THE BRAID GROUP OF A CANONICAL CHERN-SIMONS THEORY ON A RIEMANN SURFACE

We examine the problem of determining which representations of the bra id group on a Riemann surface and carried by the wave function of a qu antized Abelian Chern-Simons theory interacting with spinless non-dyna mical matter. We generalize the quantization of Chern-Simons theory to the case where the coefficient of the Chern-Simons term, k, is ration al (for a set of rational charges), the Riemann surface has arbitrary genus, and the total matter charge is non-vanishing. We find an explic it solution of the Schrodinger equation. We find that the wave functio ns carry a representation of the braid group as well as a dual project ive representation of the discrete group of large gauge transformation s. We find a Fundamental constraint which relates the charges of the p articles, 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) 199 6 Academic Press, Inc.