Cluster Expansion and Resurgence in the Polyakov Model

In the Polyakov model, a nonperturbative mass gap is formed at leading-order semiclassics by instanton effects. By using the notions of critical points at infinity, cluster expansion, and Lefschetz thimbles, we show that a third-order effect in semiclassics gives an imaginary ambiguous contribution to the mass gap, which is supposed to be real and unambiguous. This is troublesome for the original analysis, and it is difficult to resolve this issue directly in quantum field theory (QFT). However, we find a new compactification of the Polyakov model to quantum mechanics, by using a background 't Hooft flux. The compactification has the merit of remembering the monopole instantons of the full QFT within Born-Oppenheimer approximation, while the periodic compactification does not. In the quantum mechanical limit, we prove the resurgent cancellation of the ambiguity in three-instanton sector against ambiguity in the Borel resummation of the perturbation theory around one instanton. Assuming that this result holds in QFT, we provide a large-order asymptotics of perturbation theory around perturbative vacuum and instanton.