  We construct a compactification of Polyakov model on $T^2 \times \mathbb R $
down to quantum mechanics which remembers non-perturbative aspects of field
theory even at an arbitrarily small area. Standard compactification on small
$T^2 \times \mathbb R $ possesses a unique perturbative vacuum (zero magnetic
flux state), separated parametrically from higher flux states, and the
instanton effects do not survive in the Born-Oppenheimer approximation. By
turning on a background magnetic GNO flux in co-weight lattice corresponding to
a non-zero 't Hooft flux, we show that $N$-degenerate vacua appear at small
torus, and there are $N-1$ types of flux changing instantons between them. We
construct QM instantons starting with QFT instantons using the method of
replicas. For example, $SU(2)$ gauge theory with flux reduces to the
double-well potential where each well is a fractional flux state. Despite the
absence of a mixed anomaly, the vacuum structure of QFT and the one of QM are
continuously connected. We also compare the quantum mechanical reduction of the
Polyakov model with the deformed Yang-Mills, by coupling both theories to
TQFTs. In particular, we compare the mass spectrum for dual photons and energy
spectrum in the QM limit. We give a detailed description of critical points at
infinity in the semi-classical expansion, and their role in resurgence
structure.
