Genus zero Whitham hierarchy via Hurwitz--Frobenius manifolds

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.