Superspace First Order Formalism, Trivial Symmetries and Electromagnetic Interactions of Linearized Supergravity

We introduce a first order description of linearized non-minimal ($n=-1$) supergravity in superspace, using the unconstrained prepotential superfield instead of the conventionally constrained super one forms. In this description, after integrating out the connection-like auxiliary superfield of first-order formalism, the superspace action is expressed in terms of a single superfield which combines the prepotential and compensator superfields. We use this description to construct the supersymmetric cubic interaction vertex $3\mathrm{/}2-3\mathrm{/}2-1\mathrm{/}2$ which describes the electromagnetic interaction between two non-minimal supergravity multiplets (superspin $\mathsf{\text{Y}}=3\mathrm{/}2$ which contains a spin 2 and a spin 3/2 particles) and a vector multiplet (superspin $\mathsf{\text{Y}}=1\mathrm{/}2$ contains a spin 1 and a spin 1/2 particles). Exploring the trivial symmetries emerging between the two $\mathsf{\text{Y}}=3\mathrm{/}2$ supermultiplets, we show that this cubic vertex must depend on the vector multiplet superfield strength. This result generalize previous results for non-supersymmetric electromagnetic interactions of spin 2 particles. The constructed cubic interaction generates non-trivial deformations of the gauge transformations.