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Dihedral symmetry in $SU(N)$ Yang-Mills theory

Kyle Aitken, Aleksey Cherman, Mithat Ünsal

TL;DR

This work reveals that pure $SU(N)$ Yang-Mills theory possesses non-Abelian dihedral discrete symmetries arising from the noncommutation of center symmetry with charge conjugation and coordinate reflections. On $\mathbb{R}^3\times S^1$, the discrete symmetry group is $D_{2N}$ for generic $\theta$, enhanced to a central extension $D_{4N}$ at $\theta=\pi$ due to 't Hooft anomaly considerations, with parity/time-reversal factors also involved. A simple quantum-mechanical $T_N$ model is used to illustrate the $\theta$-dependent representation theory and the origin of degeneracies. The analysis is mirrored in a semiclassical YM setup on small circles, where monopole-instanton effects generate a $\theta$-dependent potential for dual photons and realize the same dihedral symmetry structure, linking anomaly constraints to vacuum structure and representation theory. Together, these results provide a concrete, calculable framework connecting discrete symmetry, anomalies, and the vacuum landscape in gauge theory.

Abstract

We point out that charge conjugation and coordinate reflection symmetries do not commute with the center symmetry of $SU(N)$ YM theory when $N>2$. As a result, for generic values of the $θ$ angle, the group of discrete zero-form symmetries of YM theory on e.g. the spacetime manifold $\mathbb{R}^3\times S^1$ includes the dihedral group $D_{2N}$ which is non-Abelian for $N>2$. At $θ= π$, the non-Abelian factor in the symmetry group is enhanced to $D_{4N}$ due to discrete 't Hooft anomaly considerations. We illustrate these results in YM theory as well as in a simple quantum mechanical model, where we study representation theory as a function of $θ$ angle.

Dihedral symmetry in $SU(N)$ Yang-Mills theory

TL;DR

This work reveals that pure Yang-Mills theory possesses non-Abelian dihedral discrete symmetries arising from the noncommutation of center symmetry with charge conjugation and coordinate reflections. On , the discrete symmetry group is for generic , enhanced to a central extension at due to 't Hooft anomaly considerations, with parity/time-reversal factors also involved. A simple quantum-mechanical model is used to illustrate the -dependent representation theory and the origin of degeneracies. The analysis is mirrored in a semiclassical YM setup on small circles, where monopole-instanton effects generate a -dependent potential for dual photons and realize the same dihedral symmetry structure, linking anomaly constraints to vacuum structure and representation theory. Together, these results provide a concrete, calculable framework connecting discrete symmetry, anomalies, and the vacuum landscape in gauge theory.

Abstract

We point out that charge conjugation and coordinate reflection symmetries do not commute with the center symmetry of YM theory when . As a result, for generic values of the angle, the group of discrete zero-form symmetries of YM theory on e.g. the spacetime manifold includes the dihedral group which is non-Abelian for . At , the non-Abelian factor in the symmetry group is enhanced to due to discrete 't Hooft anomaly considerations. We illustrate these results in YM theory as well as in a simple quantum mechanical model, where we study representation theory as a function of angle.

Paper Structure

This paper contains 13 sections, 78 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Non-Abelian global symmetry of YM theory
  3. Physical consequences
  4. Dihedral symmetries in a quantum mechanical model
  5. Symmetry group as a function of $\theta$
  6. Representations of the dihedral group for $\theta=0$ and $\theta=\pi$
  7. Dihedral symmetries in Yang-Mills theory on $\mathbb{R}^3\times S^1$
  8. Weak-coupling setup
  9. Extrema and symmetries as a function of $\theta$
  10. Summary
  11. Path integral formulation of the $T_N$ model
  12. Representations of the dihedral group
  13. Discrete symmetries of YM on $\mathbb{R}^3 \times S^1$

Figures (3)

  • Figure 1: [Color Online.] A summary of the symmetries of $SU(N)$ YM theory (right) and of a related $T_N$ toy model from quantum mechanics (left), as a function of $\theta$.
  • Figure 2: An illustration of the energy levels of the $T_N$ model for $N=5$ and $N=6$. At $\theta=0$, the ground state is unique, and fits into the one-dimensional ${\bf A_{1}}$ representation of $D_{2N}$, while the excited states fit into either the ${\bf E_{k}}$ representations (which are all two-dimensional) or into the ${\bf B_{1}}$ representation, which is one-dimensional. At $\theta = \pi$, on the other hand, the ground state is always in the two-dimensional $\bf {\widetilde{E} }_{1}$ representation of $D_{4N}$.
  • Figure 3: A sketch of how the states of the $T_N$ model with $N=5$ and $\theta = 0$ and $\theta =\pi$ fit into the dihedral group $D_{10}$ and $D_{20}$ representations. The Bloch states $|k \rangle$ are defined in \ref{['eq:bloch_states']}.