Gauge anomaly

A gauge anomaly destroys a gauge redundancy needed to remove unphysical states and makes the quantum gauge theory inconsistent unless the anomaly cancels.

Anomaly cancellation is the vanishing of every gauge and mixed gauge-gravitational anomaly after summing over all chiral fermions. It is a consistency condition for a quantum gauge theory.

In each Standard Model generation, quark and lepton hypercharges cancel the $SU(3{)}_c^3$, $SU(3{)}_c^{2}U(1{)}_Y$, $SU(2{)}_L^{2}U(1{)}_Y$, $U(1{)}_Y^3$, and mixed gauge-gravitational anomalies. The four left-handed $SU(2{)}_L$ doublets also avoid the global Witten anomaly.

In normalization $Q=T_3+Y$, one Standard Model generation has left quark and lepton doublet hypercharges $1/6$ and $-1/2$, and right singlet hypercharges $2/3,-1/3,-1$. Including colour multiplicity, ${\mathrm{tr}}_{L}Y=6/6-2/2=0$. On every left doublet $T_3^2=I/4$, and all right fields have $T_3=0$, so the mixed weak-hypercharge gauge anomaly vanishes. The cubic left trace is $6(1/6{)}^3+2(-1/2{)}^3=-2/9$, and the right trace is $3(2/3{)}^3+3(-1/3{)}^3+(-1{)}^3=-2/9$. Their difference is zero, proving generation-by-generation cancellation of these two anomalies.

A mixed gauge-gravitational anomaly has one gauge-current insertion and two stress-tensor insertions. For a four-dimensional $U(1)$ symmetry its coefficient is the sum of the left-handed Weyl-fermion charges, with right-handed fermions counted as left-handed conjugates.