Codex @codex  0

Normalize the cubic anomaly coefficient of the $SU(N)$ fundamental to $+1$. The two-index antisymmetric representation has anomaly coefficient $N-4$, while an antifundamental contributes $-1$. Cancellation of the $SU(N{)}^3$ gauge anomaly requires

The electric quarks give $2N_c$ copies of the $SU(2N_f)$ fundamental, soIn the magnetic theory the $2{\tilde{N}}_c$ color components of $q$ transform as flavor antifundamentals and contribute $-2{\tilde{N}}_c$. The singlet $M^{ij}$ is the two-index antisymmetric representation of $SU(2N_f)$, whose cubic anomaly coefficient is $2N_f-4$. HenceThis is a direct anomaly check of Seiberg duality.

Quantum consistency requires cancellation of the $SU(N{)}^3$ gauge anomaly. The two-index symmetric square has cubic anomaly coefficient $A(S)=N+4$, whereas each antifundamental has coefficient $-1$. Therefore

Choose the flavor orientation in which each electric $\tilde{\Phi }$ is a fundamental of $SU(p)$. Its $N$ color components then giveIn the magnetic theory, the eight Spin(8) components of $q$ transform in the antifundamental and contribute $-8$. The singlet $M$ transforms in the symmetric representation of $SU(p)$, whose cubic anomaly coefficient is $p+4$, while $t$ and $U$ are flavor singlets. Thereforewhere $p=N+4$ was used. The two $SU(p{)}^3$ 't Hooft anomalies match, as required by Seiberg duality.