Only the $p$ Spin(8) vectors $q$ and the one spinor $t$ contribute to the magnetic gauge beta function. The supplied indices give
$$
b_0^{\rm magnetic}
=\frac32(6)-\frac12(p+1)
=\frac{17-p}{2}=\frac{13-N}{2}.
$$
The magnetic theory is infrared free when $b_0^{\rm magnetic}<0$, namely
$$
\boxed{N>13}.
$$
At $N=13$ 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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 307 / 4 / vi / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-307/4/vi/solution