Brief review on higher spin black holes

TL;DR

The work analyzes higher spin black holes in three-dimensional gravity formulated as a Chern-Simons theory with two copies of $sl(3,\mathbb{R})$, emphasizing the role of asymptotic conditions and chemical potentials in defining global charges and thermodynamics. By implementing the Hamiltonian framework and Brown-Henneaux–type boundaries, it shows that pure gravity yields two Virasoro algebras with central charge $c=\frac{3l}{2G}$ and reproduces BTZ thermodynamics from holonomies. Extending to $sl(3,\mathbb{R})$ yields two copies of the nonlinear $W_3$ algebra, with spin-2 and spin-3 charges described by $\mathcal{L}_{\pm}$ and $\mathcal{W}_{\pm}$; however, naïve constant-charge fall-offs lead to ambiguities in charges and entropy. The introduction of extended asymptotics with nonzero spin-3 data and chemical potentials $\tilde{\mu}$ resolves these puzzles, though it reveals that consistent identifications of global charges depend on the chosen boundary conditions; when chemical potentials are consistently included in the time component, the $W_3$ symmetry is preserved and thermodynamics becomes unambiguous. Overall, the paper shows that classifying higher spin black holes requires careful treatment of fall-off and chemical potentials, yielding inequivalent solution classes and a coherent framework for their charges and entropy.

Abstract

We review some relevant results in the context of higher spin black holes in three-dimensional spacetimes, focusing on their asymptotic behaviour and thermodynamic properties. For simplicity, we mainly discuss the case of gravity nonminimally coupled to spin-3 fields, being nonperturbatively described by a Chern-Simons theory of two independent sl(3,R) gauge fields. Since the analysis is particularly transparent in the Hamiltonian formalism, we provide a concise discussion of their basic aspects in this context; and as a warming up exercise, we briefly analyze the asymptotic behaviour of pure gravity, as well as the BTZ black hole and its thermodynamics, exclusively in terms of gauge fields. The discussion is then extended to the case of black holes endowed with higher spin fields, briefly signaling the agreements and discrepancies found through different approaches. We conclude explaining how the puzzles become resolved once the fall off of the fields is precisely specified and extended to include chemical potentials, in a way that it is compatible with the asymptotic symmetries. Hence, the global charges become completely identified in an unambiguous way, so that different sets of asymptotic conditions turn out to contain inequivalent classes of black hole solutions being characterized by a different set of global charges.