Natural differential operators and graph complexes

We show how the machine invented by S. Merkulov [S.A. Merkulov, Operads, deformation theory and F-manifolds, in: Frobenius Manifolds, Aspects Math., vol. E36, Vieweg, Wiesbaden, 2004, pp. 213–251; S.A. Merkulov, PROP profile of deformation quantization, Preprint, math.QA/0412257, December 2004; S.A. Merkulov, PROP profile of Poisson geometry, Comm. Math. Phys. 262 (1) (February 2006) 117–135] can be used to study and classify natural operators in differential geometry. We also give an interpretation of graph complexes arising in this context in terms of representation theory. As application, we prove several results on classification of natural operators acting on vector fields and connections.