Abstract
The BRST algebra of supergravity is characterized by two different bilinears of the commuting supersymmetry ghosts: a vector γμ and a scalar ϕ, the latter valued in the Yang-Mills Lie algebra. We observe that under BRST transformations γ and ϕ 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.
Highlights
Our starting point will be the BRST formulation of supergravity
The BRST algebra of supergravity is characterized by two different bilinears of the commuting supersymmetry ghosts: a vector γμ and a scalar φ, the latter valued in the Yang-Mills Lie algebra
In the rest of this paper we apply to supersymmetric localization the topological structure that we have discovered sitting inside the supergravity BRST algebra
Summary
In the BRST framework one introduces ghost fields of ghost number +1 in correspondence to each of the local symmetries. Ψi = ψμi dxμ are the Majorana gravitinos and γμ is the following vector bilinear of the commuting ghosts γμ. It was noted in [5] that the BRST transformations (2.1) and (2.4) imply that the vector ghost bilinear γμ transforms as follows s γμ = −Lξ γμ (2.6) This transformation law coincides precisely with the BRST transformation rule for the superghost of topological gravity [7]. Where gμν is the metric, ψμν is the topological gravitino and γμ the vector superghost From this it is apparent that the supergravity BRST transformation rules (2.5) match the topological gravity ones, once the topological gravity superghost is identified with the supergravity ghost bilinear γμ according to (2.3).
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