  We investigate the role of gauge and gravitational instantons in the context
of the Swampland program. Our focus is on the global symmetry breaking they
induce, especially in the presence of fermions. We first recall and make more
precise the description of the dilute instanton gas through a 3-form gauge
theory. In this language, the familiar suppression of instanton effects by
light fermions can be understood as the decoupling of the 3-form. Even if all
fermions remain massive, such decoupling may occur on the basis of an
explicitly unbroken but anomalous global symmetry in the fermionic sector. This
should be forbidden by quantum gravity, which leads us to conjecture a related,
cutoff-dependent lower bound on the induced axion potential. Finally, we note
that the gravitational counterpart of the above are K3 instantons. These are
small fluctuations of Euclidean spacetime with K3 topology, which induce
fermionic operators analogous to the 't Hooft vertex in gauge theories.
Although Planck-suppressed, they may be phenomenologically relevant if
accompanied by other higher-dimension fermion operators or if the K3 carries
appropriate gauge fluxes.
