Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 307 / 4 / ii / Solution

The mixed $SU(N{)}^{2}U(1{)}_R$ anomaly receives $I(\mathrm{adj})=2N$ from the gaugino. A chiral multiplet of R-charge $r$ contains a fermion of charge $r-1$. Withthe symmetric-tensor fermion contributeswhile every antifundamental fermion contributes zero. The total is $2N-2N=0$, so this is a nonanomalous R-symmetry of the quantum theory.