Applying Witten’s formula for global gauge and gravitational anomalies to 6-dimensional supergravities, we find: (a) The perturbatively anomaly free N=4 chiral supergravity coupled to 21 tensor multiplets is global anomaly free for any choice of space-time manifold with vanishing third Betti number (b 3 ); (b) The perturbatively anomaly free matter coupled N=2 chiral super-gravities with arbitrary number of tensor multiplets, whose Yang-Mills gauge groups do not include G 2 , SU(2), or SU(3) are free of global anomalies if the theory is formulated on S 6 . In the case of 9 tensor multiplets coupled to supergravity, this result holds for any space-time with vanishing b 3 . (c) The N=6 chiral supergravity has perturbative gravitational anomalies and therefore the global anomalies need not be considered in this case.
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Bergshoeff et al. (1986) studied this question.