Deformations of Frobenius structures on Hurwitz spaces

TL;DR

This paper constructs deformations of Frobenius structures on Hurwitz spaces, linking them to Painlevé-VI solutions and exploring their properties across different genera.

Contribution

It introduces new deformations of Hurwitz Frobenius manifolds depending on complex parameters and relates them to Painlevé equations, expanding the understanding of their geometric structures.

Findings

Deformations depend on g(g+1)/2 complex parameters.

In genus one, the metric relates to Painlevé-VI solutions.

Real double deformations depend on g(g+1)/2 real parameters.

Abstract

Deformations of Dubrovin's Hurwitz Frobenius manifolds are constructed. The deformations depend on $g(g+1)/2$ complex parameters where $g$ is the genus of the corresponding Riemann surface. In genus one, the flat metric of the deformed Frobenius manifold coincides with a metric associated with a one-parameter family of solutions to the Painlev\'e-VI equation with coefficients $(1/8,-1/8,1/8,3/8).$ Analogous deformations of the real doubles of the Hurwitz Frobenius manifolds are also found; these deformations depend on $g(g+1)/2$ real parameters.