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The composite ${\chi }^{ij}$ is in the two-index symmetric representation of $SU(p)$ and hasFor that representation,Its anomalies are consequentlyin the same order as part iii. Every anomaly matches, so confinement to the proposed massless composite is consistent with 't Hooft anomaly matching.

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.

A gauge anomaly breaks a redundancy required to remove unphysical polarizations. It violates the Ward identities and makes the quantum gauge theory inconsistent, so all gauge anomalies must cancel. A chiral anomaly, or Adler-Bell-Jackiw anomaly, instead breaks a classically conserved global axial symmetry; it is physically allowed and explains effects such as anomalous pseudoscalar decays and instanton-induced charge violation. A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is invariant under renormalization-group flow, so 't Hooft anomaly matching constrains the infrared theory to reproduce it through massless fields, symmetry breaking, or a topological sector.

Normalize a fundamental cubic anomaly to $A({\mathbf{N}}_{\mathbf{f}})=1$ and its Dynkin index to $I({\mathbf{N}}_{\mathbf{f}})=1$. The two gauge colours giveThe left and charge-conjugated right fermions have opposite vector charges and equal multiplicities, soIf the theory confined with unbroken chiral symmetry, gauge-invariant operators contain an even number of $SU(2)$ fundamentals and are bosonic, so no massless composite fermions could match the nonzero chiral anomalies. 't Hooft anomaly matching therefore forces chiral symmetry breaking, with its infrared Goldstone theory carrying the anomaly.

The composite ${\chi }^{ij}={\psi }^{(i}(\lambda {\psi }^{j)})$ is a flavour-symmetric left-handed Weyl fermion. Its $U(1)$ charge isBecause $p=N-4$, the supplied symmetric-representation data giveThusThere are $p(p+1)/2=(N-4)(N-3)/2$ components, sowhich exactly equal the ultraviolet values. The proposed confined spectrum therefore satisfies 't Hooft anomaly matching.

The three Weyl components form one fundamental representation of the global $SU(3)$, so coupling $G$ to background gauge fields reveals a nonzero $SU(3{)}^3$ 't Hooft anomaly. This does not destroy the global symmetry in flat space or make the theory inconsistent; it constrains any infrared description through 't Hooft anomaly matching. If $G$ were instead intended as a dynamical gauge symmetry, the same uncancelled gauge anomaly would make the model inconsistent and additional chiral matter would be required. Ordinary quantum corrections cannot generate the forbidden superpotential, and the non-renormalization theorem protects its absence.