  In Polyakov model, a non-perturbative mass gap is formed at leading order
semi-classics 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 semi-classics gives an imaginary ambiguous contribution to 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 QFT.
However, we find a new compactification of Polyakov model to quantum mechanics,
by using a background 't Hooft flux (or coupling to TQFT). The compactification
has the merit of remembering the monopole-instantons of the full QFT within
Born-Oppenheimer (BO) approximation, while the periodic compactification does
not. In QM, we prove the resurgent cancellation of the ambiguity in 3-instanton
sector against ambiguity in the Borel resummation of the perturbation theory
around 1-instanton. Assuming that this result holds in QFT, we provide a
large-order asymptotics of perturbation theory around perturbative vacuum and
instanton.
