Conical Defects, Black Holes and Higher Spin (Super-)Symmetry

TL;DR

This work shows that in AdS$_3$ higher spin (super-)gravity, the symmetry content of classical solutions is encoded by the spatial-holonomy around the circle, with trivial holonomy yielding maximally symmetric smooth conical defects and nontrivial holonomy producing partially symmetric black holes. The authors construct and classify maximally symmetric backgrounds across various gauge groups ($SL(N)$, $Sp(2N)$, $SO(2N)$, $G_2$) and their superextensions, establishing a precise one-to-one map between these defects and highest-weight representations of the dual groups, including a CFT-inspired Euclidean perspective via $SL(N,\, ext{C})$ and Young diagrams. They extend the analysis to higher spin supergravity, showing that maximality requires holonomies in the bosonic center and consistency with fermionic boundary conditions, linking maximally symmetric defects to chiral primaries in Kazama–Suzuki models. The paper then analyzes black holes (e.g., in $ ext{osp}(3|2)$ and $ ext{sl}(3|2)$) and demonstrates that they are generally only partially symmetric, with entropy and supersymmetry determined by holonomy data, allowing both extremal and non-extremal supersymmetric configurations. Overall, these results illuminate the spectrum and symmetry structure of 3D HS/CFT dualities and provide a concrete framework for identifying and classifying symmetric configurations via holonomy.

Abstract

We study the (super-)symmetries of classical solutions in the higher spin (super-)gravity in AdS$_3$. We show that the symmetries of the solutions are encoded in the holonomy around the spatial circle. When the spatial holonomies of the solutions are trivial, they preserve maximal symmetries of the theory, and are actually the smooth conical defects. We find all the smooth conical defects in the $sl(N), so(2N+1),sp(2N), so(2N), g_2$, as well as in $sl(N|N-1)$ and $osp(2N+1|2N)$ Chern-Simons gravity theories. In the bosonic higher spin cases, there are one-to-one correspondences between the smooth conical defects and the highest weight representations of Lie group. Furthermore we investigate the higher spin black holes in $osp(3|2)$ and $sl(3|2)$ higher spin (super-)gravity and find that they are only partially symmetric. In general, the black holes break all the supersymmetries, but in some cases they preserve part of the supersymmetries.