A Comparative Study of Laplacians and Schroedinger-Lichnerowicz-Weitzenboeck Identities in Riemannian and Antisymplectic Geometry

Abstract

We introduce an antisymplectic Dirac operator and antisymplectic gamma matrices. We explore similarities between, on one hand, the Schroedinger-Lichnerowicz formula for spinor bundles in Riemannian spin geometry, which contains a zeroth-order term proportional to the Levi-Civita scalar curvature, and, on the other hand, the nilpotent, Grassmann-odd, second-order Δoperator in antisymplectic geometry, which in general has a zeroth-order term proportional to the odd scalar curvature of an arbitrary antisymplectic and torsionfree connection that is compatible with the measure density. Finally, we discuss the close relationship with the two-loop scalar curvature term in the quantum Hamiltonian for a particle in a curved Riemannian space.

55 pages, LaTeX. v2: Subsection 3.10 expanded. v3: Reference added. v4: Published version

References in corpus (5)

• On Generalized Gauge-Fixing in the Field-Antifield Formalism
• Odd Scalar Curvature in Anti-Poisson Geometry
• Odd Scalar Curvature in Field-Antifield Formalism
• Semidensities, Second-Class Constraints and Conversion in Anti-Poisson Geometry
• A Note on Semidensities in Antisymplectic Geometry

Cited by in corpus (5)

• On the Riemann Tensor in Double Field Theory
• 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
• Reparametrization-Invariant Effective Action in Field-Antifield Formalism
• External Sources in Field-Antifield Formalism