Remarks on the Green–Schwarz Terms of Six-Dimensional Supergravity Theories

Abstract

We construct the Green–Schwarz terms of six-dimensional supergravity theories on spacetimes with non-trivial topology and gauge bundle. We prove the cancellation of all global gauge and gravitational anomalies for theories with gauge groups given by products of U(n), SU(n) and Sp(n) factors, as well as for E8. For other gauge groups, anomaly cancellation is equivalent to the triviality of a certain 7-dimensional spin topological field theory. We show in the case of a finite Abelian gauge group that there are residual global anomalies imposing constraints on the 6d supergravity. These constraints are compatible with the known F-theory models. Interestingly, our construction requires that the gravitational anomaly coefficient of the 6d supergravity theory is a characteristic element of the lattice of string charges, a fact true in six-dimensional F-theory compactifications but that until now was lacking a low-energy explanation. We also discover a new anomaly coefficient associated with a torsion characteristic class in theories with a disconnected gauge group.

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Acknowledgements

We would like to thank Daniel Park for discussions that led to this project.We also thank Dan Freed,Mike Hopkins, Graeme Segal,Wati Taylor, Andrew Turner, Nathan Seiberg and Edward Witten for useful discussions. G.M. is supported by the DOE under grant DOE-SC0010008 to Rutgers University. S.M. is supported in part by the grant MODFLAT of the European Research Council, SNSF grants No. 152812, 165666, and by NCCR SwissMAP, funded by the Swiss National Science Foundation.

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Monnier, S., Moore, G.W. Remarks on the Green–Schwarz Terms of Six-Dimensional Supergravity Theories. Commun. Math. Phys. 372, 963–1025 (2019). https://doi.org/10.1007/s00220-019-03341-7

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