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

Classically the $p$ identical fields ${\psi }^i$ have $U(p)=SU(p)\times U(1{)}_{\psi }$, and $\lambda$ has an independent $U(1{)}_{\lambda }$. One linear combination has an $SU(N{)}^{2}U(1)$ anomaly. For the orthogonal combination chooseIndeed,The continuous quantum global symmetry is thereforeup to discrete identifications. The anomalous orthogonal axial rotation does not survive as a continuous quantum symmetry.