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

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 $\mathcal{N}=2$ 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, $\mathcal{N}$ real spin-3/2 gauge fields and at most one real spinor field also requires gravity and yields $\mathcal{N}=3$ pure supergravity as well as $\mathcal{N}=1$ pure supergravity coupled to a vector supermultiplet, with or without cosmological terms. Independently from 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.

LaTeX, 31 + 1 pages, no figure. v2: Extended discussion at the end of Section 3, corrected typos and references added