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

We find an explicit solution of the Schrodinger equation for a Chern-S imons theory coupled to charged particles on a Riemann surface, when t he coefficient of the Chern-Simons term is a rational number (rather t han an integer) and where the total charge is zero. We find that the w ave functions carry a projective representation of the group of large gauge transformations. We also examine the behavior of the wave functi on under braiding operations which interchange particle positions. We find that the representation of both the braiding operations and large gauge transformations involve unitary matrices which mix the componen ts of the wave function. The set of wave functions are expressed in te rms of appropriate Jacobi theta functions. The matrices form a finite dimensional representation of a particular (well known to mathematicia ns) version of the braid group on the Riemann surface. We find a const raint which relates the charges of the particles, q, the coefficient o f the Chern-Simons term, k, and the genus of the manifold, g: q2(g - 1 )/k must be an integer. We discuss a duality between large gauge trans formations and braiding operations.