# Fractional $\theta$ angle, 't Hooft anomaly, and quantum instantons in charge-$q$ multi-flavor Schwinger model

## TL;DR

This paper investigates the non-perturbative dynamics of the charge-$q$ multi-flavor Schwinger model using anomaly matching, semi-classics, and bosonization, revealing fractional theta dependence and quantum instantons that explain vacuum structure and chiral symmetry breaking.

## Contribution

It introduces the concept of quantum instantons in the Schwinger model, connecting anomaly matching with semi-classical analysis and fractional theta dependence, expanding understanding of non-perturbative effects.

## Key findings

- Different boundary conditions realize different anomalies.
- Under twisted boundary conditions, there are $Nq$ vacua with chiral symmetry breaking.
- Quantum instantons explain vacuum structure beyond usual instantons.

## Abstract

This work examines non-perturbative dynamics of a $2$-dimensional QFT by using discrete 't Hooft anomaly, semi-classics with circle compactification and bosonization. We focus on charge-$q$ $N$-flavor Schwinger model, and also Wess-Zumino-Witten model. We first apply the recent developments of discrete 't Hooft anomaly matching to theories on $\mathbb{R}^2$ and its compactification to $\mathbb{R} \times S^1_L$. We then compare the 't Hooft anomaly with dynamics of the models by explicitly constructing eigenstates and calculating physical quantities on the cylinder spacetime with periodic and flavor-twisted boundary conditions. We find different boundary conditions realize different anomalies. Especially under the twisted boundary conditions, there are $Nq$ vacua associated with discrete chiral symmetry breaking. Chiral condensates for this case have fractional $\theta$ dependence $\mathrm{e}^{\mathrm{i} \theta/Nq}$, which provides the $Nq$-branch structure with soft fermion mass. We show that these behaviors at a small circumference cannot be explained by usual instantons but should be understood by "quantum" instantons, which saturate the BPS bound between classical action and quantum-induced effective potential. The effects of the quantum-instantons match the exact results obtained via bosonization within the region of applicability of semi-classics. We also argue that large-$N$ limit of the Schwinger model with twisted boundary conditions satisfy volume independence.