  The Wess-Zumino-Witten term was first introduced in the low energy
sigma-model which describes pions, the Goldstone bosons for the broken flavor
symmetry in quantum chromodynamics. We introduce a new definition of this term
in arbitrary gravitational backgrounds. It matches several features of the
fundamental gauge theory, including the presence of fermionic states and the
anomaly of the flavor symmetry. To achieve this matching we use a certain
generalized differential cohomology theory. We also prove a formula for the
determinant line bundle of special families of Dirac operators on 4-manifolds
in terms of this cohomology theory. One consequence is that there are no global
anomalies in the Standard Model (in arbitrary gravitational backgrounds).
