Trace conservation laws in $T^2/Z_m$ orbifold gauge theories

TL;DR

This paper introduces trace conservation laws as necessary conditions to classify boundary conditions in orbifold gauge theories, enabling a complete classification of equivalence classes on $T^2/Z_m$ orbifolds.

Contribution

It proposes trace conservation laws that classify boundary conditions in orbifold gauge theories without explicit gauge transformations, revealing off-diagonal equivalence classes.

Findings

Trace conservation laws classify boundary conditions in orbifold gauge theories.

Existence of off-diagonal equivalence classes on $T^2/Z_4$ and $T^2/Z_6$.

Exact enumeration of equivalence classes achieved.

Abstract

Gauge theory compactified on an orbifold is defined by gauge symmetry, matter contents, and boundary conditions. There are equivalence classes (ECs), each of which consists of physically equivalent boundary conditions. We propose the powerful necessary conditions, trace conservation laws (TCLs), which achieve a sufficient classification of ECs in U(N) and SU(N) gauge theories on $T^2/Z_m$ orbifolds $(m=2,3,4,6)$. The TCLs yield the equivalent relations between the diagonal boundary conditions without relying on an explicit form of gauge transformations. The TCLs also show the existence of off-diagonal ECs, which consist only of off-diagonal matrices, on $T^2/Z_4$ and $T^2/Z_6$. After the sufficient classification, the exact numbers of ECs are obtained.