2007/08/31 by Igor A. Batalin, Klaus Bering · 17 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Bundle #Connection (principal bundle) #Curvature #Differential operator #Formalism (music) #Geometric Analysis and Curvature Flows #Homogeneous function #Homotopy and Cohomology in Algebraic Topology #Operator (biology) #Scalar (mathematics) #hep-th #math-ph #math.MP #math.SG
paper · pdf · doi:10.1063/1.2835485
published in Journal of Mathematical Physics 49(3) (American Institute of Physics) · 23 pages, LaTeX. v2: More material added. v3: Reference added. v4: Grant number added. v5: Minor changes. v6: Stylistic changes
arxiv created 2008/01/24 · openalex publication_date 2008/03/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider the possibility of adding a Grassmann-odd function ν to the odd Laplacian. Requiring the total Δ operator to be nilpotent leads to a differential condition for ν, which is integrable. It turns out that the odd function ν is 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 ν term is that it makes compatibility relations between E and ρ obsolete. We give a geometric interpretation of ν as (minus 18 times) the odd scalar curvature of an arbitrary antisymplectic, torsion-free, and ρ-compatible connection. We show that the total Δ operator is a ρ-dressed version of Khudaverdian’s ΔE operator, which takes semidensities to semidensities. We also show that the construction generalizes to the situation where ρ is replaced by a nonflat line bundle connection F. This generalization is implemented by breaking the nilpotency of Δ with an arbitrary Grassmann-even second-order operator source.