Isometry-preserving boundary conditions in the Kerr/CFT correspondence

TL;DR

The work investigates isometry-preserving boundary conditions for near-horizon geometries of extremal Kerr-type black holes and maps how these choices control Virasoro-type asymptotic symmetries. It shows that there exist extensive families of boundary conditions that enhance either the $U(1)$ or $SL(2,\mathbb{R})$ isometries to a Virasoro algebra, with the standard Kerr/CFT case yielding a central charge $c=12J/\hbar$. In particular, some strong boundary-condition choices produce a centreless Virasoro algebra even for the $U(1)$ enhancement, and a careful construction yields a centreless Virasoro realization containing the $SL(2,\mathbb{R})$ subalgebra, though naive charges may vanish. The analysis also highlights spurious asymptotic symmetries (vanishing charges) and demonstrates that, while multiple Virasoro-like directions may appear, nontrivial, well-defined charges can arise only for at most one such extension, underscoring subtle issues related to back-reaction and linearization stability in the Kerr/CFT context.

Abstract

The near-horizon geometries of the extremal Kerr black hole and certain generalizations thereof are considered. Their isometry groups are all given by SL(2,R) x U(1). The usual boundary conditions of the Kerr/CFT correspondence enhance the U(1) isometry to a Virasoro algebra. Various alternatives to these boundary conditions are explored. Partial classifications are provided of the boundary conditions enhancing the SL(2,R) isometries or separately the U(1) isometry to a Virasoro algebra. In the case of SL(2,R)-enhancing boundary conditions of a near-horizon geometry of the type considered, the conserved charges associated to the generators of the asymptotic Virasoro symmetry form a centreless Virasoro algebra.