Unstable instantons and localization in two-dimensional gauge theories

The success of localization in the non-perturbative evaluation of path integrals of gauge theories on compact manifolds has established the localization principle as an essential tool in quantum field theory. Despite this success, the conventional supersymmetric localization prescription fails to capture the necessary contribution of unstable instantons to the path integral of two-dimensional pure Yang-Mills (YM2) theories in the A-model, that is, topologically A-twisted N = (2, 2) supersymmetric gauge theory on a compact Riemannian manifold. At least two alternative approaches to localization remedy this failure, one of which invokes the Jeffrey-Kirwan residue theorem, the other of which draws from the original proof of non-abelian localization. This research aims to derive, from the failures and successes of different approaches to localization of YM2 theories in the A-model, more general insights about non-perturbative phenomena in gauge theories and the scope of supersymmetric localization.