Anomaly cancellation in M–theory on orbifolds

We present calculation of the anomaly cancellation in M–theory on orbifolds S/ZZ2 and T /ZZ2 in the upstairs approach. The main requirement that allows one to uniquely define solutions to the modified Bianchi identities in this case is that the field strength G be globally defined on S1 or T 5 and properly transforming under ZZ2. We solve for general G that satisfies these requirements and explicitly construct anomaly–free theories in the upstairs approach. We also obtain the solutions in the presence of five–branes. All these constructions show equivalence of the downstairs and upstairs approaches. For example in the S/ZZ2 case the ten–dimensional gauge coupling and the anomaly cancellation at each wall are the same as in the downstairs approach.