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On holography with ADE singularities

Sunjin Choi, Yuji Tachikawa

TL;DR

This work analyzes AdS/CFT for $ ext{N}=4$ $U(N)$ SYM on $S^3/9Γ$ with $9Γ$ an ADE subgroup of $SU(2)$, uncovering a large holonomy-induced vacuum degeneracy tied to an ADE bulk singularity. It argues that bulk topological degrees of freedom living on the ADE singularity are captured holographically by a three-dimensional pure Chern-Simons theory at level $N$ with gauge group $_G$ determined by $9Γ$ via the McKay correspondence, and extends this to class S theories of type $U(N)$ by relating the degeneracy to CS vacua on a Riemann surface and to affine representations at level $N$. The paper also analyzes how the electric and magnetic one-form symmetries of the 4d theories map to the one-form symmetry of the CS side, showing consistency of symmetry actions under the duality and S-duality via the CS $S$-matrix. By matching the vacuum structures across the duality, it provides a concrete holographic realization of topological degrees of freedom arising from ADE singularities and clarifies how higher-form symmetries are realized in holographic CS theories. These results illuminate the interplay between topology, ADE singularities, and higher-form symmetries in holography and suggest further checks in less supersymmetric settings and broader bulk geometries.

Abstract

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

On holography with ADE singularities

TL;DR

This work analyzes AdS/CFT for SYM on with an ADE subgroup of , uncovering a large holonomy-induced vacuum degeneracy tied to an ADE bulk singularity. It argues that bulk topological degrees of freedom living on the ADE singularity are captured holographically by a three-dimensional pure Chern-Simons theory at level with gauge group determined by via the McKay correspondence, and extends this to class S theories of type by relating the degeneracy to CS vacua on a Riemann surface and to affine representations at level . The paper also analyzes how the electric and magnetic one-form symmetries of the 4d theories map to the one-form symmetry of the CS side, showing consistency of symmetry actions under the duality and S-duality via the CS -matrix. By matching the vacuum structures across the duality, it provides a concrete holographic realization of topological degrees of freedom arising from ADE singularities and clarifies how higher-form symmetries are realized in holographic CS theories. These results illuminate the interplay between topology, ADE singularities, and higher-form symmetries in holography and suggest further checks in less supersymmetric settings and broader bulk geometries.

Abstract

We study aspects of the AdS/CFT correspondence for super Yang-Mills theory on , where 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 . 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 , showing that these topological degrees of freedom admit an effective description in terms of a three-dimensional level- Chern-Simons theory, whose gauge group is determined by . Finally, we discuss how the one-form symmetries of the super Yang-Mills theory are realized on the Chern-Simons theory side.
Paper Structure (42 sections, 134 equations, 2 figures, 2 tables)

This paper contains 42 sections, 134 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and summary
  2. Overview
  3. Detailed introduction
  4. Historical context.
  5. Duality to three-dimensional Chern-Simons theory.
  6. One-form symmetry.
  7. Organization of the paper
  8. Supersymmetric Casimir energy
  9. $\mathcal{N}{=}4$ super Yang-Mills theory on $S^3$
  10. Supersymmetric Casimir energy on $S^3/\Gamma$
  11. Duality
  12. Holographic Chern-Simons dual
  13. On 1-form symmetries
  14. 1-form symmetries of $\mathcal{N}{=}4$ super Yang-Mills theory and more general class S theories.
  15. 1-form symmetry of the 7d gauge theory.
  16. ...and 27 more sections

Figures (2)

  • Figure 1: The line operator $\mathcal{B}(z)$ acting on the state $\ket{\lambda}$.
  • Figure 2: The space $X_{\mathbb{Z}_k}$, consisting of $k$ patches $U_i \simeq\mathbb{C}^2$ with coordinates $(x_i,y_i)$, $(i=1,\ldots, k)$, where we draw complex planes as real lines. We also indicated $k$ 2-spheres $C_{1,\ldots, k}$, and one non-compact 2-cycle $D_1$.