Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 307 / 3 / Solution

A gauge anomaly breaks a redundancy needed to remove unphysical states and makes a quantum gauge theory inconsistent unless it cancels. A chiral anomaly is instead the quantum violation of a classical global axial current; the theory remains consistent, but processes such as ${\pi }^0\to \gamma \gamma$ and instanton-induced charge violation become possible. A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is compatible with a consistent theory but is invariant under renormalization-group flow, so any infrared phase must reproduce it through massless fields, spontaneous symmetry breaking, or topological degrees of freedom.