EXPLICIT QUANTIZATION OF THE CHERN-SIMONS ACTION

We quantize the three-dimensional Chern-Simons action explicitly. We found that the geometric quantization of the action strongly depends on the topology of the (fixed-time) Riemann surface. On the disk the phase space and the symplectic structure are the same as those of the (chiral) Wess-Zumino-Witten model. On the torus the Hilbert space is the vector space of characters of Kac-Moody algebras. The fusion rules of the primary fields are derived from the classical matching condition of the holonomy. In general case, the wave-functional of the theory is the generating function of the current insertion in Wess-Zumino-Witten model. © 1990 Springer-Verlag.