  In its original version, the Thomson problem consists of the search for the
minimum-energy configuration of a set of point-like electrons that are confined
to the surface of a two-dimensional sphere (${\cal S}^2$) that repel each other
according to Coulomb's law, in which the distance is the Euclidean distance in
the embedding space of the sphere, {\em i.e.}, $\mathbb{R}^3$. In this work, we
consider the analogous problem where the electrons are confined to an
$n$-dimensional flat Clifford torus ${\cal T}^n$ with $n = 1, 2, 3$. Since the
torus ${\cal T}^n$ can be embedded in the complex manifold $\mathbb{C}^n$, we
define the distance in the Coulomb law as the Euclidean distance in
$\mathbb{C}^n$, in analogy to what is done for the Thomson problem on the
sphere. The Thomson problem on a Clifford torus is of interest because
super-cells with the topology of Clifford torus can be used to describe
periodic systems such as Wigner crystals. In this work we numerically solve the
Thomson problem on a square Clifford torus. To illustrate the usefulness of our
approach we apply it to Wigner crystals. We demonstrate that the equilibrium
configurations we obtain for a large numbers of electrons are consistent with
the predicted structures of Wigner crystals. Finally, in the one-dimensional
case we analytically obtain the energy spectrum and the phonon dispersion law.
