Topological degeneracy of quantum Hall fluids

We present a simple approach to calculate the degeneracy and the structure of the ground states of non-Abelian quantum Hall (QH) liquids on the torus. Our approach can be applied to any QH liquids (Abelian or non-Abelian) obtained from the parton construction. We explain our approach by studying a series of examples of increasing complexity. When the effective theory of a non-Abelian QH liquid is a non-Abelian Chern-Simons (CS) theory, our approach reproduces the well-known results for the ground-state degeneracy of the CS theory. However, our approach also applies to non-Abelian QH liquids whose effective theories are not known and which cannot be written as a non-Abelian CS theory. We find that the ground states on a torus of all non-Abelian QH liquids obtained from the parton construction can be described by points on a lattice inside a ``folded unit cell.'' The folding is generated by reflection, rotations, etc. Thus the ground-state structures on the torus described by the ``folded unit cells'' provide a way to (at least partially) classify non-Abelian QH liquids obtained from the parton construction.