On holography with ADE singularities

Abstract

We study aspects of the AdS/CFT correspondence for $\mathcal{N}=4$ $U(N)$ super Yang-Mills theory on $S^3/\hat{I}“$, where $\hat{I}“\subset SU(2)$ is a finite subgroup, leading to an ADE singularity in the bulk AdS geometry. We show that a large vacuum degeneracy arises from the choice of gauge holonomy on $S^3/\hat{I}“$. On the gravity side, we argue that the bulk ADE singularity supports topological degrees of freedom responsible for this degeneracy. We then provide a holographic derivation of a corresponding large vacuum degeneracy for class S theories of type $U(N)$, showing that these topological degrees of freedom admit an effective description in terms of a three-dimensional level-$N$ Chern-Simons theory, whose gauge group $\mathsf{G}$ is determined by $\hat{I}“$. Finally, we discuss how the one-form symmetries of the $\mathcal{N}=4$ super Yang-Mills theory are realized on the Chern-Simons theory side.

31 pages + appendices, 2 figures; v2: references added, minor correction