  A non-perturbative renormalization of a many-body problem, where
non-relativistic bosons living on a two dimensional Riemannian manifold
interact with each other via the two-body Dirac delta potential, is given by
the help of the heat kernel defined on the manifold. After this renormalization
procedure, the resolvent becomes a well-defined operator expressed in terms of
an operator (called principal operator) which includes all the information
about the spectrum. Then, the ground state energy is found in the mean field
approximation and we prove that it grows exponentially with the number of
bosons. The renormalization group equation (or Callan-Symanzik equation) for
the principal operator of the model is derived and the $\beta$ function is
exactly calculated for the general case, which includes all particle numbers.
