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 $D\rightarrow 2$. 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