Punctures for Theories of Class $\mathcal{S}_\Gamma$

TL;DR

This paper explores the structure of punctures in class S__ theories derived from M5-branes probing ADE singularities, generalizing known boundary conditions and Nahm pole data to a broader algebraic framework.

Contribution

It introduces a new description of punctures via boundary conditions in 5D quiver gauge theories, extending the concept of Nahm poles and nilpotent orbits to class S__ theories associated with ADE singularities.

Findings

General boundary conditions for punctures are determined.

Solutions with first order poles are explicitly studied.

The connection between nilpotent orbits and su(2) representations is generalized.

Abstract

With the aim of understanding compactifications of 6D superconformal field theories to four dimensions, we study punctures for theories of class $\mathcal{S}_{\Gamma}$. The class $\mathcal{S}_{\Gamma}$ theories arise from M5-branes probing $\mathbb{C}^2 / \Gamma$, an ADE singularity. The resulting 4D theories descend from compactification on Riemann surfaces decorated with punctures. We show that for class $\mathcal{S}_{\Gamma}$ theories, a puncture is specified by singular boundary conditions for fields in the 5D quiver gauge theory obtained from compactification of the 6D theory on a cylinder geometry. We determine general boundary conditions and study in detail solutions with first order poles. This yields a generalization of the Nahm pole data present for $1/2$ BPS punctures for theories of class $\mathcal{S}$. Focusing on specific algebraic structures, we show how the standard discussion of nilpotent orbits and its connection to representations of $\mathfrak{su}(2)$ generalizes in this broader context.