Exploring Three-dimensional Higher-Spin Supergravity based on sl(N |N - 1) Chern-Simons theories

TL;DR

The paper investigates three-dimensional higher-spin AdS supergravity described by Chern-Simons theory with the finite-dimensional superalgebra $sl(N|N-1)$, focusing on $N=3$ as the principal example to reveal supersymmetric structure beyond the bosonic higher-spin theories.It derives Killing spinor equations, analyzes asymptotic symmetries, and demonstrates that each chiral sector supports an ${ m cal N}=2$ superconformal algebra, with a Sugawara-like redefinition linking the energy-momentum tensor to the extended currents.Holonomy conditions are employed to define higher-spin black holes and conical defects, revealing integrability relations that mirror known results in $sl(N)$ theories, while supersymmetric solutions with higher-spin fields are constrained and classified.The framework is then extended to general finite $N$ and to the infinite-dimensional $shs[\lambda]$ limit, highlighting potential holographic dualities with ${ m cal N}=(2,2)$ superconformal field theories and outlining future directions for a deeper holographic dictionary.

Abstract

We investigate various aspects of higher-spin anti-de Sitter supergravity in three dimensions as described by Chern-Simons theory based on the finite-dimensional superalgebra sl(N |N - 1), with the particular case of N = 3 as our prime example. This class of theories serves as a natural supersymmetrization of the higher-spin gravity theory based on sl(N) Chern-Simons theories. We demonstrate explicitly that the asymptotic symmetry algebra contains the N = 2 superconformal algebra in each sector. The appropriate Killing spinor equations are derived and used to classify existing and new classical solutions. We also discuss holonomy conditions, higher-spin black holes and conical defect spacetimes in this class of theories.