Hybrid Gauge Theory

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: $2\sin(n\pi/N),\ n\in {\mathbb Z}_N$.