Semi-classical unitarity in 3-dimensional higher-spin gravity for non-principal embeddings

TL;DR

This work challenges the prevailing belief that non-principal embeddings in 3D higher-spin gravity cannot support a semi-classical unitarity regime. By leveraging Feigin–Semikhatov W_N^{(2)} algebras and a careful analysis of the non-principal embeddings, the authors construct unitary higher-spin theories with central charges that scale linearly with N, up to the bound c <= N/4 - 1/8 - O(1/N) at certain rational k_CS values. Unitarity is achieved by tuning parameters so that the G^± sector becomes null (lambda = 0) while keeping kappa >= 0 and c >= 0; this yields discrete rational alpha values, and as N grows, arbitrarily large c can be approached. The results provide a concrete path to semi-classical holography in non-principal higher-spin gravity, expanding the landscape of unitary, topological models with sizable central charges and potential CFT duals.

Abstract

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 c = N/4 - 1/8 - O(1/N), thereby confirming a recent speculation by us 1209.2860.

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