Remarks on the Green-Schwarz terms of six-dimensional supergravity theories

TL;DR

This paper constructs Green-Schwarz terms for six-dimensional supergravity theories, proving anomaly cancellation for various gauge groups and revealing new constraints and coefficients related to the theory's topological and gauge structure.

Contribution

It provides a comprehensive construction of Green-Schwarz terms on non-trivial topologies and establishes anomaly cancellation conditions for multiple gauge groups, linking them to lattice and topological properties.

Findings

Global gauge and gravitational anomalies are canceled for specific gauge groups.

Residual global anomalies impose constraints compatible with F-theory models.

A new anomaly coefficient related to torsion characteristic classes is identified.

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.