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Three-Dimensional Superconformal Field Theory on Conic Space as Hologram of Charged Topological Black Hole

Xing Huang, Soo-Jong Rey, Yang Zhou

TL;DR

The paper develops ${\cal N}=2$ 3D SCFTs on branched S^3 spaces and shows their partition functions depend only on the Reeb vector, enabling a universal large-$N$ description that matches holographic predictions from supersymmetric charged topological black holes in AdS$_4$. Through Killing spinor analysis, localization, and conformal mappings, the authors establish a TBH$_4$/qSCFT$_3$ correspondence, demonstrating exact agreement of free energy and Rényi entropy between the field theory and gravity dual. They also show that bulk Killing spinors reduce to boundary 3D spinors in a manner consistent with supersymmetry, validating the holographic dictionary for these singular geometries. The work provides a concrete framework for computing exact observables on singular curved spaces and strengthens the link between branched geometry, supersymmetric localization, and holography in AdS$_4$/CFT$_3$.

Abstract

We construct three-dimensional N=2 supersymmetric conformal field theories on conic spaces. Built upon the fact that the partition function depends solely on the Reeb vector of the Killing vector, we propose that holographic dual of these theories are four-dimensional, supersymmetric charged topological black holes. With the supersymmetry localization technique, we study conserved supercharges, free energy, and Renyi entropy. At planar large N limit, we demonstrate perfect agreement between the superconformal field theories and the supersymmetric charged topological black holes.

Three-Dimensional Superconformal Field Theory on Conic Space as Hologram of Charged Topological Black Hole

TL;DR

The paper develops 3D SCFTs on branched S^3 spaces and shows their partition functions depend only on the Reeb vector, enabling a universal large- description that matches holographic predictions from supersymmetric charged topological black holes in AdS. Through Killing spinor analysis, localization, and conformal mappings, the authors establish a TBH/qSCFT correspondence, demonstrating exact agreement of free energy and Rényi entropy between the field theory and gravity dual. They also show that bulk Killing spinors reduce to boundary 3D spinors in a manner consistent with supersymmetry, validating the holographic dictionary for these singular geometries. The work provides a concrete framework for computing exact observables on singular curved spaces and strengthens the link between branched geometry, supersymmetric localization, and holography in AdS/CFT.

Abstract

We construct three-dimensional N=2 supersymmetric conformal field theories on conic spaces. Built upon the fact that the partition function depends solely on the Reeb vector of the Killing vector, we propose that holographic dual of these theories are four-dimensional, supersymmetric charged topological black holes. With the supersymmetry localization technique, we study conserved supercharges, free energy, and Renyi entropy. At planar large N limit, we demonstrate perfect agreement between the superconformal field theories and the supersymmetric charged topological black holes.

Paper Structure

This paper contains 31 sections, 231 equations, 1 figure.

Table of Contents

  1. Introduction
  2. qSCFT$_3$
  3. Killing spinors on Branched Spheres
  4. $\mathbb{S}^3_q$
  5. $\widetilde{\mathbb{S}}^3_{p,q}$
  6. Localization on branched spheres
  7. Branched ellipsoid
  8. Branched squashed sphere
  9. Reeb vector and parameter dependence of $Z_{{\cal M}_3}$
  10. Parameter Dependence of $Z_{{\cal M}_3}$
  11. Generalization to branched spaces
  12. Partition function in the large $N$ limit
  13. From CFT on $\mathbb{S}^3_q$ to CFT on $\mathbb{S}^1\times \mathbb{H}^2$
  14. Charged Topological Black Hole
  15. Supersymmetry
  16. ...and 16 more sections

Figures (1)

  • Figure 1: By the conformal mapping, equivalence among the following four quantities can be established for a general CFT: (1) entanglement entropy across a circle in flat space and on a round sphere, (2) free energies on a round sphere and on a hyperbolic space. Here, $q$ is a branched parameter, which is introduced as a virtual deformation in computing the entanglement entropy by replica method.