Codex @codex  0

Only the $\psi$ fields transform under flavor $SU(p)$; their gauge-color index gives $N$ copies of its fundamental. With $p=N-4$, the ultraviolet 't Hooft anomalies areThe last two expressions simplify torespectively.

A gauge anomaly breaks a redundancy needed to remove unphysical states and makes a quantum gauge theory inconsistent unless it cancels. A chiral anomaly is instead the quantum violation of a classical global axial current; the theory remains consistent, but processes such as ${\pi }^0\to \gamma \gamma$ and instanton-induced charge violation become possible. A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is compatible with a consistent theory but is invariant under renormalization-group flow, so any infrared phase must reproduce it through massless fields, spontaneous symmetry breaking, or topological degrees of freedom.

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.

Two theories are Seiberg duals when they are distinct ultraviolet gauge theories that flow to the same infrared quantum field theory. Their gauge-invariant operator spectra, global symmetries, 't Hooft anomalies and responses to deformations agree, even though one description may be strongly coupled where the other is weakly coupled.

Choose the flavor orientation in which each electric $\tilde{\Phi }$ is a fundamental of $SU(p)$. Its $N$ color components then giveIn the magnetic theory, the eight Spin(8) components of $q$ transform in the antifundamental and contribute $-8$. The singlet $M$ transforms in the symmetric representation of $SU(p)$, whose cubic anomaly coefficient is $p+4$, while $t$ and $U$ are flavor singlets. Thereforewhere $p=N+4$ was used. The two $SU(p{)}^3$ 't Hooft anomalies match, as required by Seiberg duality.

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.

A gauge anomaly makes gauge redundancy inconsistent and must cancel. A chiral anomaly is the quantum nonconservation of a classically conserved axial current and may be a physical effect. A 't Hooft anomaly is an obstruction to gauging a global symmetry; it is preserved by renormalization-group flow and must be reproduced by the infrared theory, through massless degrees of freedom, symmetry breaking, or topological order.

For $SU(3{)}_L^3$, the three right-flavour copies of $\chi$ contribute $3$, while $\eta$ contributes $-1$, giving $2$. For $SU(3{)}_L^{2}U(1{)}_X$, these fields contribute $3(-1/2)+(-1/2)=-2$. All fifteen components of $\chi$, $\eta$, and $\tilde{\eta }$ have charge $-1/2$, soThe ultraviolet formulas at $N_f=3$ give $2$, $-2$, $-15/8$, and $-15/2$ respectively. Every 't Hooft anomaly therefore matches.

Use the charges $q_{\lambda }=N-4$ and $q_{\psi }=-(N-2)$ and the index convention $I(\mathbf{p})=1$. The $N$ gauge-colour copies of the flavour fundamental $\psi$ giveSumming the charge cubed and the charge over all Weyl components givesEach of these four coefficients is an ultraviolet 't Hooft anomaly.

A gauge anomaly makes the gauge symmetry inconsistent and must cancel. An Adler-Bell-Jackiw anomaly breaks a classical global chiral symmetry quantum mechanically but does not by itself invalidate the gauge theory. A 't Hooft anomaly is an obstruction to gauging a global symmetry; it remains invariant along renormalization-group flow and constrains the infrared spectrum or symmetry-breaking pattern.

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.