Anomaly cancellation in M-theory on orbifolds

We present calculation of the anomaly cancellation in M-theory on orbifolds S-1/Z(2) and T-5/Z(2) 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 S-1 or T-5 and properly transforming under Z(2). We solve for general G that satisfie s these requirements and explicitly construct anomaly-free theories in the upstairs approach. We also obtain the solutions in the presence of five-bra nes. All these constructions show equivalence of the downstairs and upstair s approaches. For example in the S1/Z2 case the ten-dimensional gauge coupl ing and the anomaly cancellation at each wall are the same as in the downst airs approach. (C) 2000 Elsevier Science B.V. All rights reserved.