Renormalization of Crumpled Manifolds

TL;DR

This paper rigorously defines and constructs a renormalization procedure for a D-dimensional tethered manifold with excluded volume interactions, extending field theory techniques to continuous extended objects.

Contribution

It provides the first explicit mathematical construction and renormalization for an interacting extended object with continuous internal dimension.

Findings

Renormalization is achieved to all orders for the model.

A rigorous definition of the analytic continuation of the perturbative expansion is provided.

The work extends field theory methods to higher-dimensional manifolds.

Abstract

We consider a model of D-dimensional tethered manifold interacting by excluded volume in R^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, 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.