A bstract We classify all the possible parity-invariant, non-Abelian couplings involving a partially massless spin-2 field and a pair of massless spin-3/2 fields around anti-de Sitter spacetime AdS 4. We re-derive a non-Abelian vertex recently found in 48 by Yu. M. Zinoviev following a different approach, and make explicit the structure of the non-Abelian gauge algebra, very similar to N N = 2 supergravity around AdS 4. We show that this vertex is obstructed at higher order in deformation, and that the graviton plus a vector gauge field are required to eliminate most, if not all, of the obstructions. The resulting gauge algebra produces all the structures of the super algebra su (2, 2 | 1), suggesting that the only consistent theory that couples a partially massless spin-2 field with massless gravitini is N N = 1 conformal supergravity expanded around AdS 4.
Boulanger et al. (2025) studied this question.