Generalized black holes in three-dimensional spacetime

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,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathbb{R} $\end{document}). This permits an immediate generalization which may be interpreted as containing gravity and a finite number of higher spin fields. The generalization is achieved by replacing sl (2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathbb{R} $\end{document}) by sl (3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathbb{R} $\end{document}) or, more generally, by sl (N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathbb{R} $\end{document}). The asymptotic symmetries are then two copies of the so-called WN algebra, which contains the Virasoro algebra as a subalgebra. The question then arises as to whether there exists a generalization of the standard pure gravity (2 + 1) black hole which would be endowed with all the WN charges. Since the generalized Chern-Simons theory does not admit a direct metric interpretation, one must define the black hole in Euclidean spacetime through its thermal properties, and then continue to Lorentzian spacetime. The original pioneering proposal of a black hole along this line for N = 3 turns out, as shown in this paper, to actually belong to the so called “diagonal embedding” of sl (2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathbb{R} $\end{document}) in sl (3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathbb{R} $\end{document}), 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 W3 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. The relationship between the asymptotic symmetries and the chemical potentials is exhibited, and the equivalence of the different thermodynamical ensembles is discussed. A self-contained account of the background necessary to substantiate the claims made in the paper is included.