Supersymmetry of classical solutions in Chern-Simons higher spin supergravity

TL;DR

The paper addresses how to characterize supersymmetry for classical solutions in three-dimensional higher spin supergravity based on $sl(N|N-1)$. It derives the Killing spinor equations from a Chern-Simons action, computes explicit spinors for backgrounds in the $sl(3|2)$ theory (conical defects and BTZ-like black holes), and establishes a holonomy-based, gauge-invariant condition that determines supersymmetry by pairing holonomy eigenvalues with the odd roots of the superalgebra. A key result is that for $N\ge 4$ the theory admits smooth supersymmetric conical defects, while for $sl(3|2)$ such defects are not both smooth and supersymmetric; the framework is then argued to generalize to arbitrary $sl(N|N-1)$. The findings connect the bulk holonomy data to supersymmetric primaries in the dual supersymmetric minimal models, offering a concrete criterion to identify SUSY backgrounds and guiding holographic tests in AdS$_3$/CFT$_2$. Overall, the work provides a robust, gauge-invariant method to diagnose SUSY in higher spin AdS$_3$ theories and highlights new SUSY structures in conical defects and black holes with potential CFT duals.

Abstract

We construct and study classical solutions in Chern-Simons supergravity based on the superalgebra sl(N|N-1). The algebra for the N=3 case is written down explicitly using the fact that it arises as the global part of the super conformal W_3 superalgebra. For this case we construct new classical solutions and study their supersymmetry. Using the algebra we write down the Killing spinor equations and explicitly construct the Killing spinor for conical defects and black holes in this theory. We show that for the general sl(N|N-1) theory the condition for the periodicity of the Killing spinor can be written in terms of the products of the odd roots of the super algebra and the eigenvalues of the holonomy matrix of the background. Thus the supersymmetry of a given background can be stated in terms of gauge invariant and well defined physical observables of the Chern-Simons theory. We then show that for N\geq 4, the sl(N|N-1) theory admits smooth supersymmetric conical defects.