Generalized Black Holes in Three-dimensional Spacetime

TL;DR

The paper studies generalized black holes in three-dimensional AdS gravity with higher spins by a Chern-Simons formulation based on sl(N,ℝ). It clarifies the role of inequivalent sl(2,ℝ) embeddings (principal vs diagonal) and derives, in the principal embedding, a black hole carrying spins 2 and 3 with two copies of the W3 asymptotic algebra, computing entropy from on-shell holonomies; it also analyzes the diagonal embedding yielding a W3(2) black hole with spins 1, 3/2, and 2, resolving prior entropy ambiguities. By extending to higher N, it outlines how W_N asymptotic symmetry arises and how entropy generalizes via holonomies, all within a strictly Euclidean CS approach that avoids gauge-dependent metric constructs. The work provides a consistent thermodynamic framework for higher-spin black holes, clarifies previous proposals (e.g., GK black holes), and sets the stage for systematic extensions to arbitrary N and associated holographic interpretations. Overall, it demonstrates that generalized black holes in 3D AdS gravity can be robustly defined and analyzed through horizon holonomies and boundary conditions in CS language, with entropy expressible as a boundary term and tied to the underlying W-algebras.

Abstract

Three-dimensional spacetime with a negative cosmological constant has proven to be a remarkably fertile ground for the study of gravity and higher spin fields. The theory is topological and, since there are no propagating field degrees of freedom, the asymptotic symmetries become all the more crucial. For pure (2+1) gravity they consist of two copies of the Virasoro algebra. There exists a black hole which may be endowed with all the corresponding charges. The pure (2+1) gravity theory may be reformulated in terms of two Chern-Simons connections for sl(2,R). An immediate generalization containing gravity and a finite number of higher spin fields may be achieved by replacing sl(2,R) by sl(3,R) or, more generally, by sl(N,R). The asymptotic symmetries are then two copies of the so-called W_N algebra, which contains the Virasoro algebra as a subalgebra. The question then arises as to whether there exists a generalization of the standard pure gravity (2+1) black hole which would be endowed with all the W_N charges. The original pioneering proposal of a black hole along this line for N=3 turns out, as shown in this paper, to actually belong to the so called "diagonal embedding" of sl(2,R) in sl(3,R), and it is therefore endowed with charges of lower rather than higher spins. In contradistinction, we exhibit herein the most general black hole which belongs to the "principal embedding". It is endowed with higher spin charges, and possesses two copies of W_3 as its asymptotic symmetries. The most general diagonal embedding black hole is studied in detail as well, in a way in which its lower spin charges are clearly displayed. The extension to N>3 is also discussed. A general formula for the entropy of a generalized black hole is obtained in terms of the on-shell holonomies.