The superform construction of supergravity actions, christened the “ectoplasm method,” is based on the use of a closed super $d$-form in the case of $d$ space-time dimensions. In known examples, such superforms are obtained by iteratively solving nontrivial cohomological problems. The latter usually makes this scheme no less laborious than the normal coordinate method for deriving component actions for matter-coupled supergravity. In this paper we present an alternative procedure to generate required superforms in four space-time dimensions, which makes use of self-dual vector multiplets. It provides the shortest derivation of chiral actions in two different theories: (i) $\mathcal{N}=1$ old minimal supergravity; and (ii) $\mathcal{N}=2$ conformal supergravity. The $\mathcal{N}=2$ superform construction is developed here for the first time. Although our consideration is restricted to the case of four dimensions, a generalization to higher dimensions is plausible.

• Received 30 September 2009

DOI:https://doi.org/10.1103/PhysRevD.80.125015