Odd Scalar Curvature in Field-Antifield Formalism

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.

23 pages, LaTeX. v2: More material added. v3: Reference added. v4: Grant number added. v5: Minor changes. v6: Stylistic changes

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Cited by in corpus (16)

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• Semidensities, Second-Class Constraints and Conversion in Anti-Poisson Geometry
• Fedosov and Riemannian supermanifolds
• Gauge Independence in a Higher-Order Lagrangian Formalism via Change of Variables in the Path Integral
• Geometry of differential operators of second order, the algebra of densities, and groupoids
• Bone in vivo: Surface mapping technique
• A Comparative Study of Laplacians and Schroedinger-Lichnerowicz-Weitzenboeck Identities in Riemannian and Antisymplectic Geometry
• Path Integral Formulation with Deformed Antibracket
• Reparametrization-Invariant Effective Action in Field-Antifield Formalism
• Does the nontrivially deformed field-antifield formalism exist?
• External Sources in Field-Antifield Formalism
• General quantum-mechanical setting for field-antifield formalism as a hyper-gauge theory
• Symplectic connections and Fedosov's quantization on supermanifolds
• Symplectic scalar curvature on supermanifolds
• Supermanifolds, symplectic geometry and curvature