  It is widely believed and in part established that exact global symmetries
are inconsistent with quantum gravity. One then expects that approximate global
symmetries can be quantitatively constrained by quantum gravity or swampland
arguments. We provide such a bound for an important class of global symmetries:
Those arising from a gauged $U(1)$ with the vector made massive via Higgsing
with an axion. The latter necessarily couples to instantons, and their action
can be constrained, using both the electric and magnetic version of the axionic
weak gravity conjecture, in terms of the cutoff of the theory. As a result,
instanton-induced symmetry breaking operators with a suppression factor not
smaller than $\exp(-M_{\rm P}^2/\Lambda^2)$ are present, where $\Lambda$ is a
cutoff of the 4d effective theory. We provide a general argument and clarify
the meaning of $\Lambda$. Simple 4d and 5d models are presented to illustrate
this, and we recall that this is the standard way in which things work out in
string compactifications with brane instantons. The relation of our constraint
to bounds that can be derived from wormholes or gravitational instantons and to
those motivated by black-hole effects at finite temperature are discussed, and
we present a generalization of the Giddings-Strominger wormhole solution to the
case of a gauge-derived $U(1)$ global symmetry. Finally, we discuss potential
loopholes to our arguments.
