On anomalies in orbifold theories

We study the issue of gauge invariance in five-dimensional theories compactified on an orbifold S1/(Z2×Z′2) in the presence of an external U(1) gauge field. From the four-dimensional point of view the theory contains a tower of Kaluza–Klein Dirac fermions with chiral couplings and it looks anomalous at the quantum level. We show that this “anomaly” is cancelled by a topological Chern–Simons term which is generated in the effective action when the gauge theory is regularized introducing a Pauli–Villars fermion with an odd mass term. In the presence of a classical background gauge field, the fermionic current acquires a vacuum expectation value, thus generating the suitable Chern–Simons term and a gauge invariant theory.