Punctures for theories of class S Γ $$ {\mathcal{S}}_{\varGamma } $$

Abstract With the aim of understanding compactifications of 6D superconformal field theories to four dimensions, we study punctures for theories of class S Γ $$ {\mathcal{S}}_{\varGamma } $$ . The class S Γ $$ {\mathcal{S}}_{\varGamma } $$ theories arise from M5-branes probing ℂ 2/Γ, an ADE singularity. The resulting 4D theories descend from compactification on Riemann surfaces decorated with punctures. We show that for class S Γ $$ {\mathcal{S}}_{\varGamma } $$ 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 S $$ \mathcal{S} $$ . Focusing on specific algebraic structures, we show how the standard discussion of nilpotent orbits and its connection to representations of s u 2 $$ \mathfrak{s}\mathfrak{u}(2) $$ generalizes in this broader context