Abstract: By careful use of the Fefferman-Graham ambient construction, we construct conformally invariant operators from k-forms to k-form densities in even dimensions (for k less than n/2), that admit factorizations of the form L=δ Qd. Here Q is a differential operator of order n-2k-2 which is given by a universal formula in a pseudo-Riemannian metric, but which is not conformally invariant. Nevertheless, the operator Q has special conformal properties. For example, the conformal deformation of the k-form Q on closed forms may be written in terms of δ, d, and the (k+1)-form Q, establishing a kind of interlocking of the operators at various form orders. The operators L are not elliptic, but admit gauge companion operators G with the property that the pair (L,G) is injectively elliptic. One special case of this was known previously, namely the Maxwell operator L=δ d on (n/2-1)-forms, with the Eastwood-Singer gauge operator G=δ dδ+(lower order) as its gauge companion. In contrast to the situation for this low-order special case, one cannot reasonably pursue classical tensor formulas for the higher-order (L,G) directly; thus the need for the full force of the Fefferman-Graham construction.

In the compact Riemannian signature case, moving along the beginning of the de Rham complex with coboundaries d, then moving by the k-form operator L for some k, and continuing with coboundaries δ gives an elliptic complex which we call a detour complex. Each such complex admits a generalization of the Cheeger half-torsion; in fact, the Cheeger half-torsion is the Maxwell operator special case of this construction. These half-torsions admit Polyakov formulas, and the validity of the construction persists for arbitrary regular conformal Bernstein-Gelfand-Gelfand (BGG) diagrams, when they are elliptic complexes. The coupling constants for the functional determinants of Laplacians at various points of the BGG diagram depend on the weights of the modules involved.

The detour complexes also provide generalizations of the Q-curvature, in the form of compressions of the operators Q (from the formulas L=δ Qd above) acting between natural subquotients of form and form-density section spaces -- the cohomologies of the detour complexes, and related harmonic spaces. It is these compressions (rather than the full operator Q from the formula L=δ Qd) that are conformally invariant. This generalizes the invariance of the original (scalar density) Q-curvature modulo the range of the critical Graham-Jenne-Mason-Sparling operator.