1 Grand Unified Models on S1/(Z2 × Z ′ 2) Orbifold ∗)

We suggest two grand unified models where the fifth dimensional coordinate is compactified on an S1/(Z2 × Z ′ 2) orbifold. Before showing the models, let us see the compactification of the fifth dimension. We denote the five dimensional coordinate as y, which is compactified on an S1/(Z2 × Z ′ 2) orbifold. Under the parity transformation of Z2 and Z ′ 2, which transform y → −y and y ′ → −y (y = y + πR/2), respectively, a field φ(x, y) transforms as φ(x, y) → φ(x,−y) = Pφ(x, y) and φ(x, y) → φ(x,−y) = P φ(x, y), where P and P ′ are operators of Z2 and Z ′ 2 transformations, respectively. Two walls at y = 0 (πR) and πR/2 (−πR/2) are fixed points under Z2 and Z ′ 2 transformations, respectively. The physical space can be taken to be 0 ≤ y ≤ πR/2, since the walls at y = πR and −πR/2 are identified with those at y = 0 and πR/2, respectively. On this orbifold, the field φ(x, y) is divided into four eigenfunctions according to the eigenvalues (±,±) of the parity (Z2, Z ′ 2). We take the Z2 parity operator as P = diag.(1, 1, 1, 1, 1) and the Z ′ 2 parity operator as P ′ = diag.(−1,−1,−1, 1, 1) acting on a 5 representation in SU(5) 1) 2) 3) 4).