Geometry of differential operators of second order, the algebra of densities, and groupoids

TL;DR

This paper explores the geometry of second-order differential operators on a density algebra, revealing connections with groupoids, supermanifold structures, and classical geometric operators like Laplacians and Sturm-Liouville operators.

Contribution

It introduces a geometric framework for second-order differential operators on densities, linking them to groupoids and connections, with applications to supermanifolds and classical geometry.

Findings

Operators depend on equivalence classes of connections.

Identifies singular values related to geometric structures.

Analyzes examples like the Batalin-Vilkovisky Laplacian and Sturm-Liouville operator.

Abstract

In our previous works, we introduced, for each (super)manifold, a commutative algebra of densities. It is endowed with a natural invariant scalar product. In this paper, we study geometry of differential operators of second order on this algebra. In the more conventional language they correspond to certain operator pencils. We consider the self-adjoint operators and analyze the operator pencils that pass through a given operator acting on densities of a particular weight. There are `singular values' for pencil parameters. They are related with interesting geometric picture. In particular, we obtain operators that depend on certain equivalence classes of connections (instead of connections as such). We study the corresponding groupoids. From this point of view we analyze two examples: the canonical Laplacian on an odd symplectic supermanifold appearing in Batalin--Vilkovisky geometry and the Sturm--Liouville operator on the line, related with classical constructions of projective geometry. We also consider the canonical second order semi-density arising on odd symplectic supermanifolds, which has some similarity with mean curvature of surfaces in Riemannian geometry.