Renormalization theory for interacting crumpled manifolds

Abstract

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 δ-potential (but without self-avoidance interactions). Except for D = 1 (the polymer case), this model cannot be mapped onto a loca; 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 bulk space dimension d smaller that the upper critical dimension d ★ = 2D (2−D) , the perturbative expansion is ultraviolet finite, while ultraviolet divergences occur as pole at d = d ★. The standard proof of perturbative renormalizability for local field theories (the Bogoliubov-Parasiuk-Hepp theorem) does not apply to this model. We prove perturbative renormalizability to all orders by constructing a subtraction operator R 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 ϵ-expansion à la Wilson-Fisher 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 study 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.