  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 \Delta operator 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.
