Laughlin states on higher genus Riemann surfaces

Considering quantum Hall states on geometric backgrounds has proved over the past few years to be a useful tool for uncovering their less evident properties, such as gravitational and electromagnetic responses, topological phases and novel geometric adiabatic transport coefficients. One of the transport coefficients, the central charge associated with the gravitational anomaly, appears as a Chern number for the adiabatic transport on the moduli spaces of higher genus Riemann surfaces. This calls for a better understanding of the QH states on these backgrounds. Here we present a rigorous definition and give a detailed account of the construction of Laughlin states on Riemann surfaces of genus ${\rm g}>1$. By the first principles construction we prove that the dimension of the vector space of Laughlin states is at least $\beta^{\rm g}$ for the filling fraction $\nu=1/\beta$. Then using the path integral for the 2d bosonic field compactified on a circle, we reproduce the conjectured $\beta^{\rm g}$-degeneracy as the number of independent holomorphic blocks. We also discuss the lowest Landau level, integer QH state and its relation to the bosonization formulas on higher genus Riemann surfaces.