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Resurgence theory implies that the non-perturbative (NP) and perturbative (P)data in a QFT are quantitatively related, and that detailed information aboutnon-perturbative saddle point field configurations of path integrals can beextracted from perturbation theory. Traditionally, only stable NP saddle pointsare considered in QFT, and homotopy group considerations are used to classifythem. However, in many QFTs the relevant homotopy groups are trivial, and evenwhen they are non-trivial they leave many NP saddle points undetected.Resurgence provides a refined classification of NP-saddles, going beyondconventional topological considerations. To demonstrate some of these ideas, westudy the $SU(N)$ principal chiral model (PCM), a two dimensionalasymptotically free matrix field theory which has no instantons, because therelevant homotopy group is trivial. Adiabatic continuity is used to reach aweakly coupled regime where NP effects are calculable. We then use resurgencetheory to uncover the existence and role of novel `fracton' saddle points,which turn out to be the fractionalized constituents of previously observedunstable `uniton' saddle points. The fractons play a crucial role in thephysics of the PCM, and are responsible for the dynamically generated mass gapof the theory. Moreover, we show that the fracton-anti-fracton events are theweak coupling realization of 't Hooft's renormalons, and argue that therenormalon ambiguities are systematically cancelled in the semi-classicalexpansion. Our results motivate the conjecture that the semi-classicalexpansion of the path integral can be geometrized as a sum over Lefschetzthimbles.

Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles