The Kerr/CFT Correspondence and its Extensions

Abstract

We present a first-principles derivation of the main results of the Kerr/CFT correspondence and its extensions using only tools from gravity and quantum field theory, filling a few gaps in the literature when necessary. Firstly, we review properties of extremal black holes that imply, according to semi-classical quantization rules, that their near-horizon quantum states form a centrally-extended representation of the one-dimensional conformal group. This motivates the conjecture that the extremal Kerr and Reissner-Nordström black holes are dual to the chiral limit of a two-dimensional CFT. We also motivate the existence of an SL(2, ℤ) family of two-dimensional CFTs, which describe in their chiral limit the extremal Kerr-Newman black hole. We present generalizations in anti-de Sitter spacetime and discuss other matter-coupling and higher-derivative corrections. Secondly, we show how a near-chiral limit of these CFTs reproduces the dynamics of near-superradiant probes around near-extremal black holes in the semi-classical limit. Thirdly, we review how the hidden conformal symmetries of asymptotically-flat black holes away from extremality, combined with their properties at extremality, allow for a microscopic accounting of the entropy of non-extremal asymptotically-flat rotating or charged black holes. We conclude with a list of open problems.

New entropy formula for Kerr black holes

“…which coincides with the near horizon extremal Kerr result [47,48]. Note, however, that the boundary conditions of [6] on which our construction is based solely apply to non-extremal Kerr black holes, so one has to be careful with this limit.…”

Particle collisions on near horizon extremal Kerr background

“…First of all, it should be understood that the results obtained within the conventional Kerr geometry and the NHEK setup can not be related by a coordinate transformation. It is true that the derivation of the NHEK metric involves the coordinate transformation (14). However, it becomes singular in the near horizon limit ǫ → 0.…”

“…Thus, not only the energy parameters of individual particles are not directly comparable in the two pictures but the quantities (10) and (17) have different meanings as well. In fact, it is straightforward to verify that the coordinate transformation (14) applied to (9) and followed by ǫ → 0 yields a divergent expression even before fixing any critical value for the particle parameters.…”

“…In particular, the near horizon extremal Kerr (NHEK) geometry is a cornerstone of the Kerr/CFT-correspondence [13] which is being extensively investigated nowadays (for a review see [14]). It is then important to confront the NHEK setup with the BSW analysis and to investigate whether the ultrahigh energy collisions take place within the context of the NHEK geometry.…”

Integrable models associated with Myers–Perry–AdS–dS black hole in diverse dimensions

“…The first reason to be concerned about the near horizon geometries is the duality between the near horizon Kerr black hole and a conformal field theory suggested in [2] (for a review see [3]). As was proved in [4], the duality holds true also in higher dimensions and in the presence of a cosmological constant.…”