We consider a class of quantum field theories and quantum mechanics, which we couple to topological QFTs, in order to classify non-perturbative effects in the original theory. The TQFT structure arises naturally from turning on a classical background field for a discrete global symmetry. In SU(N) Yang-Mills theory coupled to TQFT, the non-perturbative expansion parameter is exp[-S_I/N] both in the semi-classical weak coupling domain and strong coupling domain, corresponding to a fractional topological charge and action configurations. To classify the non-perturbative effects in original SU(N) theory, we must use PSU(N) bundle and lift configurations (critical points at infinity) for which there is no obstruction back to SU(N). These provide a refinement of instanton sums: integer topological charge, but crucially fractional action configurations contribute, providing a TQFT protected generalization of resurgent semiclassical expansion to strong coupling.
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