  There exist non-trivial stationary points of the Euclidean action for an
axion particle minimally coupled to Einstein gravity, dubbed wormholes. They
explicitly break the continuos global shift symmetry of the axion in a
non-perturbative way, and generate an effective potential that may compete with
QCD depending on the value of the axion decay constant. In this paper, we
explore both theoretical and phenomenological aspects of this issue. On the
theory side, we address the problem of stability of the wormhole solutions, and
we show that the spectrum of the quadratic action features only positive
eigenvalues. On the phenomenological side, we discuss, beside the obvious
application to the QCD axion, relevant consequences for models with ultralight
dark matter, black hole superradiance, and the relaxation of the electroweak
scale. We conclude discussing wormhole solutions for a generic coset and the
potential they generate.
