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

TL;DR

This paper proves the uniqueness of pure supergravity theories with specific supersymmetry counts in four dimensions by employing BRST-BV deformation methods, establishing constraints on possible gaugings and coupling structures.

Contribution

It demonstrates the exclusive nature of ${ m N}=2$ and ${ m N}=3$ pure supergravities in 4D through a systematic deformation analysis, clarifying their foundational uniqueness.

Findings

Proves the impossibility of non-minimal coupling of a real Rarita-Schwinger field to electromagnetism.

Shows that coupling complex Rarita-Schwinger fields to electromagnetism necessitates gravity, leading to ${ m N}=2$ supergravity.

Derives quadratic constraints on gaugings for theories with multiple gauge fields.

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 ${\cal N}=2$ 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, ${\cal N}$ real spin-3/2 gauge fields and at most one real spinor field also requires gravity and yields ${\cal N}=3$ pure supergravity as well as ${\cal N}=1$ 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.