  In an earlier publication, we have introduced a method to obtain, at large N,
the effective action for d-dimensional manifolds in a N-dimensional disordered
environment. This allowed to obtain the Functional Renormalization Group (FRG)
equation for N=infinity and was shown to reproduce, with no need for
ultrametric replica symmetry breaking, the predictions of the Mezard-Parisi
solution. Here we compute the corrections at order 1/N. We introduce two novel
complementary methods, a diagrammatic and an algebraic one, to perform the
complicated resummation of an infinite number of loops, and derive the
beta-function of the theory to order 1/N. We present both the effective action
and the corresponding functional renormalization group equations. The aim is to
explain the conceptual basis and give a detailed account of the novel aspects
of such calculations. The analysis of the FRG flow, comparison with other
studies, and applications, e.g. to the strong-coupling phase of the
Kardar-Parisi-Zhang equation are examined in a subsequent publication.
