Codimension two defects and the Springer correspondence

TL;DR

This paper links codimension two defects in 6D SCFTs to the Springer correspondence, providing a framework to understand S-duality and boundary conditions in 4D gauge theories through representation theory.

Contribution

It introduces a novel association between defects and Springer correspondence, enabling classification of boundary conditions and analysis of symmetry breaking in gauge theories.

Findings

Constructs functors relating boundary conditions with broken and unbroken gauge symmetry.

Provides a new perspective on S-duality via Springer correspondence.

Facilitates classification of boundary conditions in $ ext{N}=4$ SYM.

Abstract

One can associate an invariant to a large class of regular codimension two defects of the six dimensional $(0,2)$ SCFT $\mathscr{X}[j]$ using the classical Springer correspondence. Such an association allows a simple description of S-duality of associated Gaiotto-Witten boundary conditions in $\mathcal{N}=4$ SYM for arbitrary gauge group and by extension, a determination of certain local aspects of class $\mathcal{S}$ constructions. I point out that the problem of \textit{classifying} the corresponding boundary conditions in $\mathcal{N}=4$ SYM is intimately tied to possible symmetry breaking patterns in the bulk theory. Using the Springer correspondence and the representation theory of Weyl groups, I construct a pair of functors between the class of boundary conditions in the theory in the phase with broken gauge symmetry and those in the phase with unbroken gauge symmetry.