Solutions to Open WDVV Equations for the Universal Whitham Hierarchy

TL;DR

This paper constructs explicit solutions to the open WDVV equations linked to the universal Whitham hierarchy, expanding the understanding of Frobenius manifolds and their hierarchies in integrable systems.

Contribution

It provides new explicit solutions to open WDVV equations for infinite-dimensional Frobenius manifolds and their reductions, connecting to known singularity cases.

Findings

Constructed solutions for infinite-dimensional Frobenius manifolds.

Demonstrated compatibility with finite-dimensional reductions.

Recovered polynomial solutions for A- and D-type singularities.

Abstract

In this paper, we construct a pair of solutions to the open WDVV equations associated with the infinite-dimensional Frobenius manifolds that underlie the genus-zero universal Whitham hierarchy, and for the resulting flat F-manifolds, we explicitly construct their principal hierarchies. We further demonstrate that this construction is compatible with finite-dimensional reductions, yielding solutions for Frobenius manifolds associated with general rational superpotentials and those subject to a $\mathbb{Z}_{2}$-symmetry reduction. In particular, the polynomial solutions derived by Basalaev and Buryak via open Saito theory for A- and D-type singularities are recovered as special cases.