Hidden Conformal Symmetry of Extremal Black Holes

TL;DR

This paper reveals a hidden conformal symmetry in extremal black holes by introducing new coordinates, showing their scalar Laplacian relates to an $SL(2,R)$ Casimir, and matching scattering amplitudes with dual CFT predictions.

Contribution

It introduces a new set of conformal coordinates for extremal black holes, demonstrating their scalar Laplacian can be expressed via $SL(2,R)$ Casimir, supporting dual CFT descriptions.

Findings

Scalar Laplacian expressed in terms of $SL(2,R)$ Casimir.

Dual CFT temperatures can be directly read from coordinates.

Scattering amplitudes match CFT predictions.

Abstract

We study the hidden conformal symmetry of the extremal black holes. We introduce a new set of conformal coordinates to write the $SL(2,R)$ generators. We find that the Laplacian of the scalar field in many extremal black holes could be written in terms of the $SL(2,R)$ quadratic Casimir. This suggests that there exist dual CFT descriptions of these black holes. From the conformal coordinates, the temperatures of the dual CFTs could be read directly. For the extremal black hole, the Hawking temperature is vanishing. Correspondingly, only the left (right) temperature of the dual CFT is non-vanishing and the excitations of the other sector are suppressed. In the probe limit, we compute the scattering amplitudes of the scalar off the extremal black holes and find perfect agreement with the CFT prediction.