Vacuum structure of bifundamental gauge theories at finite topological angles

TL;DR

The paper uses mixed 't Hooft anomaly constraints obtained by gauging the center $\mathbb{Z}_n$ one-form symmetry to bound the vacuum structure of $SU(n)\times SU(n)$ gauge theories with bifundamental matter at finite topological angles $\theta_1,\theta_2$. By analyzing how CP and one-form symmetries behave under gauging, it derives that CP must be spontaneously broken at $(\pi,0)$ and $(0,\pi)$ (for generic $n\ge3$ under reasonable assumptions), and it carefully analyzes the CP properties at $(\pi,\pi)$, showing that global consistency requires either CP breaking or a phase boundary in the $\theta_1$-$\theta_2$ plane. The authors construct phase diagrams consistent with these constraints and interpret them through a dual superconductor picture of confinement, also commenting on the special case $n=2$. This work connects topological aspects of gauge theories with nonperturbative vacuum structure and provides a framework for understanding how bifundamental matter and discrete theta angles shape confinement dynamics.

Abstract

We discuss possible vacuum structures of $SU(n)\times SU(n)$ gauge theories with bifundamental matters at finite $θ$ angles. In order to give a precise constraint, a mixed 't Hooft anomaly is studied in detail by gauging the center $\mathbb{Z}_n$ one-form symmetry of the bifundamental gauge theory. We propose phase diagrams that are consistent with the constraints, and also give a heuristic explanation of the result based on the dual superconductor scenario of confinement.

Figures (6)

• Figure 1: Two different paths connecting $(\theta_1,\theta_2)=(0,0)$ and $(\theta_1,\theta_2)=(\pi,\pi)\sim (-\pi,\pi)$.
• Figure 2: Possible phase boundaries of $SU(n)\times SU(n)$ bifundamental gauge theories in the $\theta_1$-$\theta_2$ plane ($n\ge 3$).
• Figure 3: Schematic figure on the ground state energy $E(\theta)$ of the $SU(n)$ Yang-Mills theory based on the dual superconductor model for $n=3$. There are $n$ different branches labeled by the condensed charge $(\theta/2\pi -k,1)$ of dyons ($k=0,1,\ldots,n-1$), and each branch is $2\pi n$ periodic.
• Figure 4: Energies of the quasi-vacua of the $SU(n)\times SU(n)$ gauge theory in the limit of $m\to \infty$ when $\theta=\theta_1=\theta_2$.
• Figure 5: Two possibilities of the mixing of states due to dynamical bifundamental matter fields when $\theta=\theta_1=\theta_2$ are around $\pi$.