  We study the replica field theory which describes the pinning of elastic
manifolds of arbitrary internal dimension d in a random potential, with the aim
of bridging the gap between mean field and renormalization theory. The full
effective action is computed exactly in the limit of large embedding space
dimension N. The second cumulant of the renormalized disorder obeys a closed
self-consistent equation. It is used to derive a Functional Renormalization
Group (FRG) equation valid in any dimension d, which correctly matches the
Balents-Fisher result to first order in epsilon=4-d. We analyze in detail the
solutions of the large-N FRG for both long-range and short-range disorder, at
zero and finite temperature. We find consistent agreement with the results of
Mezard Parisi (MP) from the Gaussian variational method (GVM) in the case where
full replica symmetry breaking (RSB) holds there. We prove that the cusplike
non-analyticity in the large N FRG appears at a finite scale, corresponding to
the instability of the replica symmetric solution of MP. We show that the FRG
exactly reproduces, for any disorder correlator and with no need to invoke
Parisi's spontaneous RSB, the non-trivial result of the GVM for small overlap.
A formula is found yielding the complete RSB solution for all overlaps. Since
our saddle-point equations for the effective action contain both the MP
equations and the FRG, it can be used to describe the crossover from FRG to
RSB. A qualitative analysis of this crossover is given, as well as a comparison
with previous attempts to relate FRG to GVM. Finally, we discuss applications
to other problems and new perspectives.
