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The CFT-interpolating Black Hole in Three Dimensions

K. Hotta, Y. Hyakutake, T. Kubota, T. Nishinaka, H. Tanida

TL;DR

The paper constructs a new exact extremal black hole in three-dimensional gravity coupled to a scalar field, yielding $AdS_3$ geometries at both spatial infinity and the horizon and thereby hosting two dual $CFT_2$s with different central charges. Using Brown-Henneaux techniques and near-horizon boundary conditions, it derives $c_{UV}=\frac{3L}{2G_N}$ and $c_{IR}=\frac{3\ell}{2G_N}$ with $c_{IR}<c_{UV}$, and shows the IR entropy from Cardy matches the Bekenstein-Hawking entropy while the UV entropy is larger. The holographic RG flow, analyzed via the Hamilton-Jacobi formalism, yields a superpotential $W(\phi)$, a monotone c-function $C(\phi)=\frac{3}{G_N W(\phi)}$, and a Callan-Symanzik framework for the dual $CFT_2$, confirming Zamolodchikov's $c$-theorem in this gravity setup. Altogether, the work provides a concrete AdS$_3$/CFT$_2$ realization of flow between two fixed points connected by a scalar-driven interpolating black hole and offers a platform for exploring holographic RG in lower dimensions.

Abstract

We present a new exact black hole solution in three dimensional Einstein gravity coupled to a single scalar field. This is one of the extended solutions of the BTZ black hole and has in fact $\textrm{AdS}_3$ geometries both at the spatial infinity and at the event horizon. An explicit derivation of Virasoro algebras for $\textrm{CFT}_2$ at the two boundaries is shown to be possible à la Brown and Henneaux's calculation. If we regard the scalar field as a running coupling in the dual two dimensional field theory, and its flow in the bulk as the "holographic" renormalization group flow, our black hole should interpolate the two $\textrm{CFT}_2$ living at the infinity and at the horizon. Following the Hamilton-Jacobi analysis by de Boer, Verlinde and Verlinde, we calculate the central charges $c_{\textrm{UV}}$ and $c_{\textrm{IR}}$ for the $\textrm{CFT}_2$ on the infinity and the horizon, respectively. We also confirm that the inequality $c_{\textrm{IR}} < c_{\textrm{UV}}$ is satisfied, which is consistent with the Zamolodchikov's c-theorem.

The CFT-interpolating Black Hole in Three Dimensions

TL;DR

The paper constructs a new exact extremal black hole in three-dimensional gravity coupled to a scalar field, yielding geometries at both spatial infinity and the horizon and thereby hosting two dual s with different central charges. Using Brown-Henneaux techniques and near-horizon boundary conditions, it derives and with , and shows the IR entropy from Cardy matches the Bekenstein-Hawking entropy while the UV entropy is larger. The holographic RG flow, analyzed via the Hamilton-Jacobi formalism, yields a superpotential , a monotone c-function , and a Callan-Symanzik framework for the dual , confirming Zamolodchikov's -theorem in this gravity setup. Altogether, the work provides a concrete AdS/CFT realization of flow between two fixed points connected by a scalar-driven interpolating black hole and offers a platform for exploring holographic RG in lower dimensions.

Abstract

We present a new exact black hole solution in three dimensional Einstein gravity coupled to a single scalar field. This is one of the extended solutions of the BTZ black hole and has in fact geometries both at the spatial infinity and at the event horizon. An explicit derivation of Virasoro algebras for at the two boundaries is shown to be possible à la Brown and Henneaux's calculation. If we regard the scalar field as a running coupling in the dual two dimensional field theory, and its flow in the bulk as the "holographic" renormalization group flow, our black hole should interpolate the two living at the infinity and at the horizon. Following the Hamilton-Jacobi analysis by de Boer, Verlinde and Verlinde, we calculate the central charges and for the on the infinity and the horizon, respectively. We also confirm that the inequality is satisfied, which is consistent with the Zamolodchikov's c-theorem.

Paper Structure

This paper contains 14 sections, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Black Hole Solution
  3. Virasoro Algebras at the Spatial Infinity and the Horizon
  4. Review of Brown-Henneaux's canonical formalism
  5. Central charges at the spatial infinity
  6. Central charges at the event horizon
  7. Holographic Renormalization Group Flow
  8. Review of Hamilton-Jacobi equation
  9. Beta function and c-function from Hamilton-Jacobi equation
  10. Summary and Discussion
  11. Supplementary Calculations on Central Extensions
  12. At the spatial infinity
  13. At the horizon
  14. Coordinate Transformation

Figures (2)

  • Figure 1: Shape of the potential $V(\phi)$.
  • Figure 2: Shape of the "superpotential" $W(\phi)$.