  We explain the physical role of non-perturbative saddle points of path
integrals in theories without instantons, using the example of the
asymptotically free two-dimensional principal chiral model (PCM). Standard
topological arguments based on homotopy considerations suggest no role for
non-perturbative saddles in such theories. However, resurgence theory, which
unifies perturbative and non-perturbative physics, predicts the existence of
several types of non-perturbative saddles associated with features of the
large-order structure of perturbation theory. These points are illustrated in
the PCM, where we find new non-perturbative `fracton' saddle point field
configurations, and give a quantum interpretation of previously discovered
`uniton' unstable classical solutions. The fractons lead to a semi-classical
realization of IR renormalons in the circle-compactified theory, and yield the
microscopic mechanism of the mass gap of the PCM.
