Localization and resummation of unstable instantons in 2d Yang-Mills

We compute the exact all-orders perturbative expansion for the partition function of 2d $\mathrm{SU}(2)$ Yang-Mills theory on closed surfaces around higher critical points. We demonstrate that the expansion can be derived from the lattice partition function for all genera using a distributional generalization of the Poisson summation formula. We then recompute the expansion directly, using a stationary phase version of supersymmetric localization. The result of localization is a novel effective action which is itself a distribution rather than a function of the supersymmetric moduli. We comment on possible applications to A-twisted models and their analogs in higher dimensions.