Type I anomaly cancellation revisited

We revisit the issue of how the perturbative and global fermion anomaly of Type I string theory in ten dimensions is cancelled by the Green-Schwarz mechanism using the RR fields. This will be done by realising the RR fields as boundary modes of an eleven-dimensional bulk theory described in terms of a quadratic refinement of the differential KO-theory pairing. We will then generalise this analysis to Sugimoto's $usp(32)$ string and Sagnotti's $u(32)$ string. We also discuss in a more general setting the procedures which need to be followed when we try to cancel fermion anomalies in terms of $p$-form fields based on differential K-theory classes. This we illustrate by performing an analysis of the mod-2 anomaly cancellation in nine dimensions arising from the $S^1$ compactification of the Type I theory.