Superspace approaches to $\mathcal{N}=1$ supergravity

TL;DR

This paper reviews the superspace formalism for $ abla=1$ supergravity in four dimensions, unifying three approaches and discussing off-shell formulations, equations of motion, and solutions within conformal supergravity frameworks.

Contribution

It provides a comprehensive, unified overview of the three superspace approaches to $ abla=1$ supergravity, including prepotential formulations and applications to off-shell theories.

Findings

Unified description of three superspace approaches

Derivation of superfield equations of motion

Demonstration that solutions of off-shell equations also solve conformal supergravity equations

Abstract

The superspace formalism for $\mathcal{N}=1$ supergravity in four dimensions is a powerful geometric setting to engineer off-shell supergravity-matter theories, including higher-derivative couplings. This review provides a unified description of the three superspace approaches to $\mathcal{N}=1$ conformal supergravity: (i) conformal superspace; (ii) $\mathsf{U}(1)$ superspace; and (iii) the Grimm-Wess-Zumino formalism. The prepotential formulation for the latter is discussed. We briefly describe the known off-shell formulations for Poincar\'e and anti-de Sitter supergravity theories as conformal supergravity coupled to certain compensators. As simple applications of the formalism, we present the superfield equations of motion for various off-shell formulations for pure Poincar\'e and anti-de Sitter supergravity, and show that every solution of these equations is also a solution of the equations of motion for conformal supergravity.