Classification of 1D and 2D Orbifolds

Abstract

We present a complete classification of all 1D and 2D orbifold compactifications. There exist 2 one-dimensional and 17 two-dimensional orbifolds. The classification includes orbifolds such as S^1/Z_2 or T^2/Z_n, as well as less familiar ones like T^2/D_n or the Mobius strip. We derive the explicit form of the basis functions and prove their orthonormality and completeness. Our study is based on the classification of space groups, which is well-known from crystallography. We define these groups in a novel, purely algebraic way. That enables us to determine all possible parities that can be defined on the orbifolds. We discuss field theories on T^2/Z_n with brane kinetic terms, and describe the derivation of their mass eigenstate bases.