Local supersymmetry: Variations on a theme by Volkov and Soroka

Abstract: We revisit the work by Volkov and Soroka on spontaneously broken local supersymmetry. It is demonstrated for the first time that, for specially chosen parameters of the theory, the Volkov-Soroka action is invariant under two different local supersymmetries. One of them is present for arbitrary values of the parameters and acts on the Goldstino, while the other supersymmetry emerges only in a special case and leaves the Goldstino invariant. The former can be used to gauge away the Goldstino, and then the resulting action coincides with that proposed by Deser and Zumino for consistent supergravity in the first-order formalism. In this sense, pure $\mathcal{N} = 1$ supergravity is a special case of the Volkov-Soroka theory, although it was not discovered by these authors. We also explain how the Volkov-Soroka approach allows one to naturally arrive at the 1.5 formalism. Our analysis provides a nonlinear realisation approach to construct unbroken $\mathcal{N} = 1$ Poincar\'e supergravity.