Anyons on a torus

We show that the states of an anyon system on a torus are not completely determined by the positions of the anyons. There are q states for each fixed anyon configuration if the statistics of the anyons is given by θ=pπ/q. We explicitly construct the lattice Hamiltonian for the anyon system on the torus. The Hamiltonian is shown, both analytically and numerically, to respect the translation symmetries and the rotation symmetries. The flux of the anyon system is found to be quantized in units of 2π/q, without any shift. We also write down the effective Hamiltonian for the holons (with θ=π/2) in the chiral spin state