Fermionic Shift Symmetries in (Anti) de Sitter Space

We study extended shift symmetries that arise for fermionic fields on anti-de Sitter (AdS) space and de Sitter (dS) space for particular values of the mass relative to the curvature scale. We classify these symmetries for general mixed-symmetry fermionic fields in arbitrary dimension and describe how fields with these symmetries arise as the decoupled longitudinal modes of massive fermions as they approach partially massless points. For the particular case of AdS$_4$, we look for non-trivial Lie superalgebras that can underly interacting theories that involve these fields. We study from this perspective the minimal such theory, the Akulov--Volkov theory on AdS$_4$, which is a non-linear theory of a spin-$1/2$ Goldstino field that describes the spontaneous breaking of ${\cal N}=1$ supersymmetry on AdS$_4$ down to the isometries of AdS$_4$. We show how to write the nonlinear supersymmetry transformation for this theory using the fermionic ambient space formalism. We also study the Lie superalgebras of candidate multi-field examples and rule out the existence of a supersymmetric special galileon on AdS$_4$.