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

Write every fermion as a left-handed Weyl field. One generation isThe perturbative anomaly coefficients vanish as follows:The common Dynkin-index factors have been suppressed in the mixed non-Abelian lines. An $SU(2{)}^3$ perturbative anomaly vanishes because the doublet is pseudoreal, while mixed anomalies containing one non-Abelian generator vanish because that generator is traceless. There are $3+1=4$ left-handed $SU(2)$ doublets after color multiplicity, so the nonperturbative Witten SU(2) anomaly also cancels. The gauge-singlet right-handed neutrino changes none of these sums.