Nonperturbative double scaling limits

Recently, the author has proposed a generalization of the matrix and vector models approach to the theory of random surfaces and polymers. The idea is to replace the simple matrix or vector (path)-integrals by gauge theory or nonlinear a model (path)integrals. We explain how this solves one of the most fundamental limitations of the classic approach: we automatically obtain nouperturbative definitions in non-Borel summable cases. This is exemplified in the simplest possible examples involving O(N) symmetric nonlinear a models with N-dimensional target spaces, for which we construct (multi)critical metrics. The nonperturbative definitions of the double scaled, manifestly positive, partition functions rely on remarkable identities involving (path)-integrals.