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).