Unitarity Bounds in AdS_3 Higher Spin Gravity

TL;DR

This work analyzes unitarity bounds for AdS$_3$ higher-spin gravity in the Chern-Simons formulation with SL(N,$\mathbb{R}$)×SL(N,$\mathbb{R}$) gauge symmetry. By performing the Drinfeld-Sokolov reduction for different SL(2,$\mathbb{R}$) embeddings into SL(N,$\mathbb{R}$), the authors show that non-principal embeddings generically produce spin-1 currents whose Kac-Moody subalgebras acquire negative level at large central charge $c$, leading to negative-norm states and thus non-unitary representations. Consequently, unitarity strongly constrains the viable theories, suggesting the principal embedding (which yields a $\mathcal{W}_N$ algebra without problematic spin-1 sectors) as the only consistent framework in the semiclassical limit. The paper provides explicit OPEs and norm arguments for sum and product embeddings (including the ${\mathcal W}^{(2)}_3$ and ${\mathcal W}^{(2,2)}_4$ cases) to support this unitarity selection principle, with broader implications for holography and the space of AdS$_3$ higher-spin duals.

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.