On representation matrices of boundary conditions in $SU(n)$ gauge theories compactified on two-dimensional orbifolds

TL;DR

This paper investigates the structure of boundary condition representation matrices in $SU(n)$ gauge theories on orbifolds, showing that diagonal representatives always exist for some orbifolds, while others allow non-diagonal matrices with implications for symmetry breaking.

Contribution

It provides a detailed analysis of the conditions under which boundary condition matrices can be diagonalized in orbifold compactifications, revealing the role of discrete Wilson line phases in symmetry breaking.

Findings

Diagonal representatives exist on $T^2/\mathbb{Z}_2$ and $T^2/\mathbb{Z}_3$.

Non-diagonal matrices appear on $T^2/\mathbb{Z}_4$ and $T^2/\mathbb{Z}_6$.

Discrete parameters in non-diagonal matrices can induce rank-reducing symmetry breaking.

Abstract

We study the existence of diagonal representatives in each equivalence class of representation matrices of boundary conditions in $SU(n)$ or $U(n)$ gauge theories compactified on the orbifolds $T^2/{\mathbb Z}_N$ ($N = 2, 3, 4, 6$). We suppose that the theory has a global $G' = U(n)$ symmetry. Using constraints, unitary transformations and gauge transformations, we examine whether the representation matrices can simultaneously become diagonal or not. We show that at least one diagonal representative necessarily exists in each equivalence class on $T^2/{\mathbb Z}_2$ and $T^2/{\mathbb Z}_3$, but the representation matrices on $T^2/{\mathbb Z}_4$ and $T^2/{\mathbb Z}_6$ can contain not only diagonal matrices but also non-diagonal $2 \times 2$ ones and non-diagonal $3 \times 3$ and $2 \times 2$ ones, respectively, as members of block-diagonal submatrices. These non-diagonal matrices have discrete parameters, which means that the rank-reducing symmetry breaking can be caused by the discrete Wilson line phases.