Second order operators on the algebra of densities and a groupoid of connections

TL;DR

This paper explores the geometry of second order linear operators on the algebra of densities on supermanifolds, introducing a groupoid of connections and analyzing their geometric and algebraic properties with applications in supergeometry and classical geometry.

Contribution

It introduces a new geometric framework for second order operators on densities, including a groupoid of connections and analysis of canonical operator pencils and their singularities.

Findings

Operators depend on equivalence classes of connections

Groupoid of connections organizes these classes

Applications to Batalin-Vilkovisky geometry and Sturm-Liouville operators

Abstract

We consider the geometry of second order linear operators acting on the commutative algebra of densities on a (super)manifold introduced in our previous work. In the conventional language, operators on the algebra of densities correspond to operator pencils. This algebra has a natural invariant scalar product. We consider self-adjoint operators on the algebra of densities and analyze the corresponding "canonical operator pencils" passing through a given operator on densities of a particular weight. There are singular values for the pencil parameters. This leads to an interesting geometrical picture. In particular we obtain operators that depend on equivalence classes of connections and we study a groupoid of connections such that the orbits of this groupoid are these equivalence classes. Based on this point of view we analyze two examples: the second order canonical operator on an odd symplectic supermanifold appearing in the Batalin-Vilkovisky geometry and the Sturm-Liouville operator on the line related with classical constructions of projective geometry. We also consider a canonical second order semidensity arising on odd symplectic supermanifolds, which has some resemblance with mean curvature in Riemannian geometry.