The Stability Limit of Prepotentials for Hurwitz-Frobenius Manifolds: An Infinite-Dimensional Approach

TL;DR

This paper provides a geometric proof linking the stability of prepotentials in Hurwitz-Frobenius manifolds to the tau-structure of the Whitham hierarchy, extending understanding of integrable hierarchies in infinite-dimensional settings.

Contribution

It offers a direct geometric proof of the correspondence between prepotential stability and the Whitham hierarchy within infinite-dimensional Frobenius manifolds, extending to open WDVV solutions.

Findings

Stability of prepotentials is intrinsic to the tau-structure of the Whitham hierarchy.

The identification is extended to hierarchies from open WDVV equations.

Provides a geometric explanation for the genus-zero case.

Abstract

The stability of prepotential derivatives for Frobenius manifolds associated with A_N and D_N singularities has been utilized to construct (2+1)-dimensional dispersionless integrable hierarchies. Although the generalization of this construction to genus-zero Hurwitz-Frobenius manifolds was shown to yield the genus-zero Whitham hierarchy, a direct geometric explanation of this correspondence has been lacking. In this note, we provide a direct proof of this identification within the framework of infinite-dimensional Frobenius manifolds. We demonstrate that the stability of prepotentials is an intrinsic property of the tau-structure of the Whitham hierarchy. Furthermore, we extend this identification to the hierarchies arising from the stability of solutions to the open WDVV equations with the extensions of the Whitham hierarchy.