Interacting Crumpled Manifolds: Exact Results to all Orders of Perturbation Theory

TL;DR

This paper presents an exact summation of the perturbation series for a model of crumpled manifolds interacting with a point, providing insights into the behavior of polymers and tethered membranes through advanced field theory techniques.

Contribution

It offers the first exact summation of perturbation expansion for a tethered membrane model in the limit D -> 2, connecting to polymer physics.

Findings

Exact summation of perturbation series at D -> 2

Expansion results up to order (2-D)^4

Progress towards solving the self-avoiding manifold problem

Abstract

In this letter, we report progress on the field theory of polymerized tethered membranes. For the toy-model of a manifold repelled by a single point, we are able to sum the perturbation expansion in the strength g of the interaction exactly in the limit of internal dimension D -> 2. This exact solution is the starting point for an expansion in 2-D, which aims at connecting to the well studied case of polymers (D=1). We here give results to order (2-D)^4, where again all orders in g are resummed. This is a first step towards a more complete solution of the self-avoiding manifold problem, which might also prove valuable for polymers.