The electric quarks give $2N_c$ copies of the $SU(2N_f)$ fundamental, so
$$
\mathcal A_{\rm el}[SU(2N_f)^3]=2N_c.
$$
In the magnetic theory the $2\widetilde N_c$ color components of $q$ transform as flavor antifundamentals and contribute $-2\widetilde N_c$. The singlet $M^{ij}$ is the <two-index antisymmetric representation> of $SU(2N_f)$, whose <cubic anomaly coefficient> is $2N_f-4$. Hence
$$
\begin{aligned}
\mathcal A_{\rm mag}[SU(2N_f)^3]
&=-2(N_f-N_c-2)+(2N_f-4)\\
&=\boxed{2N_c}
=\mathcal A_{\rm el}.
\end{aligned}
$$
This is a direct anomaly check of <Seiberg duality>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 307 / 4 / viii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-307/4/viii/solution