On holography with ADE singularities

TL;DR

This paper explores the holographic duality involving ADE singularities in AdS geometries, revealing a large vacuum degeneracy linked to topological degrees of freedom described by Chern-Simons theory, with implications for one-form symmetries.

Contribution

It demonstrates how ADE singularities in the bulk lead to topological degrees of freedom and vacuum degeneracy, connecting super Yang-Mills theory, holography, and Chern-Simons theory.

Findings

Large vacuum degeneracy from gauge holonomy on orbifolded spheres

Topological degrees of freedom supported by ADE singularities in AdS

Effective description of these degrees via three-dimensional Chern-Simons theory

Abstract

We study aspects of the AdS/CFT correspondence for $\mathcal{N}=4$ $U(N)$ super Yang-Mills theory on $S^3/\Gamma$, where $\Gamma \subset SU(2)$ 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 $S^3/\Gamma$. 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 $\Gamma$. 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.