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.

24 pages, based on talks, given at Workshop on combinatorics of moduli spaces, Hurwitz numbers, and cluster algebras; Abel Symposium 2008; and Geometry and integrability in mathematical physics 08

References in corpus (5)

• Extended Seiberg-Witten Theory and Integrable Hierarchy
• On Microscopic Origin of Integrability in Seiberg-Witten Theory
• Seiberg-Witten Theory and Extended Toda Hierarchy
• Non Abelian gauge theories, prepotentials and Abelian differentials
• The equivariant Toda lattice, I

Cited by in corpus (6)

• On non-conformal limit of the AGT relations
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Zamolodchikov asymptotic formula and instanton expansion in N=2 SUSY N_f=2N_c QCD
• On relation between Nekrasov functions and BS periods in pure SU(N) case
• Non Abelian gauge theories, prepotentials and Abelian differentials
• On two-dimensional quantum gravity and quasiclassical integrable hierarchies