  The main new result here is the cancellation of global anomalies in the Type
I superstring, with and without D-branes. Our argument here depends on a
precise interpretation of the 2-form abelian gauge field using KO-theory; then
the anomaly cancellation follows from a geometric form of the full
Atiyah-Singer index theorem for families of Dirac operators. This is a refined
version of the Green-Schwarz mechanism. It seems that a geometric
interpretation of this mechanism-the cancellation of local and global fermion
anomalies against local and global anomalies in the electric coupling of an
abelian gauge field-always proceeds in a similar manner. For example, a
previous paper with M. Hopkins (hep-th/0002027) explains the cancellation of
anomalies in Type II with D-branes in these terms. The focal point of this
paper is a general discussion about abelian gauge fields and Dirac charge
quantization. Namely, we argue that quantization of charge is implemented in
the functional integral by interpreting abelian gauge fields as cochains in a
generalized differential cohomology theory. Our exposition includes elementary
examples as well as examples from superstring theory. The mathematical
underpinnings of differential cohomology are currently under development; we
only give a sketch here. The anomaly cancellation in Type I depends on
properties of a certain quadratic form in KO-theory, which we analyze in an
appendix written jointly with M. Hopkins. In particular, the usual equation
``Tr R^2 = Tr F^2'' is refined to an equation in the KO-theory of spacetime.
