the family. Furthermore, we introduce the notion of a polynomial primitive form and show that the corresponding class of Frobenius manifolds contains the Hurwitz Frobenius manifolds of Dubrovin. Finally, we apply our theory to investigate the relation between the Eynard--Orantin recursion and Frobenius manifolds.