Polyakov Model in 't Hooft flux background: A quantum mechanical reduction with memory

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.