  The landscape of six-dimensional supergravities is dramatically constrained
by the cancellation of gauge and gravitational anomalies, but the full extent
of its implications has not been uncovered. We explore the cancellation of
global anomalies of the Dai-Freed type in this setting with abelian and simply
laced gauge groups, finding novel constraints. In particular, we exclude
arbitrarily large abelian charges in an infinite family of theories for certain
types of quadratic refinements, including a specific one defined in the
literature. We also show that the Gepner orientifold with no tensor multiplets
is anomaly-free for a different choice, as well as a number of heterotic models
with and without spacetime supersymmetry in six dimensions. The latter analysis
extends previous results in ten dimensions to some lower-dimensional settings
in the heterotic landscape.
