# Phase structure of the twisted $SU(3)/U(1)^2$ flag sigma model on $\mathbb{R}\times S^1$

## TL;DR

This paper explores the phase structure of a twisted SU(3)/U(1)^2 flag sigma model on a cylinder, revealing quantum phase transitions, vacua, and instanton effects, and demonstrating adiabatic continuity with the uncompactified theory.

## Contribution

It provides a semiclassical analysis of the model's phase diagram using dilute instanton gas approximation, connecting anomaly considerations with explicit instanton solutions.

## Key findings

- Identified classical vacua and fractional instantons connecting them.
- Discovered quantum phase transitions and triple degeneracy points.
- Estimated instanton–anti-instanton contributions and their ambiguities.

## Abstract

We investigate the phase structure of the compactified $2$-dimensional nonlinear $SU(3)/U(1)^2$ flag sigma model with respect to two $\theta$-terms. Based on the circle compactification with the ${\mathbb Z}_{3}$-twisted boundary condition, which preserves an 't Hooft anomaly of the original uncompactified theory, we perform the semiclassical analysis based on the dilute instanton gas approximation (DIGA). We clarify classical vacua of the theory and derive fractional instanton solutions connecting these vacua. The resulting phase structure based on DIGA exhibits the quantum phase transitions and triple degeneracy at special points in the $(\theta_1,\theta_2)$-plane, which is consistent with the phase diagram obtained from the anomaly matching and global inconsistency conditions. This result indicates the adiabatic continuity between the flag sigma models on ${\mathbb R}^{2}$ and ${\mathbb R}\times S^{1}$ with small compactification radius. We further estimate contributions from instanton--anti-instanton configuration (bion) and show the existence of the imaginary ambiguity, which is expected to be cancelled by that of the perturbative Borel resummation.