The Wess-Zumino-Witten term was first introduced in the low
energy σ-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).