Non Abelian gauge theories, prepotentials and Abelian differentials

TL;DR

This paper explores solutions of integrable systems related to Gromov-Witten invariants and their connection to non-Abelian gauge theories through complex curves and Abelian differentials, advancing understanding in mathematical physics.

Contribution

It introduces new solutions for higher genus curves using extended Abelian differentials, linking integrable systems with gauge theories and mirror symmetry.

Findings

Explicit solutions for genus zero and higher curves.

Identification of generating functions with prepotentials of complex manifolds.

Extension of Abelian differentials to include singularities at branch points.

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.