Supersymmetric higher spin theories

Abstract

We revisit the higher spin extensions of the anti de Sitter algebra in four dimensions that incorporate internal symmetries and admit representations that contain fermions, classified long ago by Konstein and Vasiliev. We construct the dS4, Euclidean and Kleinian version of these algebras, as well as the corresponding fully nonlinear Vasiliev type higher spin theories, in which the reality conditions we impose on the master fields play a crucial role. The supersymmetric higher spin theory in dS4, on which we elaborate further, is included in this class of models. A subset of the Konstein–Vasiliev algebras are the minimal higher spin extensions of the AdS4 superalgebra with mod 4, whose R-symmetry can be realized using fermionic oscillators. We tensor these algebras with appropriate internal symmetry algebras, namely u(n) for mod 4 and so(n) or usp(n) for mod 4. We show that the mod 4 higher spin algebras are isomorphic to those with mod 4. We describe the fully nonlinear higher spin theories based on these algebras, including the coupling between the adjoint and twisted-adjoint master fields. We elaborate further on the model in AdS4, and provide two equivalent descriptions one of which exhibits manifestly its relation to the model.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.