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

TL;DR

This paper reduces the Polyakov model with 't Hooft flux to quantum mechanics, revealing non-perturbative vacuum structures and instanton effects at small torus sizes, with implications for resurgence and spectrum analysis.

Contribution

It introduces a quantum mechanical reduction of the Polyakov model incorporating non-perturbative effects and flux backgrounds, connecting quantum mechanics and quantum field theory insights.

Findings

Degenerate vacua emerge with flux backgrounds at small torus.

Construction of quantum mechanical instantons from QFT instantons.

Comparison of spectra and vacuum structures with deformed Yang-Mills theory.

Abstract

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.