Canonical Quantization of the Boundary Wess-Zumino-Witten Model

We present an analysis of the canonical structure of the Wess-Zumino-Witten theory with untwisted conformal boundary conditions. The phase space of the boundary theory on a strip is shown to coincide with the phase space of the Chern-Simons theory on a solid cylinder (a disc times a line) with two Wilson lines. This reveals a new aspect of the relation between two-dimensional boundary conformal field theories and three-dimensional topological theories. A decomposition of the Chern-Simons phase space on a punctured disc in terms of the one on a punctured sphere and of coadjoint orbits of the loop group easily lends itself to quantization. It results in a description of the quantum boundary degrees of freedom in the WZW model by invariant tensors in a triple product of quantum group representations. In the action on the space of states of the boundary theory, the bulk primary fields of the WZW model are shown to combine the usual vertex operators of the current algebra with monodromy acting on the quantum group invariant tensors. We present the details of this construction for the spin 1/2 fields in the SU(2) WZW theory, establishing their locality and computing their 1-point functions.