Black holes in Truncated Higher Spin AdS$_3$ Gravity

TL;DR

The paper analyzes black holes in truncated $AdS_3$ higher spin gravity with a finite set of even spins, identifying Spin $\widetilde{4}$ (Type I/II) and $G_2$ as the simplest nontrivial theories. Using a Chern–Simons framework and holonomy conditions, it derives exact black hole solutions with a single higher spin hair (spin 4 or 6) and obtains closed-form entropy expressions governed by $W$-algebras: $W(2,4)$ for Spin $\widetilde{4}$ and $W(2,6)$ for $G_2$. It also extends to two-hair cases (Spin $\widetilde{6}$ in Type I/II) where the entropy is described by a two-parameter function $f(y,z)$ but requires perturbative solutions. The results illustrate that such truncated theories yield tractable thermodynamics and sharp features like charge-conjugation violation, providing prototypes for HS/CFT tests and motivating further exploration of finite truncations and their holographic duals.

Abstract

We study the higher spin black holes in a truncated version of higher spin gravity in $AdS_3$. This theory contains only finite number of even spins s=2,4,...,2N. We mainly focus on the simplest case, so-called (Type I and II) spin ${\tilde 4}$ gravity, which contains only spin 2 and spin 4 fields. This spin ${\tilde 4}$ gravity is as simple as spin 3 gravity, thus provides another example to test various ideas on higher spin gravity. We find that the asymptotical symmetry of this spin ${\tilde 4}$ gravity is a classical W(2,4)-symmetry. Moreover, we study the black hole solution with pure spin 4 hair and discuss its thermodynamics. One important feature of this black hole is that its entropy could be written in compact forms. Furthermore, we investigate a $G_2$ generated higher spin gravity. This higher spin gravity only contains spin 2 and spin 6 fields which makes it different from other kinds of higher spin gravity. We find the corresponding black hole with spin 6 hair, and discuss its thermodynamics analytically. It turns out that the black holes with spin 4 or spin 6 hair constructed in this paper are the only black holes with single higher spin hair, besides the spin 3 black hole found in arXiv:1103.4304.