Frobenius structures, integrable systems and Hurwitz spaces

This thesis consists of two main parts. In the first part a new family of integrable systems related to Hurwitz spaces of elliptic coverings with simple branch points is constructed. The integrable systems are closely related to Takasaki's version of the Schlesinger system on an elliptic surface. A trigonometric degeneration of the integrable systems is presented. The trigonometric version of an auxiliary system of differential equations for the images of branch points of the covering under a uniformization map with respect to branch points is derived. This system is applied to solving the Boyer-Finley equation (self-dual Einstein equation with a rotating Killing vector). Thereby, a class of implicit solutions to the Boyer-Finley equation is found in terms of objects related to the Hurwitz spaces.; The second part presents two classes of new semisimple Frobenius structures on Hurwitz spaces (spaces of ramified coverings of ( C P1). The original construction of Hurwitz Frobenius manifolds by Dubrovin is described in terms of the normalized meromorphic bidifferential W of the second kind on a Riemann surface. In Dubrovin's construction, the branch points {lcub}lambdam{rcub} of the covering play the role of canonical coordinates on the Hurwitz Frobenius manifolds. We find new Frobenius structures on Hurwitz spaces with coordinates {lcub}lambdam; lambda m{rcub} in terms of the Schiffer and Bergman kernels (bidifferentials) on a Riemann surface. We call these structures the "real doubles" of the Hurwitz Frobenius manifolds of Dubrovin.; To construct another class of new Frobenius structures on Hurwitz spaces, we introduce a g(g + 1)/2-parametric deformation of the bidifferential W, where g is the genus of the corresponding Riemann surface. Analogously to the bidifferential W its deformation defines Frobenius structures on Hurwitz spaces; these structures give a g(g + 1)/2-parametric deformation of Dubrovin's Hurwitz Frobenius manifolds. Similarly, we introduce the deformations of the Schiffer and Bergman kernels which define Frobenius structures on the Hurwitz spaces with coordinates {lcub}lambdam ; lambdam{rcub}. Thereby we obtain deformations of the real doubles of the Hurwitz Frobenius manifolds of Dubrovin. Each new Frobenius structure gives a new solution to the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) system. For the simplest Hurwitz space in genus one, the corresponding solutions are found explicitly, together with the corresponding G-function.