Unitarity Bounds in AdS_3 Higher Spin Gravity

TL;DR

This paper investigates unitarity bounds in three-dimensional higher spin gravity theories formulated as SL(N,R) Chern-Simons gauge theories, analyzing how different embeddings affect the boundary symmetry algebras and their unitarity.

Contribution

It identifies that only the W_N algebra from the principal embedding can support unitary representations at large central charge.

Findings

Only the principal embedding's W_N algebra admits unitarity.

Different embeddings lead to distinct asymptotic symmetry groups.

The structure of boundary algebras constrains the unitarity of the theory.

Abstract

We study SL(N,R) Chern-Simons gauge theories in three dimensions. The choice of the embedding of SL(2,R) in SL(N,R), together with asymptotic boundary conditions, defines a theory of higher spin gravity. Each inequivalent embedding leads to a different asymptotic symmetry group, which we map to an OPE structure at the boundary. A simple inspection of these algebras indicates that only the W_N algebra constructed using the principal embedding could admit a unitary representation for large values of the central charge.