Covariant superspace approaches to ${\cal N}=2$ supergravity

TL;DR

This work presents a unified, covariant framework for ${\cal N}=2$ conformal supergravity in four dimensions by detailing three equivalent superspace formulations—conformal, ${\mathsf{U}}(2)$, and ${\mathsf{SU}}(2)$—and their realization of general supergravity–matter systems through covariant projective multiplets. It systematically develops rigid and local superspace structures, action principles, and multiplet constructions (vector, tensor, and hyper multiplets), and shows how to perform off-shell reductions to component Weyl multiplets. The paper also discusses higher-derivative and topological invariants, including Gauss–Bonnet constructions and super-Weyl anomalies, and outlines two compensator-based off-shell supergravity formulations. Together, these results provide versatile tools for constructing and analyzing ${\cal N}=2$ supergravity theories with matter couplings across multiple geometric formalisms, with implications for both formal structure and practical model building.

Abstract

We provide a unified description of the three covariant superspace approaches to ${\cal N}=2$ conformal supergravity in four dimensions: (i) conformal superspace; (ii) $\mathsf{U}(2)$ superspace; and (iii) $\mathsf{SU}(2)$ superspace. Each of them can be used to formulate general supergravity-matter systems, although conformal superspace has the largest structure group and is intimately related to the superconformal tensor calculus. We review the structure of covariant projective multiplets and demonstrate how they are used to describe pure and matter-coupled supergravity, including locally superconformal off-shell sigma models. Higher-derivative invariants, topological invariants and super-Weyl anomalies are also briefly discussed.