Past exam of the groups and representations course of the University of Oxford / 2023 / 1 / e / Solution

Normalize the circle group as in solution and count all Weyl fermions as left-handed; a right-handed Weyl fermion is replaced by its left-handed charge conjugate. The restriction of a representation givesThe triangle coefficient of the cubic U(1) gauge anomaly is proportional to ${\sum }_a{Q}_a^3$. In a theory with this single gauge group, cancellation of its local pure gauge anomaly requiresIf absence of the mixed gauge-gravitational anomaly is also required, as when coupling to a general gravitational background, one additionally needs ${\sum }_a{Q}_a=0$. Additional gauge group factors would require their mixed gauge anomalies to cancel as well.

Choose integer representatives and write $Q_a=q_a+n{k}_a$. Expanding the cubic givesHence the inherited necessary congruence, and its gravitational counterpart when imposed, areThese statements concern the same complete fermion spectrum used in the parent theory. They are consequences of an anomaly-free integer lift, rather than a full classification of intrinsic discrete gauge anomalies or a sufficient test for such a lift. They also do not imply an ordinary equality for the chosen residues: for $n=3$, the anomaly-free U(1) charges $(1,-1)$ have residues $(1,2)$, whose cubes sum to $9$, rather than zero. Integrating out heavy fermions can require extra contributions to the effective discrete gauge anomaly accounting.