On diagonal representatives in boundary condition matrices on orbifolds

TL;DR

This paper investigates the existence of diagonal representatives in boundary condition matrices on various orbifolds, providing alternative proofs and identifying cases where diagonal representatives do or do not exist.

Contribution

It offers an alternative proof for the existence of diagonal representatives on $S^1/Z_2$ and analyzes their existence on other orbifolds, revealing new structural insights.

Findings

Diagonal representatives exist on $S^1/Z_2$

They do not necessarily exist on $T^2/Z_2$, $T^2/Z_3$, and $T^2/Z_4$

All classes on $T^2/Z_6$ have diagonal representatives

Abstract

We study diagonal representatives of boundary condition matrices on the orbifolds $S^1/Z_2$ and $T^2/Z_m$ ($m=2, 3, 4, 6$). We give an alternative proof of the existence of diagonal representatives in each equivalent class of boundary condition matrices on $S^1/Z_2$, using a matrix exponential representation, and show that they do not necessarily exist on $T^2/Z_2$, $T^2/Z_3$, and $T^2/Z_4$. Each equivalence class on $T^2/Z_6$ has a diagonal representative, because its boundary conditions are determined by a single unitary matrix.