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

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.