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.

For left-handed Weyl fermions in representations $(R_i,Q_i)$ of a simple semisimple factor times $U(1)$, the displayed traces cancel the local cubic non-Abelian, mixed non-Abelian–Abelian, and cubic Abelian gauge anomalies. Include spectator multiplicities in every trace. A triangle with one non-Abelian insertion and two Abelian insertions vanishes by tracelessness. Coupling to gravity also requires ${\sum }_i\dim (R_i)Q_i=0$, and global anomalies must be checked separately. For a conjugate representation the cubic coefficient changes sign but the Dynkin index $T(R)$ does not.

In four spacetime dimensions, the pure local gauge anomaly for a circle group is proportional to the sum of cubed U(1) charges over left-handed Weyl fermions, counting a right-handed Weyl fermion through its left-handed charge conjugate. A consistent anomaly-free gauge theory requires this sum to vanish, absent an additional anomaly-cancelling mechanism. Cancellation of the mixed gauge-gravitational anomaly additionally requires ${\sum }_a{Q}_a=0$. If the same spectrum is restricted to a finite cyclic group, $q_a=Q_a\mathrm{mod}n$ implies ${\sum }_a{q}_a^3=0\mathrm{mod}n$ and, when imposed, ${\sum }_a{q}_a=0\mathrm{mod}n$. These elementary consequences alone do not classify intrinsic discrete gauge anomalies or guarantee an anomaly-free integer lift.

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.