Punctures for Theories of Class $\mathcal{S}_Γ$

TL;DR

This work develops a framework to study punctures in class $\mathcal{S}_{\Gamma}$ theories, arising from M5-branes on $\mathbb{C}^2/\Gamma$, by encoding punctures as singular boundary data in the 5D affine ADE quiver obtained from circle reduction. It generalizes Nahm pole data to include commuting pairs of nilpotent matrices $Q$ and $\widetilde{Q}$ subject to orbifold constraints, with regular punctures corresponding to first-order poles; special cases recover the familiar class $\mathcal{S}$ puncture structure via $\Gamma$-trivial limits. The authors present several constructive ansätze—$\mathfrak{su}(2)_{Q}$, $\mathfrak{su}(2)_{Q} \times \mathfrak{su}(2)_{\widetilde{Q}}$, and $\mathfrak{su}(2)^{l}$ directed paths—to generate and classify punctures for A-, D-, and E-type singularities, finding nontrivial solutions for A- and D-type but not for E-type in the two-su(2) ansatz. These results extend the nilpotent-orbit/Nahm-pole framework of class $\mathcal{S}$ to $(1,0)$ theories and connect to $\Gamma$-equivariant Hilbert schemes and flavor-symmetry structures, with potential applications to the anomaly polynomial and superconformal index of the resulting 4D theories.

Abstract

With the aim of understanding compactifications of 6D superconformal field theories to four dimensions, we study punctures for theories of class $\mathcal{S}_Γ$. The class $\mathcal{S}_Γ$ theories arise from M5-branes probing $\mathbb{C}^2 / Γ$, an ADE singularity. The resulting 4D theories descend from compactification on Riemann surfaces decorated with punctures. We show that for class $\mathcal{S}_Γ$ 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.

Figures (1)

• Figure 1: $\widehat{D_4}$ quiver diagram. Each link is labeled by the bifundamental field mapping between representation spaces. The four row vectors $\vec{a}^t, \vec{b}^t, \vec{g}^t, \vec{h}^t$ are in the representation $(2, \overline{1})$ of U(2) $\times$ U(1), while the four column vectors $\vec{c}, \vec{d}, \vec{e}, \vec{f}$ are in the representation $(1,\overline 2)$ of U(1) $\times$ U(2). Since all of the bifundamental hypermultiplets are organized in the $8 \times 8$ matrix $Q$, the matrix $\widetilde{Q}$ provides redundant information.