The CFT-interpolating Black Hole in Three Dimensions

TL;DR

This paper introduces an exact three-dimensional black hole solution with scalar fields that interpolates between two AdS geometries, providing insights into holographic RG flows and dual CFTs at the boundary and horizon.

Contribution

It presents a new exact black hole solution in 3D gravity with scalar fields that connects two AdS regions and analyzes the dual CFTs' central charges at both boundaries.

Findings

The black hole interpolates between two AdS$_3$ geometries.

The central charges at the boundary and horizon satisfy the c-theorem.

Explicit Virasoro algebras are derived at both boundaries.

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 \`{a} 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.