Genus zero Whitham hierarchy via Hurwitz--Frobenius manifolds

TL;DR

This paper establishes a connection between genus zero Hurwitz--Frobenius manifolds and the genus zero Whitham hierarchy, showing that the associated PDE system stabilizes and can be extended to the multicomponent KP hierarchy.

Contribution

It proves that the Frobenius potentials of genus zero Hurwitz--Frobenius manifolds define an infinite integrable PDE system equivalent to the genus zero Whitham hierarchy.

Findings

The PDE system stabilizes and defines an infinite set of commuting equations.

The system has both Fay and coordinate-free Lax forms.

Extension to multicomponent KP hierarchy via ar-deformation is possible.

Abstract

B. Dubrovin introduced the structure of a Dubrovin--Frobenius manifold on a space of ramified coverings of a sphere by a genus $g$ Riemann surface with the prescribed ramification profile. This is now known as a genus $g$ Hurwitz--Frobenius manifold. We investigate the genus zero Hurwitz--Frobenius manifolds and their connection to the integrable hierarchies. In particular, we prove that the Frobenius potentials of the genus zero Hurwitz--Frobenius manifolds stabilize and therefore define an infinite system of commuting PDEs. We show that this system of PDEs is equivalent to the genus zero Whitham hierarchy of I. Krichever. Our result shows that this system of PDEs has both Fay form, depending heavily on the flat structure fo the Hurwitz--Frobenius manifold and coordinatefree Lax form. We also show how to extend this system of PDEs to the multicomponent KP hierarchy via the $\hbar$--deformation of the differential operators.