The braid group of a canonical Chern-Simons theory on a Riemann surface

We examine the problem of determining which representations of the braid group on a Riemann surface and carried by the wave function of a quantized Abelian Chern-Simons theory interacting with spinless non-dynamical matter. We generalize the quantization of Chern-Simons theory to the case where the coefficient of the Chern-Simons term, k, is rational (for a set of rational charges), the Riemann surface has arbitrary genus, and the total matter charge is non-vanishing. We find an explicit solution of the Schrodinger equation. We find that the wave functions carry a representation of the braid group as well as a dual projective representation of the discrete group of large gauge transformations. We find a Fundamental constraint which relates the charges of the particles, q(i), the coefficient k and the genus of the manifold, g:q(i)(Q + q(i)(g - 1))/k is integer (where Q is the total charge). (C) 1996 Academic Press, Inc.