Interacting crumpled manifolds: Exact results to all orders of perturbation theory

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 g₀ 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)², where again all orders in g₀ 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.