The composite $\chi^{ij}$ is in the <two-index symmetric representation> of $SU(p)$ and has
$$
q_\chi=2q_\psi+q_\lambda=-N.
$$
For that representation,
$$
A(\mathrm{sym})=p+4=N,
\quad I(\mathrm{sym})=p+2=N-2,
\quad \dim(\mathrm{sym})=\frac{p(p+1)}2.
$$
Its anomalies are consequently
$$
N,qquad -N(N-2),qquad
-\frac12p(p+1)N^3,qquad
-\frac12p(p+1)N,
$$
in the same order as part iii. Every anomaly matches, so confinement to the proposed massless composite is consistent with <'t Hooft anomaly matching>.

Solution

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

{title2=$\operatorname{Sym}^2V$}

The two-index symmetric representation of $SU(N)$ is the symmetric square of its fundamental representation. It has dimension $N(N+1)/2$, Dynkin index $N+2$, and cubic anomaly coefficient $N+4$ when the fundamental index and anomaly are normalized to one.


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