  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
non-linear sigma model (path) integrals. We explain how this solves one of the
most fundamental limitation of the classic approach: we automatically obtain
non-perturbative definitions in non-Borel summable cases. This is exemplified
on the simplest possible examples involving O(N) symmetric non-linear sigma
models with N-dimensional target spaces, for which we construct (multi)critical
metrics. The non-perturbative definitions of the double scaled, manifestly
positive, partition functions rely on remarkable identities involving (path)
integrals.
