Boundary WZW, G/H, G/G and CS theories

TL;DR

This work extends the canonical analysis of the WZW theory to bulk and boundary coset models $G/H$ and $G/G$, showing that the resulting phase spaces on both closed and open geometries align with those of a double Chern-Simons theory on suitably chosen 3-manifolds. It provides explicit symplectic isomorphisms between the 2D coset/WZW phase spaces and the 3D CS phase spaces, including detailed treatments of Wilson lines and boundary data, and demonstrates how the boundary $G/G$ coset yields a quantizable two-dimensional boundary topological field theory. The quantization of these boundary theories leads to spaces of conformal blocks and a Verlinde-like fusion structure, situating boundary CFTs within a concrete TFT framework that connects loop-space K-theory and fusion algebras. Overall, the paper gives a comprehensive CS–WZW/coset dictionary and a concrete path to boundary TFT quantization for coset models.

Abstract

We extend the analysis of the canonical structure of the Wess-Zumino-Witten theory to the bulk and boundary coset G/H models. The phase spaces of the coset theories in the closed and in the open geometry appear to coincide with those of a double Chern-Simons theory on two different 3-manifolds. In particular, we obtain an explicit description of the canonical structure of the boundary G/G coset theory. The latter may be easily quantized leading to an example of a two-dimensional topological boundary field theory.