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The topological structure of supergravity: an application to supersymmetric localization. (English) Zbl 1391.83127

Summary: 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 \(d=4\).

MSC:

83E50 Supergravity
81T60 Supersymmetric field theories in quantum mechanics
81T45 Topological field theories in quantum mechanics
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
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