The topological structure of supergravity: an application to supersymmetric localization

TL;DR

This paper reveals a universal topological and cohomological structure within supergravity theories, connecting BRST algebra, supersymmetric backgrounds, and equivariant cohomology, with detailed analysis for specific supersymmetry cases.

Contribution

It uncovers a topological framework inside supergravity that leads to universal cohomological equations, extending understanding of supersymmetric backgrounds and their geometric properties.

Findings

Identifies topological structures in supergravity BRST algebra.

Derives universal equivariant cohomological equations for supersymmetric backgrounds.

Analyzes specific cases like N=(2,2) in d=2 and N=2 in d=4.

Abstract

The BRST algebra of supergravity is characterized by two different bilinears of the commuting supersymmetry ghosts: a vector $\gamma^\mu$ and a scalar $\phi$, the latter valued in the Yang-Mills Lie algebra. We observe that under BRST transformations $\gamma$ and $\phi$ transform as the superghosts of, respectively, topological gravity and topological Yang-Mills coupled to topological gravity. This topological structure sitting inside any supergravity leads to universal equivariant cohomological equations for the curvatures 2-forms which hold on supersymmetric bosonic backgrounds. Additional equivariant cohomological equations can be derived for supersymmetric backgrounds of supergravities for which certain gauge invariant scalar bilinears of the commuting ghosts exist. Among those, $N= (2,2)$ in $d=2$, which we discuss in detail in this paper, and $N=2$ in $d=4$.