A global anomaly obstructs invariance of a quantum theory under a topologically nontrivial symmetry transformation even when infinitesimal local anomaly tests vanish. The Witten SU(2) anomaly is one example; global gravitational anomalies are another class.
The Witten SU(2) anomaly is a four-dimensional global gauge anomaly present for an odd number of left-handed Weyl fermion doublets. A consistent SU(2) gauge theory must have an even number modulo the corresponding representation index.
