Frobenius manifolds were introduced by Dubrovin to give a geometric
reformulation of the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde)
system of differential equations, which describes deformations of
topological field theories. The Frobenius manifold structures appear
in various actively studied branches of mathematics revealing
remarkable relationships between different theories such as the theory
of Gromov-Witten invariants, singularity theory, differential
geometry of the orbit spaces of reflection groups, and the Hamiltonian
theory of integrable hierarchies.
Frobenius structures on Hurwitz spaces (moduli spaces of functions
over Riemann surfaces) constitute an important class of Frobenius
manifolds. These structures admit an explicit description in terms of
meromorphic objects defined on a Riemann surface and therefore are
useful in understanding properties of a generic Frobenius manifold.
The first examples of Hurwitz Frobenius structures were discovered by
Dubrovin.
In this talk I will discuss some examples of Frobenius manifolds.
Then I will describe new Frobenius structures on Hurwitz spaces,
namely the "real doubles" and "deformations" of Dubrovin's
construction. The "real doubles" are Frobenius structures on a
Hurwitz space considered as a real manifold. There are two "real
doubles" for each Hurwitz Frobenius manifold. The "deformations" are
obtained by introducing parameters into Dubrovin's construction. In a
certain limit, the deformations reduce to the original structures.
Reproducing kernels on Riemann surfaces (the Schiffer and Bergman
kernels) are the main tools of our construction.
