  In this article we study the effect of a delta-interaction on a polymerized
membrane of arbitrary internal dimension D. Depending on the dimensionality of
membrane and embedding space, different physical scenarios are observed. We
emphasize on the difference of polymers from membranes. For the latter,
non-trivial contributions appear at the 2-loop level. We also exploit a
``massive scheme'' inspired by calculations in fixed dimensions for scalar
field theories. Despite the fact that these calculations are only amenable
numerically, we found that in the limit of D to 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. This is a novel and quite
surprising result. Finally, an attempt to go beyond D=2 is presented.
Applications to the case of self-avoiding membranes are mentioned.
