Isomonodromic deformations and supersymmetric gauge theories

TL;DR

This paper demonstrates that isomonodromy problems provide a unifying framework for understanding the integrable structures in Seiberg-Witten solutions of 4D supersymmetric gauge theories, linking them to Painlevé equations and topological sigma models.

Contribution

It introduces an isomonodromy framework to interpret Seiberg-Witten solutions and related integrability features in supersymmetric gauge theories, connecting them to Painlevé equations and multiscale analysis.

Findings

Seiberg-Witten solutions can be viewed as WKB limits of isomonodromy problems.

The underlying Whitham dynamics are explained via multiscale asymptotic methods.

The isomonodromy problem relates closely to the third Painlevé equation and topological sigma models.

Abstract

Seiberg-Witten solutions of four-dimensional supersymmetric gauge theories possess rich but involved integrable structures. The goal of this paper is to show that an isomonodromy problem provides a unified framework for understanding those various features of integrability. The Seiberg-Witten solution itself can be interpreted as a WKB limit of this isomonodromy problem. The origin of underlying Whitham dynamics (adiabatic deformation of an isospectral problem), too, can be similarly explained by a more refined asymptotic method (multiscale analysis). The case of $N=2$ SU($s$) supersymmetric Yang-Mills theory without matter is considered in detail for illustration. The isomonodromy problem in this case is closely related to the third Painlev\'e equation and its multicomponent analogues. An implicit relation to $t\tbar$ fusion of topological sigma models is thereby expected.