  We demonstrate that the three-dimensional $e_g$ orbital Hubbard model can be
generalized to arbitrary dimension $d$, and that the form of the result is
determined uniquely by the requirements that
  (i) the two-fold degeneracy of the $e_g$ orbital be retained, and (ii) the
cubic lattice be turned into a hypercubic lattice. While the local Coulomb
interaction $U$ is invariant for each basis of orthogonal orbitals, the form of
the kinetic energy depends on the orbital basis and takes the most symmetric
form for the so-called complex-orbital basis. Characteristically, with respect
to this basis, the model has two hopping channels, one that is orbital-flavor
conserving, and a second one that is orbital-flavor non-conserving. We show
that the noninteracting electronic structure consists of two nondegenerate
bands of plane-wave real-orbital single-particle states for which the orbital
depends on the wave vector. Due to the latter feature each band is unpolarized
at any filling, and has a non-Gaussian density of states at $d=\infty$. The
\textit{orbital liquid} state is obtained by filling these two bands up to the
same Fermi energy. We investigate the $e_g$ orbital Hubbard model in the limit
$d\to\infty$, treating the on-site Coulomb interaction $U$ within the
Gutzwiller approximation, thus determining the correlation energy of the
orbital liquid and the (disordered) para-orbital states. (...) We show that the
orbital liquid is the ground state everywhere in the $(n,U)$ phase diagram
except close to half-filling at sufficiently large $U$, where ferro-orbital
order with real orbitals occupied is favored. The latter feature is shown to be
specific for $d=\infty$, being of mathematical nature due to the exponential
tails in the density of states.
