On holography with ADE singularities

A bstract We study aspects of the AdS/CFT correspondence for $$ \mathcal{N}=4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> U( N ) super Yang-Mills theory on S 3 / Γ, where Γ ⊂ 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 / Γ. 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 G is determined by Γ. Finally, we discuss how the one-form symmetries of the $$ \mathcal{N}=4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> super Yang-Mills theory are realized on the Chern-Simons theory side.