Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory

TL;DR

This paper explores the connection between the Whitham-Toda hierarchy and the exact solutions of N=2 supersymmetric Yang-Mills theory, revealing how integrable hierarchies describe the theory's moduli space and effective action.

Contribution

It establishes a link between the Whitham hierarchy and the low-energy effective action of N=2 supersymmetric Yang-Mills theory, providing a new integrable systems perspective.

Findings

Identification of the solution with a homogeneous solution of the Whitham hierarchy

Clarification of the relation between the pre-potential and the Toda hierarchy's tau function

Description of the modulation of quasi-periodic solutions in the context of gauge theory

Abstract

The exact solution of $N=2$ supersymmetric $SU(N)$ Yang-Mills theory is studied in the framework of the Whitham hierarchies. The solution is identified with a homogeneous solution of a Whitham hierarchy. This integrable hierarchy (Whitham-Toda hierarchy) describes modulation of a quasi-periodic solution of the (generalized) Toda lattice hierarchy associated with the hyperelliptic curves over the quantum moduli space. The relation between the holomorphic pre-potential of the low energy effective action and the $\tau$ function of the (generalized) Toda lattice hierarchy is also clarified.