Hybrid Gauge Theory

TL;DR

This paper introduces a hybrid gauge theory where cyclic symmetry $C_N$ is gauged using a Lie group, resulting in a novel framework linking discrete symmetries with continuous gauge groups and specific coupling constant ratios.

Contribution

It proposes a new hybrid gauge theory framework combining discrete cyclic symmetry with Lie group gauge fields, specifying allowed groups and coupling constant relations.

Findings

Allowed Lie groups are SO(2), SU(3), and SU(2) depending on N.

Matter fields are in irreducible representations of $C_N$.

Coupling constant ratios relate to geometric properties of regular polygons.

Abstract

Cyclic symmetry $C_N$ is gauged in such a way that the local parametrization is provided by a Lie group: matter fields are in irreducible representations of $C_N$ while gauge fields are in the adjoint representation of a Lie group, hence "hybrid". Allowed simple Lie groups are only SO(2) for $N=2$, SU(3) for $N=3$, and SU(2) for all $N$. The implication of the local discrete symmetry $C_N$ is evident as the ratio of the coupling constant to the usual gauge theory one of the parametrization Lie group is given by that of the length between any two vertices of a regular N-polygon to the radius of the circumcircle: $2\sin(n\pi/N),\ n\in {\mathbb Z}_N$.