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arXiv · 0708.0400

Odd Scalar Curvature in Field-Antifield Formalism

Igor A. Batalin·Klaus Bering
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Abstract

We consider the possibility of adding a Grassmann-odd function νto the odd Laplacian. Requiring the total Δoperator to be nilpotent leads to a differential condition for ν, which is integrable. It turns out that the odd function νis not an independent geometric object, but is instead completely specified by the antisymplectic structure E and the density ρ. The main impact of introducing the νterm is that it makes compatibility relations between E and ρobsolete. We give a geometric interpretation of νas (minus 1/8 times) the odd scalar curvature of an arbitrary antisymplectic, torsion-free and ρ-compatible connection. We show that the total Δoperator is a ρ-dressed version of Khudaverdian's Δ_E operator, which takes semidensities to semidensities. We also show that the construction generalizes to the situation where ρis replaced by a non-flat line bundle connection F. This generalization is implemented by breaking the nilpotency of Δwith an arbitrary Grassmann-even second-order operator source.

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Igor A. Batalin, Klaus Bering. 2008-01-24. Odd Scalar Curvature in Field-Antifield Formalism. https://doi.org/10.1063/1.2835485

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