Supersymmetric Higher Spin Theories

TL;DR

This work classifications and constructs a broad family of nonlinear, fully consistent supersymmetric higher spin theories in four dimensions across AdS4, dS4, and related signatures, built from Konstein–Vasiliev algebras. It extends the bosonic Vasiliev framework by incorporating fermions and internal Yang–Mills symmetries, using oscillator realizations and a network of reality and projection conditions to define Type A and Type B parity-invariant models, as well as various N mod 4 supersymmetric sequences. A key result is the demonstration of isomorphisms between N=3 mod 4 and N=4 mod 4 algebras, along with explicit constructions of N=6 and N=8 related models, including two descriptions that illuminate the relation to higher supersymmetry. The findings provide a robust platform for exploring holography, de Sitter physics, and connections to string/M-theory, and point toward future work on matter couplings, symmetry breaking, and superspace formulations of nonlinear higher spin dynamics.

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 $dS_4$, 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 ${\cal N}=2$ supersymmetric higher spin theory in $dS_4$, on which we elaborate further, is included in this class of models. A subset of Konstein-Vasiliev algebras are the higher spin extensions of the $AdS_4$ superalgebras $osp(4|{\cal N})$ for ${\cal N}=1,2,4$ mod 4 and can be realized using fermionic oscillators. We tensor the higher superalgebras of the latter kind with appropriate internal symmetry groups and show that the ${\cal N}=3$ mod 4 higher spin algebras are isomorphic to those with ${\cal N}=4$ mod 4. We describe the fully nonlinear higher spin theories based on these algebras as well, and we elaborate further on the ${\cal N}=6$ supersymmetric theory, providing two equivalent descriptions one of which exhibits manifestly its relation to the ${\cal N}=8$ supersymmetric higher spin theory.