  The nearest-neighbor Villain, or periodic Gaussian, model is a useful tool to
understand the physics of the topological defects of the two-dimensional
nearest-neighbor $XY$ model, as the two models share the same symmetries and
are in the same universality class. The long-range counterpart of the
two-dimensional $XY$ model has been recently shown to exhibit a non-trivial
critical behavior, with a complex phase diagram including a range of values of
the power-law exponent of the couplings decay, $\sigma$, in which there are a
magnetized, a disordered and a critical phase (arXiv:2104.13217). Here we
address the issue of whether the critical behavior of the two-dimensional $XY$
model with long-range couplings can be described by the Villain counterpart of
the model. After introducing a suitable generalization of the Villain model
with long-range couplings, we derive a set of renormalization-group equations
for the vortex-vortex potential, which differs from the one of the long-range
$XY$ model, signaling that the decoupling of spin-waves and topological defects
is no longer justified in this regime. The main results are that for $\sigma<2$
the two models no longer share the same universality class. Remarkably, within
a large region of its phase diagram, the Villain model is found to behave
similarly to the one-dimensional Ising model with $1/r^2$ interactions.
