Non-Abelian gauge theories, prepotentials, and Abelian differentials

Abstract

I discuss particular solutions of the integrable systems, starting from well-known dispersionless KdV and Toda hierarchies, which define in most straightforward way the generating functions for the Gromov-Witten classes in terms of the rational complex curve. On the ``mirror'' side these generating functions can be identified with the simplest prepotentials of complex manifolds, and I present few more exactly calculable examples of them. For the higher genus curves, corresponding in this context to the non Abelian gauge theories via the topological gauge/string duality, similar solutions are constructed using extended basis of Abelian differentials, generally with extra singularities at the branching points of the curve.