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

TL;DR

The paper analyzes non-perturbative dynamics of the two-dimensional charge-$q$ $N$-flavor Schwinger model using discrete ’t Hooft anomaly, circle compactification, and bosonization. It identifies the full internal symmetry group, including a 1-form $\mathbb{Z}_q^{[1]}$ factor, and constructs a 3d bulk action that cancels the 2d anomaly via anomaly inflow, revealing a rich pattern of discrete chiral symmetry breaking. On $\mathbb{R}\times S^1$, twisted boundary conditions preserve a larger anomaly structure, yielding $Nq$ vacua and a chiral condensate with fractional $\theta$-dependence $e^{i\theta/(Nq)}$, explained through fractional (quantum) instantons saturating a BPS bound. The results are shown to be consistent with exact bosonization and imply volume independence in the large-$N$ limit for twisted setups, while thermal compactification generally does not share this property. The analysis extends to twisted Wess–Zumino–Witten models, linking 2d conformal data to ground-state degeneracies on $\mathbb{R}\times S^1$ and highlighting deep connections between anomaly, holonomy potentials, and non-perturbative saddles in low dimensions.

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 $θ$ dependence $\mathrm{e}^{\mathrm{i} θ/Nq}$, 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.

Figures (9)

• Figure 1: One-particle energy levels as a function of $a$ for $q=1$, $N=1$ with periodic boundary condition. The black solid line stands for $E^{(k)}_{\rm R}$ while the black broken line is $E^{(k)}_{\rm L}$.
• Figure 2: One-particle energy levels as a function of $a$ for $q=1, N=3$ with periodic boundary condition. Black, red and blue solid lines stands for $E^{(k)}_{f, {\rm R}}$ with $f=0,1,2$ while black, red and blue broken lines are $E^{(k)}_{f, {\rm L}}$ with $f=0,1,2$.
• Figure 3: One-particle energy levels as a function of $a$ for $q=2, N=1$ with periodic boundary condition. A black solid line stands for $E^{(k)}_{\rm R}$ while a black broken line is $E^{(k)}_{\rm L}$.
• Figure 4: One-particle energy levels as a function of $a$ for $q=2, N=3$ with periodic boundary condition. Black, red and blue solid lines stands for $E^{(k)}_{f, {\rm R}}$ with $f=0,1,2$ while black, red and blue broken lines are $E^{(k)}_{f, {\rm L}}$ with $f=0,1,2$.
• Figure 5: One-particle energy levels as a function of $a$ for $q=1, N=3$ with ${\mathbb Z}_{N}$ twisted boundary condition. Black, red and blue solid lines stands for $E^{(k)}_{f, {\rm R}}$ with $f=0,1,2$ while black, red and blue broken lines are $E^{(k)}_{f, {\rm L}}$ with $f=0,1,2$.