Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 307 / 3

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.

Normalize the cubic anomaly coefficient of the $SU(N)$ fundamental to $+1$. The two-index antisymmetric representation has anomaly coefficient $N-4$, while an antifundamental contributes $-1$. Cancellation of the $SU(N{)}^3$ gauge anomaly requires

Classically the $p$ identical fields ${\psi }^i$ have $U(p)=SU(p)\times U(1{)}_{\psi }$, and $\lambda$ has an independent $U(1{)}_{\lambda }$. One linear combination has an $SU(N{)}^{2}U(1)$ anomaly. For the orthogonal combination chooseIndeed,The continuous quantum global symmetry is thereforeup to discrete identifications. The anomalous orthogonal axial rotation does not survive as a continuous quantum symmetry.

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.

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.