  This work is a step towards a non-perturbative continuum definition of
quantum field theory (QFT), beginning with asymptotically free two dimensional
non-linear sigma-models, using recent ideas from mathematics and QFT. The ideas
from mathematics are resurgence theory, the trans-series framework, and
Borel-Ecalle resummation. The ideas from QFT use continuity on R^1 x S^1_L,
i.e, the absence of any phase transition as N \to infinity, or rapid-crossovers
for finite-N, and the small-L weak coupling limit to render the semi-classical
sector well-defined and calculable. We classify semi-classical configurations
with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where
the 2d instanton action is normalized to one. Perturbation theory possesses the
IR-renormalon ambiguity that arises due to non-Borel summability of the
large-orders perturbation series (of Gevrey-1 type), for which a microscopic
cancellation mechanism was unknown. This divergence must be present because the
corresponding expansion is on a singular Stokes ray in the complexified
coupling constant plane, and the sum exhibits the Stokes phenomenon crossing
the ray. We show that there is also a non-perturbative ambiguity inherent to
certain neutral topological molecules (neutral bions and bion-anti-bions) in
the semiclassical expansion. We find a set of "confluence equations" that
encode the exact cancellation of the two different type of ambiguities. We show
that a new notion of "graded resurgence triangle" is necessary to capture the
path integral approach to resurgence, and that graded resurgence underlies a
potentially rigorous definition of general QFTs. The mass gap and the Theta
angle dependence of vacuum energy are calculated from first principles, and are
in accord with large-N and lattice results.
