Hypersymmetry bounds and three-dimensional higher-spin black holes

TL;DR

This work analyzes how hypersymmetry in three-dimensional AdS higher-spin gravity constrains black hole and soliton solutions. Using a Chern-Simons formulation with gauge algebra $osp(1|4)\oplus osp(1|4)$, the authors derive the asymptotic $WB_2$-algebra and nonlinear hypersymmetry bounds from the anticommutators of spin-$\tfrac{5}{2}$ generators. Hypersymmetric black holes saturate the strongest bound and are extremal with $\tfrac{1}{4}$-hypersymmetry, while certain $sp(4)$-solitonic solutions either saturate or violate these bounds in a controlled way; Euclidean analysis with spin-4 chemical potentials reveals partial preservation of hypersymmetry. The results illuminate how hypersymmetry governs charge regions and thermodynamics in higher-spin AdS$_3$ and motivate exploring supersymmetric extensions for additional bounds and richer extremal structures.

Abstract

We investigate the hypersymmetry bounds on the higher spin black hole parameters that follow from the asymptotic symmetry superalgebra in higher-spin anti-de Sitter gravity in three spacetime dimensions. We consider anti-de Sitter hypergravity for which the analysis is most transparent. This is a $osp(1\vert 4) \oplus osp(1\vert 4)$ Chern-Simons theory which contains, besides a spin-$2$ field, a spin-$4$ field and a spin-$5/2$ field. The asymptotic symmetry superalgebra is then the direct sum of two-copies of the hypersymmetric extension $W_{(2,\frac52,4)}$ of $W_{(2,4)}$, which contains fermionic generators of conformal weight $5/2$ and bosonic generators of conformal weight $4$ in addition to the Virasoro generators. Following standard methods, we derive bounds on the conserved charges from the anticommutator of the hypersymmetry generators. The hypersymmetry bounds are nonlinear and are saturated by the hypersymmetric black holes, which turn out to possess $1/4$-hypersymmetry and to be "extreme", where extremality can be defined in terms of the entropy: extreme black holes are those that fulfill the extremality bounds beyond which the entropy ceases to be a real function of the black hole parameters. We also extend the analysis to other $sp(4)$-solitonic solutions which are maximally (hyper)symmetric.