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

TL;DR

The paper develops a consistent canonical framework for Einstein–Maxwell theory in $AdS_{3}$ with relaxed fall-off, showing that finite charges arise only after improving the diffeomorphism generators with a $U(1)$ term and imposing integrability. This integrability enforces a functional relation between the leading electromagnetic data, making charges boundary-condition dependent. A very special boundary condition recovers the full conformal symmetry at infinity, yielding a Virasoro$^2$ plus $U(1)$ algebra with central charge $c=3l/(2G)$ and a well-behaved energy spectrum for electrically charged rotating black holes, with implications for $AdS_{3}/CFT_{2}$ holography and related condensed-matter applications. The analysis highlights the crucial role of boundary conditions in holographic interpretations and broadens the landscape of admissible asymptotics in three-dimensional gravity coupled to electromagnetism.

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.