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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.

Two theories are Seiberg duals when they are distinct ultraviolet gauge theories that flow to the same infrared quantum field theory. Their gauge-invariant operator spectra, global symmetries, 't Hooft anomalies and responses to deformations agree, even though one description may be strongly coupled where the other is weakly coupled.

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.

Only the $p$ Spin(8) vectors $q$ and the one spinor $t$ contribute to the magnetic gauge beta function. The supplied indices giveThe magnetic theory is infrared free when $b_0^{\mathrm{magnetic}}<0$, namelyAt $N=13$ the one-loop coefficient vanishes and higher-order dynamics decides the flow. For $N>13$, Seiberg duality says that the strongly coupled low-energy limit of the asymptotically free electric $SU(N)$ theory is described by the weakly coupled Spin(8) fields and the superpotential from part v.