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

When people talk about the first superstring revolution they often mention the miraculous cancellation of anomalies via the Green-Schwarz mechanism. My question is whether such a string-theoretic mechanism is also at work when the 4D gravitational and gauge-gravitational anomalies are tackled? In this context, would it be fair to say that a possible discovery of superpartners at the LHC, which automatically implies some version of $${\cal N}=1$$ $$D=4$$ supergravity, imply that stringy couplings (higher order in $$\alpha'$$) must be present in the corresponding Lagrangian to cancel the anomalies? What type of coupling are those?

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 $H=dB=*da$.
• Thank you @Jeff Harvey! Are there anomalies when one couples the MSSM with the global (spontaneously broken) $U(1)$ Peccei-Quinn symmetry of the axion to supergravity? Is this one of the possible additional $U(1)$s you mentioned above? Feb 13 '11 at 16:44
• Dear @stringpheno, some of the 10D Green-Schwarz anomaly cancellation mechanisms are almost directly inherited in $d=4$ and are important, see e.g. sciencedirect.com/… - and many other papers by Ferrara (et al.) from the 1980s and early 1990s. Feb 13 '11 at 18:52
• The second paper: cdsweb.cern.ch/record/180015/files/CM-P00062405.pdf?version=1 - as discussed above, you need the "linear multiplets" which remember the $B$ field from 10 dimensions and the axion that it produces in $d=4$. You will find an equivalent condition if you restrict yourself to "conformal supergravities", too. The argument is not quite waterproof but it's very likely to hold. Feb 13 '11 at 18:56
In 4D, we can have an axion mechanism. We have the axion-gauge coupling $\int d^4x\, d^2\theta \, \Phi W^\alpha W_\alpha$. But there are no gauge or gravitational anomalies in MSSM!