On Integrable Systems and Supersymmetric Gauge Theories

TL;DR

This paper explores the connection between integrable systems and N=2 supersymmetric gauge theories, focusing on finite-gap solutions, complex curves, and recent exact nonperturbative solutions, highlighting their mathematical structure and physical implications.

Contribution

It presents a comprehensive formulation of finite-gap solutions for integrable equations using complex curves and applies these methods to recent exact solutions in N=2 SUSY gauge theories.

Findings

Finite-gap solutions are formulated via complex curves and differentials.

Recent exact nonperturbative solutions are interpreted through integrable systems.

Parallels between quantum field theory results and integrable systems solutions are discussed.

Abstract

The properties of the N=2 SUSY gauge theories underlying the Seiberg-Witten hypothesis are discussed. The main ingredients of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential are presented, the invariant sense of these definitions is illustrated. Recently found exact nonperturbative solutions to N=2 SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to integrable systems are discussed.