Non-Abelian gauge theories, prepotentials, and Abelian differentials

We discuss particular solutions of integrable systems (starting from the well-known dispersionless KdV and Toda hierarchies) that most directly define the generating functions for the Gromov–Witten classes in terms of a rational complex curve. From the mirror theory standpoint, these generating functions can be identified with the simplest prepotentials of complex manifolds, and we present some new exactly calculable examples of such prepotentials. For higher-genus curves, which in this context correspond to non-Abelian gauge theories via the topological string/gauge duality, we construct similar solutions using an extended basis of Abelian differentials, generally with extra singularities at the branch points of the curve.