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

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.

39 pages

References in corpus (5)

• Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory
• 6d Conformal Matter
• Tinkertoys for Gaiotto Duality
• F-Theory and N=1 SCFTs in Four Dimensions
• Boundaries and Defects of N=4 SYM with 4 Supercharges, Part I: Boundary/Junction Conditions

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• Revisiting the classifications of 6d SCFTs and LSTs
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• Defect $a$-Theorem and $a$-Maximization
• A slow review of the AGT correspondence
• From 6d flows to 4d flows
• 6d SCFTs, Center-Flavor Symmetries, and Stiefel--Whitney Compactifications
• From 6D SCFTs to Dynamic GLSMs
• Sequences of $6d$ SCFTs on generic Riemann surfaces
• Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model
• 2D CFT blocks for the 4D class ${\mathcal{S}}_k$ theories
• Green-Schwarz Automorphisms and 6D SCFTs
• The Coulomb and Higgs Branches of $\mathcal{N}=1$ Theories of Class ${\mathcal{S}}_k$
• Infinitely many 4d N=1 SCFTs with a=c
• Nilpotent Networks and 4D RG Flows
• $\mathcal{N}=1$ Curves on Generalized Coulomb Branches
• Punctures and Dynamical Systems
• Instantons on Calabi-Yau and hyper-Kähler cones
• Aspects of 4d supersymmetric dynamics and geometry