Unitarity bounds in AdS3 higher spin gravity

A bstractWe study SL(N,$\mathbb{R}$) Chern-Simons gauge theories in three dimensions. The choice of the embedding of SL(2,$\mathbb{R}$) in SL(N,$\mathbb{R}$), 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 ${{\mathcal{W}}_N}$ algebra constructed using the principal embedding could admit a unitary representation for large values of the central charge.