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 <'t Hooft anomaly>[anomalies] are
$$
\operatorname{Tr}R=3+12\left(-\frac23\right)=-5,
$$
$$
\operatorname{Tr}R^3=3+12\left(-\frac23\right)^3=-\frac59.
$$
Each 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 give
$$
\operatorname{Tr}R=15\left(-\frac13\right)=-5,
\qquad
\operatorname{Tr}R^3=15\left(-\frac13\right)^3=-\frac59.
$$
Both results agree, establishing the requested <'t Hooft anomaly matching>. This smooth composite description is the $SU(2)$, $N_f=3$ example of <s-confinement>.

Solution

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

A theory s-confines when its infrared physics is described smoothly everywhere on moduli space by gauge-invariant composites with a local superpotential and no remaining gauge group. SQCD with $N_f=N_c+1$ is the standard example.


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 Incoming links: s-confinement