Abelian and Non-Abelian Hall Liquids and Charge-Density Wave: Quantum Number Fractionalization in One and Two Dimensions

Previously, we have demonstrated that, on a torus, the Abelian quantum Hall liquid is adiabatically connected to a charge-density wave as the smaller dimension of the torus is varied. In this work, we extend this result to the non-Abelian bosonic Hall state. The outcome of these works is the realization that the paradigms of quantum number fractionalization in one dimension (polyacetylene) and two dimensions (fractional quantum Hall effect) are, in fact, equivalent.