Operator formalism for Chern-Simons theories

The operator formalism for Chern-Simons theories with gauge group G and parameter k (G=U(1), SU(2)) on an arbitrary oriented compact three-dimensional manifold is constructed. The states of the Hilbert space are obtained in a wave-functional representation which corresponds to a generalized form of the exponential of a Wess-Zumino-Witten action. It is shown explicitly that the states of a basis of the Hilbert space are in one-to-one correspondence with the characters of a Wess-Zumino-Witten model with gauge group G at level k and have their same properties under modular transformations. In addition, it is also shown that the Wilson line operators with gauge field in some distinguished representations act as creation operators in the Hilbert space and verify the fusion algebra of the corresponding conformal field theory. (orig.)