Supersymmetric higher spin theories

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 N=2 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 osp(4|N ) with N=1, 2, 4 mod 4, whose R-symmetry can be realized using fermionic oscillators. We tensor these algebras with appropriate internal symmetry algebras, namely u(n) for N= 2 mod 4 and so(n) or usp(n) for N=1,4 mod 4. We show that the N= 3 mod 4 higher spin algebras are isomorphic to those with N = 4 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 N = 6 model in AdS₄, and provide two equivalent descriptions one of which exhibits manifestly its relation to the N = 8 model.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.