Supersymmetric Higher Spin Theories

TL;DR

This paper explores the construction and classification of supersymmetric higher spin theories in various spacetime signatures, extending known algebras and formulating fully nonlinear models with internal symmetries and fermionic representations.

Contribution

It introduces new higher spin algebras in different signatures, constructs corresponding nonlinear theories, and clarifies relations among supersymmetric extensions, including ${ m N}=3$, ${ m N}=4$, ${ m N}=6$, and ${ m N}=8$ models.

Findings

Constructed $dS_4$, Euclidean, and Kleinian higher spin algebras.

Developed fully nonlinear Vasiliev-type higher spin theories.

Established isomorphisms between ${ m N}=3$ mod 4 and ${ m N}=4$ mod 4 algebras.

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.