Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 4 / ii / Solution

An uncancelled gauge anomaly would make the quantum effective action vary along a gauge orbit. The resulting failure of the Ward-Takahashi identities and BRST symmetry prevents unphysical polarizations from decoupling, so the theory loses unitarity or renormalizability and cannot define a consistent gauge theory. An uncancelled mixed gauge-gravitational anomaly would similarly conflict with simultaneous gauge-current and stress-energy conservation. A global anomaly such as the Witten SU(2) anomaly would make the fermion determinant change sign under a large gauge transformation, so even the path integral would be ill defined.