  It is widely believed that consistent theories of quantum gravity satisfy two
basic kinematic constraints: they are free from any global symmetry, and they
contain a complete spectrum of gauge charges. For compact, abelian gauge
groups, completeness follows from the absence of a 1-form global symmetry.
However, this correspondence breaks down for more general gauge groups, where
the breaking of the 1-form symmetry is insufficient to guarantee a complete
spectrum. We show that the correspondence may be restored by broadening our
notion of symmetry to include non-invertible topological operators, and prove
that their absence is sufficient to guarantee a complete spectrum for any
compact, possibly disconnected gauge group. In addition, we prove an analogous
statement regarding the completeness of twist vortices: codimension-2 objects
defined by a discrete holonomy around their worldvolume, such as cosmic strings
in four dimensions. We discuss how this correspondence is modified in various,
more general contexts, including non-compact gauge groups, Higgsing of gauge
theories, and the addition of Chern-Simons terms. Finally, we discuss the
implications of our results for the Swampland program, as well as the
phenomenological implications of the existence of twist strings.
