Anomalies, characters and strings

One-loop modular invariant heterotic and type II closed string theories are proved to be anomaly free, apart from terms proportional to TrF2-TrR2. It is shown why this conclusion holds with remarkable generality for any conceivable fermionic string theory, regardless even of conformal invariance. A detailed discussion is given of the modular properties of character valued partition functions, upon which the proof is based. The fact that TrF2-TrR2 terms remain in the fermionic contribution to the anomaly is shown to be a consequence of Quillen's holomorphic anomaly. (orig.)