Superspace Approaches to \(\mathscr {N}\,{=}\,\text{1}\) Supergravity

The superspace formalism for \(\mathscr {N}\,{=}\,1\) 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 \(\mathscr {N}\,{=}\,1\) conformal supergravity: (i) conformal superspace, (ii) 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 Poincaré 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 Poincaré 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.