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 $E_8$. 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.

76 pages. v2: Update in the references

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• Reflection groups and 3d $\mathcal{N}\ge$ 6 SCFTs
• Omega vs. pi, and 6d anomaly cancellation
• Distinguishing 6d (1,0) SCFTs: an extension to the geometric construction
• The discrete Green-Schwarz mechanism in 6D F-Theory and Elliptic Genera of Non-Critical Strings
• Charge completeness and the massless charge lattice in F-theory models of supergravity
• Anomalies as Obstructions: from Dimensional Lifts to Swampland
• Automatic Enhancement in 6D Supergravity and F-theory Models
• The Fate of Discrete 1-Form Symmetries in 6d
• Flavor Symmetries and Automatic Enhancement in the 6d Supergravity Swampland
• What bordism-theoretic anomaly cancellation can do for U
• Geometric Approach to 3D Interfaces at Strong Coupling
• On the consistency of a class of R-symmetry gauged 6D N=(1,0) supergravities
• Cancelling mod-2 anomalies by Green-Schwarz mechanism with $B_{\mu \nu }$
• Symmetry TFTs from String Theory