  In many gauge theories, the existence of particles in every representation of
the gauge group (also known as completeness of the spectrum) is equivalent to
the absence of one-form global symmetries. However, this relation does not
hold, for example, in the gauge theory of non-abelian finite groups. We refine
this statement by considering topological operators that are not necessarily
associated with any global symmetry. For discrete gauge theory in three
spacetime dimensions, we show that completeness of the spectrum is equivalent
to the absence of certain Gukov-Witten topological operators. We further extend
our analysis to four and higher spacetime dimensions. Since topological
operators are natural generalizations of global symmetries, we discuss evidence
for their absence in a consistent theory of quantum gravity.
