The uses of Whitham hierarchies

TL;DR

This paper reviews the applications of Whitham hierarchies in supersymmetric gauge theories, focusing on their role in prepotential theory, contact terms, u-plane integrals, and instanton correction techniques.

Contribution

It synthesizes various approaches connecting Whitham hierarchies with supersymmetric gauge theories and introduces new methods for extracting instanton corrections.

Findings

Clarifies the structure of contact terms in topological theories

Links Whitham hierarchies to u-plane integrals for higher rank groups

Develops techniques for instanton correction extraction using RG equations

Abstract

We review some of the uses of Whitham hierarchies in the context of the theory of the prepotential in N=2 supersymmetric gauge theories. We focus on the structure of the contact terms in the twisted topological theory, and on the connection between Whitham hierarchies and the u-plane integrals for higher rank gauge groups, trying to put together the different approaches involved in this connection. We also review two other uses of the Whitham hierarchies: the interpretation of the slow times as supersymmetry breaking parameters, and the new techniques to extract instanton corrections using the RG equations written in terms of theta functions.