Constraints on discrete symmetries from anomaly cancellation in compactified superstring theories

Compactified string theories give rise to discrete symmetries which are essential if they are to provide a realistic low energy theory. We find that in a class of four dimensional string theories these symmetries are constrained by similar conditions to those discrete anomaly cancellation conditions found in the case the discrete symmetry is a residue of a spontaneously broken gauge symmetry. Such conditions strongly constrain the allowed form of the low energy effective theory. While four dimensional string theories appear to have all the necessary ingredients needed to give the structure of the Standard Model, progress towards the construction of realistic string theory models has been hampered by the embarrassingly large number of candidate theories. Given this, any generic property of a string theory which may be used to limit the effective low-energy theory without committing the analysis to a specific example is important. In this letter we explore the possibility that the discrete symmetries that follow from compactified string theories are constrained by conditions similar to that were found [1] to result from anomaly cancellation when a local gauge symmetry is broken leaving a discrete symmetry factor unbroken [2]. The importance of these conditions stems from the fact that they involve only the states left light after the breaking of the gauge symmetry and thus constrain the low energy theory on its own. The power of these constraints is well illustrated by the study of discrete gauge symmetries in supersymmetric versions of the Standard Model where it was shown [3] that only two non-R symmetry discrete groups of low dimension satisfy the discrete anomaly cancellation conditions while suppressing SERC Advanced Fellow Supported by the European Community, Science Program