RENORMALIZATION THEORY FOR INTERACTING CRUMPLED MANIFOLDS

We consider a continuous model of D-dimensional elastic (polymerized) manifold fluctuating in d-dimensional euclidean space, interacting wit h a single impurity via an attractive or repulsive delta-potential (bu t 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 cons truction of the high-temperature perturbative expansion and for analyt ic continuation in the manifold dimension D. We study the renormalizat ion properties of the model for 0 < D < 2, and show that for bulk spac e dimension d smaller that the upper critical dimension d(star) = 2D/( 2 - D), the perturbative expansion is ultraviolet finite, while ultrav iolet divergences occur as poles at d = d(star). The standard proof of perturbative renormalizability for local field theories (the Bogoliub ov-Parasiuk-Hepp theorem) does not apply to this model. We prove pertu rbative renormalizability to all orders by constructing a subtraction operator R based on a generalization of the Zimmermann forests formali sm, and which makes the theory finite at d = d(star). This subtraction operation corresponds to a renormalization of the coupling constant o f the model (strength of the interaction with the impurity). The exist ence of a Wilson function, of an epsilon-expansion A la Wilson-Fisher around the critical dimension, of scaling laws for d < d(star) in the repulsive case, and of non-trivial critical exponents of the delocaliz ation transition for d > d(star) in the attractive case, is thus estab lished. To our knowledge, this study provides the first proof of renor malizability for a model of extended objects, and should be applicable to the study of self-avoidance interactions for random manifolds.