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

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 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.

48 pages. Invited chapter for the "Handbook of Quantum Gravity" (Eds. C. Bambi, L. Modesto and I.L. Shapiro, Springer, expected in 2023)