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

Abstract

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 T2/ℤN (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 T2/ℤ2 and T2/ℤ3, but the representation matrices on T2/ℤ4 and T2/ℤ6 can contain not only diagonal matrices but also non-diagonal 2 × 2 ones and non-diagonal 3 × 3 and 2 × 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.