Unitarity bounds in AdS3 higher spin gravity

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.