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

TL;DR

This work determines the phase structure of the twisted $SU(3)/U(1)^2$ flag sigma model on $\mathbb{R}\times S^1$ as a function of the two theta angles $(\theta_1,\theta_2)$. By imposing a ${\mathbb Z}_3$-twisted boundary condition, the authors perform a semiclassical analysis via the dilute instanton gas approximation (DIGA) at small $S^1$ circumference, classify six classical vacua and eighteen fractional instantons, and derive a DIGA-based phase diagram that exhibits phase transitions and triple degeneracies in agreement with 't Hooft anomaly matching and global inconsistency constraints. They demonstrate adiabatic continuity between the compactified and uncompactified theories and examine the resurgent structure by computing instanton–anti-instanton (bion) contributions, revealing an imaginary ambiguity expected to cancel with perturbative Borel resummation. The results illuminate nonperturbative dynamics in flag sigma models and extend the semiclassical toolkit to multi-parameter theta spaces, with potential extensions to general $SU(N)/U(1)^{N-1}$ flags and connections to PCM fracton structures. Overall, the paper provides a concrete semiclassical realization of anomaly/a priori constraints and reinforces the viability of adiabatic continuity in asymptotically free two-dimensional field theories.

Abstract

We investigate the phase structure of the compactified $2$-dimensional nonlinear $SU(3)/U(1)^2$ flag sigma model with respect to two $θ$-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 $(θ_1,θ_2)$-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.

Figures (7)

• Figure 1: The Polyakov-loop phase $P_\ell (x)$ corresponding to one of the classical vacua $\Phi_1$ in eq. \ref{['eq:6Vacua']}. Red, green, and blue curves correspond to $P_1 (x)$, $P_2(x)$ and $P_3(x)$, respectively.
• Figure 2: Spatial dependences of the Polyakov-loop phase for the fractional instanton \ref{['eq:FracInst']}.
• Figure 3: Polyakov-loop phases for two pairs of the fractional instantons starting from the classical vacuum $\Phi_1$ given in eq. \ref{['eq:6Vacua']}. Red, green, and blue curves correspond to $P_1 (x)$, $P_2(x)$ and $P_3(x)$, respectively.
• Figure 4: Polyakov-loop phases for four pairs of the fractional instantons starting from and ending at the classical vacuum $\Phi_1$ in eq. \ref{['eq:6Vacua']} with the fractional instantons \ref{['eq:FracInst']}. At this order, there are $27$ possible configurations, but we here only show $9$ of them by restricting the first jump of fractional instantons as $\Phi_1\to \Phi_5$.
• Figure 5: The ground-state energy of the $2$-dimensional twisted $SU(3)/U(1)^2$ sigma model $(\lambda=0)$ in the $(\theta_1,\theta_2)$-plane. The quantum phase-transition lines are located in the places consistent with the anomaly/global inconsistency matching. Furthermore, we also see the triple degeneracies of the ground-state energy appear at special points e.g. $(2\pi /3,-2\pi /3)$.