Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 305 / 3 / ii / Solution

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.