  We apply the geometric quantization method with real polarizations to the
quantization of a symplectic torus. By quantizing with half-densities we
canonically associate to the symplectic torus a projective Hilbert space and
prove that the projective factor is expressible in terms of the
Maslov-Kashiwara index. As in the quantization of a linear symplectic space, we
have two ways of resolving the projective ambiguity: (i) by introducing a
metaplectic structure and using half-forms in the definition of the Hilbert
space; (ii) by choosing a 4-fold cover of the Lagrangian Grassmannian of the
linear symplectic space covering the torus. We show that the Hilbert space
constructed through either of these approaches realizes a unitary
representation of the integer metaplectic group.
