Asymptotic structure of the Einstein-Maxwell theory on AdS$_{3}$

TL;DR

This paper investigates the asymptotic structure of Einstein-Maxwell theory in AdS3, revealing how boundary conditions influence symmetries and black hole energy spectra, with a focus on conformal invariance and boundary term finiteness.

Contribution

It introduces a new set of boundary conditions in AdS3 Einstein-Maxwell theory that preserve conformal symmetry and yield well-defined energy spectra for charged black holes.

Findings

Asymptotic symmetries reduce to


 symmetry group.

Special boundary conditions preserve the full conformal group and Virasoro algebra.

Energy spectrum of charged rotating black holes remains well-behaved under these conditions.

Abstract

The asymptotic structure of AdS spacetimes in the context of General Relativity coupled to the Maxwell field in three spacetime dimensions is analyzed. Although the fall-off of the fields is relaxed with respect to that of Brown and Henneaux, the variation of the canonical generators associated to the asymptotic Killing vectors can be shown to be finite once required to span the Lie derivative of the fields. The corresponding surface integrals then acquire explicit contributions from the electromagnetic field, and become well-defined provided they fulfill suitable integrability conditions, implying that the leading terms of the asymptotic form of the electromagnetic field are functionally related. Consequently, for a generic choice of boundary conditions, the asymptotic symmetries are broken down to $\mathbb{R}\otimes U\left(1\right)\otimes U\left(1\right)$. Nonetheless, requiring compatibility of the boundary conditions with one of the asymptotic Virasoro symmetries, singles out the set to be characterized by an arbitrary function of a single variable, whose precise form depends on the choice of the chiral copy. Remarkably, requiring the asymptotic symmetries to contain the full conformal group selects a very special set of boundary conditions that is labeled by a unique constant parameter, so that the algebra of the canonical generators is given by the direct sum of two copies of the Virasoro algebra with the standard central extension and $U\left(1\right)$. This special set of boundary conditions makes the energy spectrum of electrically charged rotating black holes to be well-behaved.