Classification of $T^2/Z_m$ orbifold boundary conditions in $SO(N)$ gauge theories

TL;DR

This paper classifies the boundary conditions of $SO(N)$ gauge theories on $T^2/Z_m$ orbifolds, identifying finite equivalence classes and reconstructing canonical forms using a re-orthogonalization method.

Contribution

It provides a systematic classification of orbifold boundary conditions for $SO(N)$ gauge theories, including canonical form reconstruction and equivalence relations analysis.

Findings

Number of equivalence classes for each orbifold model determined

Canonical forms of boundary conditions reconstructed

Re-orthogonalization method applied to classify BCs

Abstract

We generally classify the equivalence classes of the $T^2/Z_m$ $(m=2,3,4,6)$ orbifold boundary conditions (BCs) for the $SO(N)$ gauge group. Higher-dimensional gauge theories are defined by gauge groups, matter field contents, and the BCs. The numerous patterns of the BCs are classified into the finite equivalence classes, each of which consists of the physically equivalent BCs. In this paper, we reconstruct the canonical forms of the BCs for the $SO(N)$ gauge group through the ``re-orthogonalization method." All the possible equivalent relations between the canonical forms are examined by using the trace conservation laws. The number of the equivalence classes in each orbifold model is obtained.