Gravitational and gauge-gravitational anomalies in ${\cal N}=1$ $D=4$ supergravity coupled to a SUSY gauge theory with chiral matter

There are no purely gravitational anomalies in $D=4$. The one source of gauge-gravitational anomalies is a triangle diagram with one gauge vertex and two graviton vertices. This vanishes provided that ${\rm Tr}_L Q=0$ where the trace runs over all left-handed fermions and $Q$ is the gauge generator with the potential anomaly. In the SM the only potential nonzero contribution arises from taking $Q=Y$, where $Y$ is the generator of the $U(1)$ part of $SU(3) \times SU(2) \times U(1)$. In the SM with standard fermion assignments this trace vanishes. In string compactifications one often gets additional $U(1)$ symmetries, and sometimes one finds these are anomalous by the above criterion. In such situations one finds a version of the Green-Schwarz mechanism involving a coupling of an axion-like mode to the gauge field which ends up giving a mass to the $U(1)$ gauge field. The axion-like field arises in string theory as a two-form field $B$, but in $D=4$ one can dualize to a scalar via $H=dB=*da$.