Semi-classical unitarity in three-dimensional higher spin gravity for non-principal embeddings

Higher spin gravity in three dimensions is efficiently formulated as a Chern-Simons gauge-theory, typically with gauge algebra sl(N)⊕sl(N). The classical and quantum properties of the higher spin theory depend crucially on the embedding into the full gauge algebra of the sl(2)⊕sl(2) factor associated with gravity. It has been argued previously that non-principal embeddings do not allow for a semi-classical limit (large values of the central charge) consistent with unitarity. In this work we show that it is possible to circumvent these conclusions. Based upon the Feigin-Semikhatov generalization of the Polyakov-Bershadsky algebra, we construct infinite families of unitary higher spin gravity theories at certain rational values of the Chern-Simons level that allow arbitrarily large values of the central charge up to , thereby confirming a recent speculation by us (Afshar et al 2012 arXiv:1209.2860).