  We consider a continuous model of D-dimensional elastic (polymerized)
manifold fluctuating in d-dimensional Euclidean space, interacting with a
single impurity via an attractive or repulsive delta-potential (but without
self-avoidance interactions). Except for D=1 (the polymer case), this model
cannot be mapped onto a local field theory. We show that the use of intrinsic
distance geometry allows for a rigorous construction of the high-temperature
perturbative expansion and for analytic continuation in the manifold dimension
D. We study the renormalization properties of the model for 0<D<2, and show
that for d<d* where d*=2D/(2-D) is the upper critical dimension, the
perturbative expansion is UV finite, while UV divergences occur as poles at
d=d*. The standard proof of perturbative renormalizability for local field
theories (the BPH theorem) does not apply to this model. We prove perturbative
renormalizability to all orders by constructing a subtraction operator based on
a generalization of the Zimmermann forests formalism, and which makes the
theory finite at d=d*. This subtraction operation corresponds to a
renormalization of the coupling constant of the model (strength of the
interaction with the impurity). The existence of a Wilson function, of an
epsilon-expansion around the critical dimension, of scaling laws for d<d* in
the repulsive case, and of non-trivial critical exponents of the delocalization
transition for d>d* in the attractive case is thus established. To our
knowledge, this provides the first proof of renormalizability for a model of
extended objects, and should be applicable to the study of self-avoidance
interactions for random manifolds.
