Distributional adjoints of the main operators arising in fundamental field theory are examined, with particular regard to their direct geometric construction, in the context of generalized densities valued in a vector bundle; this notion extends the usual one of `section-distributions'.

2000 MSC: 46F10, 81R20, 81R25. 

Keywords: generalized sections, natural operators.

Generalized maps (in the distributional sense) have been usually treated in a purely analytical setting, all constructions being based on ${\bf R}^n$. However, the notion of generalized sections of a vector bundle,
in a geometrical setting, is certainly important in fundamental physics. This notion has been briefly considered by some authors, and some specific cases have been examined in greater detail.

Let us consider an arbitrary finite dimensional vector bundle, with no fixed background structure;
I will argue that, in this setting, the notion of section-distribution has to be introduced as that of `generalized density' valued in the bundle. All other cases, including currents and generalized half-densities, can be seen as particular cases of this one. If a volume form on the base manifold is fixed, then one recovers the notion of section-distribution.

The notion of distributional adjoint of a differential operator can be readily introduced in the geometrical setting. Actually, the local coordinate expression of the adjoint can be written by the standard methods. However, the intrinsic geometric meaning of this expression may not be evident.

The main goal of this paper is to examine the distributional adjoints of the main operators arising in fundamental field theory, with particular regard to their geometric construction. First, I will consider natural operators on scalar-valued currents. Then I will argue that, in some cases, the Fr\"olicher-Nijenhuis bracket yields a natural operator on bundle-valued currents, and examine these. Next, the covariant derivative of a generalized section along a given vector field, relatively to a given connection, yields a non-trivial case. The codifferential and Laplacian are considered in a setting which generalizes Lichnerowicz'. Finally, the distributional adjoint of the Dirac operator on curved background confirms the importance of spacetime torsion in this context.