We study global anomalies of discrete gauge symmetries in six-dimensional
supergravities and their realizations in F-theory. We explicitly construct a
discrete Green-Schwarz mechanism that depends on the choice of a coupling
constant and on a certain quadratic refinement in differential cohomology. By
geometrically engineering theories with G=Z3​ gauge symmetry and no
tensor multiplets, we observe that a particular choice of the quadratic
refinement is singled out in F-theory. This implies new Swampland constraints
on the discrete charge spectra of 6d supergravities. On the other hand, the
discrete Green-Schwarz coupling depends on the geometry of the Calabi-Yau. We
use anomaly inflow to relate this to a 't Hooft anomaly of the induced global
symmetry in the worldsheet theories of non-critical strings. Using topological
symmetry lines, we further relate this anomaly to the modular properties of
twisted-twined elliptic genera. We then argue that the latter are encoded in
the A-model topological string partition functions on different torus
fibrations that are equipped with a flat torsional B-field. This allows us to
derive a geometric expression for the global discrete anomaly in terms of the
height-pairing of a multi-section on a genus one fibered Calabi-Yau.Comment: 35+16 pages, 21 figure
