Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces.

The fractional quantum Hall (FQH) states are shown to have q\ifmmode \tilde{}\else \~{}\fi{} $^{\mathit{g}}\mathrm{fold}$ ground-state degeneracy on a Riemann surface of genus g, where q\ifmmode \tilde{}\else \~{}\fi{} is the ground-state degeneracy in a torus topology. The ground-state degeneracies are directly related to the statistics of the quasiparticles given by \ensuremath{\theta}=p\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{\pi}/q\ifmmode \tilde{}\else \~{}\fi{}. The ground-state degeneracy is shown to be invariant against weak but otherwise arbitrary perturbations. Therefore the ground-state degeneracy provides a new quantum number, in addition to the Hall conductance, characterizing different phases of the FQH systems. The phases with different ground-state degeneracies are considered to have different topological orders. For a finite system of size L, the ground-state degeneracy is lifted. The energy splitting is shown to be at most of order ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathit{L}/\ensuremath{\xi}}$. We also show that the Ginzburg-Landau theory of the FQH states (in the low-energy limit) is a dual theory of the U(1) Chern-Simons topological theory.