  We revisit the relation between the anomalies in four and six dimensions and
the Chern-Simons couplings one dimension below. While the dimensional reduction
of chiral theories is well-understood, the question which three and
five-dimensional theories can come from a general circle reduction, and are
hence liftable, is more subtle. We argue that existence of an anomaly
cancellation mechanism is a necessary condition for liftability. In addition,
the anomaly cancellation and the CS couplings in six and five dimensions
respectively determine the central charges of string-like BPS objects that
cannot be consistently decoupled from gravity, a.k.a. supergravity strings.
Following the completeness conjecture and requiring that their worldsheet
theory is unitary imposes bounds on the admissible theories. We argue that for
the anomaly-free six-dimensional theories it is more advantageous to study the
unitarity constraints obtained after reduction to five dimensions. In general
these are slightly more stringent and can be cast in a more geometric form,
highly reminiscent of the Kodaira positivity condition (KPC). Indeed, for the
F-theoretic models which have an underlying Calabi-Yau threefold these can be
directly compared. The unitarity constraints (UC) are in general weaker than
KPC, and maybe useful in understanding the consistent models without
F-theoretic realisation. We catalogue the cases when UC is more restrictive
than KPC, hinting at more refined hidden structure in elliptic Calabi-Yau
threefolds with certain singularity structure.
