Geometric Quantization for the Moduli Space of Vector Bundles with Parabolic Structure

G. Daskalopoulos and R. Wentworth

We initiate a study of the geometric quantization of Chern-Simons gauge theory on Riemann surfaces with punctures. We construct a moduli space of flat connections using weighted Sobolev spaces, and then by analogy with the compact case, we construct a line bundle over this moduli space. The line bundle exists only for certain holonomies which correspond in a one-to-one way with representations. When the bundle does exist, its holomorphic sections reproduce the space of states defined by Segal.