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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 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?

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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 $H=dB=*da$.

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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? –  stringpheno Feb 13 '11 at 16:44
    
@ Jeff Harvey. I agree that it's not straightforward but it's quite possible. Vacua with potentially many ultra-light string axions where all the geometric moduli are very heavy have been constructed here: springerlink.com/content/2845m53jmpw5754h –  stringpheno Feb 13 '11 at 17:16
    
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. –  Luboš Motl Feb 13 '11 at 18:52
    
Two more papers that allow you to believe that the discovery of SUSY, in a world with gravity, would pretty much imply that a string-like Green-Schwarz mechanism is necessary, and in this sense, a discovery of SUSY could also be a proof of string theory: sciencedirect.com/… –  Luboš Motl Feb 13 '11 at 18:54
    
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. –  Luboš Motl 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!

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I was not talking about the global SUSY limit, please read my question carefully! The MSSM in the rigid limit has no gauge anomalies but when embedded into N=1 D=4 supergravity, which is automatic since the super-Poincare symmetry must also be local, one generates the anomalies I was asking about. –  stringpheno Feb 13 '11 at 15:45

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