Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 307 / 3 / iv

For $N_f=3$, the infrared fields at the origin are the nine mesons $M_i{}^j$ and three baryons plus three antibaryons, for fifteen chiral multiplets. Every elementary quark superfield has R-charge $1/3$, so its fermion has charge $-2/3$. In the ultraviolet there are twelve quark Weyl fermions and three adjoint gauginos. Hence the two anomalies areEach meson or baryon superfield contains two quarks and has R-charge $2/3$, so each composite fermion has charge $-1/3$. The unconstrained infrared fields therefore giveBoth results agree, establishing the requested 't Hooft anomaly matching. This smooth composite description is the $SU(2)$, $N_f=3$ example of s-confinement.