|XXVIII Workshop on Geometric Methods in Physics||28.06-04.07.2009|
Frobenius manifolds and Riemann-Hilbert problems
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 briefly 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 (dual) Riemann-Hilbert problems naturally associated to every Frobenius manifold. It turns out that these problems are solvable in terms of bidifferentials defined on the corresponding Riemann surfaces.