CFT Duals for Extreme Black Holes

TL;DR

This work extends the Kerr/CFT framework to four-dimensional extremal Kerr-Newman-AdS-dS black holes, showing their near-horizon region possesses a left-moving Virasoro symmetry whose central charge c_L and temperature T_L yield a microscopic entropy S = (π^2/3) c_L T_L that exactly matches the Bekenstein-Hawking area law. The central charge is computed from the gravitational sector (c_grav) with c_L = (12 r_+ sqrt((3 r_+^4/ℓ^2 + r_+^2 − q^2)(1 − r_+^2/ℓ^2)))/(1 + 6 r_+^2/ℓ^2 − 3 r_+^4/ℓ^4 − q^2/ℓ^2), and T_L is obtained from the extremal thermodynamics via T_L = 1/(2π k). In the J = 0 Reissner-Nordström-AdS limit, the usual c_L description becomes singular, so the authors propose a second CFT obtained by uplifting the gauge S^1 to a fifth dimension, with central charge c_(y) = 6 q_e \bar{r}_0^2 and temperature T_e that again reproduces S_BH. The paper argues for a universal near-horizon CFT mechanism for extremal black holes and discusses generalizations to higher dimensions and more general actions.

Abstract

It is argued that the general four-dimensional extremal Kerr-Newman-AdS-dS black hole is holographically dual to a (chiral half of a) two-dimensional CFT, generalizing an argument given recently for the special case of extremal Kerr. Specifically, the asymptotic symmetries of the near-horizon region of the general extremal black hole are shown to be generated by a Virasoro algebra. Semiclassical formulae are derived for the central charge and temperature of the dual CFT as functions of the cosmological constant, Newton's constant and the black hole charges and spin. We then show, assuming the Cardy formula, that the microscopic entropy of the dual CFT precisely reproduces the macroscopic Bekenstein-Hawking area law. This CFT description becomes singular in the extreme Reissner-Nordstrom limit where the black hole has no spin. At this point a second dual CFT description is proposed in which the global part of the U(1) gauge symmetry is promoted to a Virasoro algebra. This second description is also found to reproduce the area law. Various further generalizations including higher dimensions are discussed.