Vasilisa Shramchenko (University of Oxford)
shramche@maths.ox.ac.uk
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.
Frobenius structures on Hurwitz spaces (moduli spaces of functions over Riemann surfaces) constitute an important class of Frobenius manifolds; they admit an explicit description in terms of meromorphic objects defined on a Riemann surface.
In this talk I will give a definition of a Frobenius manifold and describe Dubrovin's Frobenius structures on Hurwitz spaces and their generalizations, the "real doubles" and deformations of Hurwitz Frobenius manifolds. Then I will focus on two Riemann-Hilbert problems naturally associated to every Frobenius manifold. It turns out that, in the case of structures on Hurwitz spaces, these problems are solvable in terms of bidifferentials defined on the underlying Riemann surfaces.