  Resurgence theory implies that the non-perturbative (NP) and perturbative (P)
data in a QFT are quantitatively related, and that detailed information about
non-perturbative saddle point field configurations of path integrals can be
extracted from perturbation theory. Traditionally, only stable NP saddle points
are considered in QFT, and homotopy group considerations are used to classify
them. However, in many QFTs the relevant homotopy groups are trivial, and even
when they are non-trivial they leave many NP saddle points undetected.
Resurgence provides a refined classification of NP-saddles, going beyond
conventional topological considerations. To demonstrate some of these ideas, we
study the $SU(N)$ principal chiral model (PCM), a two dimensional
asymptotically free matrix field theory which has no instantons, because the
relevant homotopy group is trivial. Adiabatic continuity is used to reach a
weakly coupled regime where NP effects are calculable. We then use resurgence
theory to uncover the existence and role of novel `fracton' saddle points,
which turn out to be the fractionalized constituents of previously observed
unstable `uniton' saddle points. The fractons play a crucial role in the
physics of the PCM, and are responsible for the dynamically generated mass gap
of the theory. Moreover, we show that the fracton-anti-fracton events are the
weak coupling realization of 't Hooft's renormalons, and argue that the
renormalon ambiguities are systematically cancelled in the semi-classical
expansion. Our results motivate the conjecture that the semi-classical
expansion of the path integral can be geometrized as a sum over Lefschetz
thimbles.
