Quantization ambiguities for Chern-Simons topological theories

Abstract

We point out that the singularities of the gauge invariant phase space of the three-dimensional topological Chern-Simons theory with compact gauge group G give rise to ambiguities of the canonical quantization which are classified by the one-dimensional representations of the Weyl group of G. We show that the quantum Hilbert space associated with the non-trivial representation. of the Weyl group has the unitary structure of the space of current blocks of two-dimensional Wess-Zumino-Witten theory on the group manifold G. This provides a topological understanding of the “shift” of the central charge in the two-dimensional Sugawara construction.