Field theory of interacting crumpled manifolds
The effect of a (delta)-interaction on a polymerized membrane of arbitrary internal dimension D has been studied. Depending on the dimensionality of membrane and embedding space, different physical scenarios are observed: The delocalization of a membrane from an attractive defect as well as steric repulsions. The difference of polymers from membranes is emphasized. For the latter, non-trivial contributions appear at the 2-loop level. Furthermore, a "massive scheme" inspired by calculations in fixed dimensions for scalar field theories has been exploited. Despite the fact that these calculations are only amenable numerically, it has been found that in the limit of D->2 each diagram can be evaluated analytically. This property extends in fact to any order in perturbation theory, allowing for a summation of all orders. An analytically continued expression for the effective coupling of membranes in the scaling limit of large sizes as compared to the microscopic cutoff is obtained. Finally, the construction of an expansion of the effective coupling about D=2 is presented. Applications to the case of self-avoiding membranes are mentioned.