Fuchsian ODEs as Seiberg dualities

TL;DR

This paper reveals a deep connection between Fuchsian differential equations and Seiberg dualities in supersymmetric gauge theories, providing a physical interpretation and explicit duality actions in terms of representation theory.

Contribution

It establishes a novel correspondence between Fuchsian ODEs and Seiberg dualities, using physical mathematics to interpret integral solutions and connection formulae.

Findings

Fuchsian ODEs correspond to BPS dyons in gauge theories

Seiberg duality acts explicitly on Fuchsian ODEs via representation theory

The relation is explained through mirror symmetry and gauge theory connections

Abstract

The classical theory of Fuchsian differential equations is largely equivalent to the theory of Seiberg dualities for quiver SUSY gauge theories. In particular: all known integral representations of solutions, and their connection formulae, are immediate consequences of (analytically continued) Seiberg duality in view of the dictionary between linear ODEs and gauge theories with 4 supersymmetries. The purpose of this divertissement is to explain "physically'' this remarkable relation in the spirit of Physical Mathematics. The connection goes through a "mirror-theoretic'' identification of irreducible logarithmic connections on $\mathbb{P}^1$ with would-be BPS dyons of 4d $\mathcal{N}=2$ $SU(2)$ SYM coupled to a certain Argyres-Douglas "matter''. When the underlying bundle is trivial, i.e. the log-connection is a Fuchs system, the world-line theory of the dyon simplifies and the action of Seiberg duality on the Fuchsian ODEs becomes quite explicit. The duality action is best described in terms of Representation Theory of Kac-Moody Lie algebras (and their affinizations).