Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 / 1 / a / ii / Solution

The vacuum is unique, so $N_0=1$. The stress tensor is the spin-$2$ irreducible representation of $SU(2)$ and has $2j+1=5$ states. At the next level, a translation of spin $1$ givesStress-tensor conservation makes the spin-$1$ divergence a null conformal descendant, leaving dimensions $7+5$. Hence