Frobenius structures on double Hurwitz spaces

TL;DR

This paper constructs Frobenius structures on double Hurwitz spaces, generalizing Dubrovin's work, and explores their applications to integrable hierarchies like the q-deformed Gelfand-Dickey and sine-Gordon equations.

Contribution

It introduces Frobenius structures of dual type on moduli spaces of ramified coverings, extending existing frameworks and linking to integrable systems and WDVV equations.

Findings

Constructed Frobenius structures on double Hurwitz spaces.

Derived hierarchies of hydrodynamic type from deformed flat connections.

Computed solutions to WDVV equations for specific integrable hierarchies.

Abstract

We construct Frobenius structures of "dual type" on the moduli space of ramified coverings of $\mathbb{P}^1$ with given ramification type over two points, generalizing a construction of Dubrovin. A complete hierarchy of hydrodynamic type is obtained from the corresponding deformed flat connection. This provides a suitable framework for the Whitham theory of an enlarged class of integrable hierarchies; we treat as examples the q-deformed Gelfand-Dickey hierarchy and the sine-Gordon equation, and compute the corresponding solutions of the WDVV equations.