Uniqueness of $$ \mathcal{N} $$ = 2 and 3 pure supergravities in 4D

Abstract

Abstract After proving the impossibility of consistent non-minimal coupling of a real Rarita-Schwinger gauge field to electromagnetism, we re-derive the necessity of introducing the graviton in order to couple a complex Rarita-Schwinger gauge field to electromagnetism, with or without a cosmological term, thereby obtaining $$ \mathcal{N} $$ N = 2 pure supergravity as the only possibility. These results are obtained with the BRST-BV deformation method around the flat and (A)dS backgrounds in 4 dimensions. The same method applied to n v vectors, $$ \mathcal{N} $$ N real spin-3/2 gauge fields and at most one real spinor field also requires gravity and yields $$ \mathcal{N} $$ N = 3 pure supergravity as well as $$ \mathcal{N} $$ N = 1 pure supergravity coupled to a vector supermultiplet, with or without cosmological terms. Independently of the matter content, we finally derive strong necessary quadratic constraints on the possible gaugings for an arbitrary number of spin-1 and spin-3/2 gauge fields, that are relevant for larger supergravities.