We consider a model of a D-dimensional tethered manifold interacting by excluded volume in ${\mathit{openR}}^{\mathit{d}}$ with a single point. By use of intrinsic distance geometry, we first provide a rigorous definition of the analytic continuation of its perturbative expansion for arbitrary D, 0<D<2. We then construct explicitly a renormalization operation ssR, ensuring renormalizability to all orders. This is the first example of mathematical construction and renormalization for an interacting extended object with continuous internal dimension, encompassing field theory.

• Received 18 December 1992

DOI:https://doi.org/10.1103/PhysRevLett.70.2205