A Superspace Normal Coordinate Derivation of the Density Formula

TL;DR

This paper presents a purely superspace derivation of the component density formula using a modified normal-coordinate expansion, enabling covariant extraction of component actions from supergravity–matter actions. By expanding the inverse vielbein and the superdeterminant around fermionic normal coordinates, the authors derive density formulas in several theories, including 2D (1,0), (1,1), (2,0) supergravities and 4D N=1 supergravity, often via a two-step chiral decomposition. The work demonstrates that a covariant, assumption-light procedure exists for obtaining the component actions, clarifying the role of chiral reductions and density projectors in superspace. The technique offers a general framework applicable to a wide class of supergravity theories, with implications for systematic component-action derivations in supersymmetric backgrounds.

Abstract

Using normal coordinate expansions we derive by purely superspace methods the density formula giving the component action corresponding to a superspace supergravity-matter action.