On structure constants and fusion rules in the $SL(2,\BC)/SU(2)$ WZNW model

TL;DR

This work develops a noncompact CFT framework for the $H_3^+=SL(2,\mathbb{C})/SU(2)$ WZNW model, deriving an exact closed form for the three-point structure constants by leveraging degenerate-field constraints and a Liouville-like functional structure. By coupling canonical quantization, current-algebra Ward identities, and a generalized bootstrap, the authors extract a reflection amplitude $R(j)$ and show how crossing symmetry fixes the structure constants up to a cosmological normalization, with the Liouville theory guiding the analytic structure. They propose fusion rules consistent with spectral decompositions into principal-series representations, supported by normalizability criteria and analytic-continuation arguments that align with degenerate-field decoupling. The results enable an unambiguous definition of genus-zero $n$-point functions in the $H_3^+$ WZNW model and illuminate deep connections to Liouville theory and euclidean black hole CFTs, offering a path toward a rigorous duality framework via conformal-block and quantum-group techniques.

Abstract

A closed formula for the structure constants in the SL(2,C)/SU(2) WZNW model is derived by a method previously used in Liouville theory. With the help of a reflection amplitude that follows from the structure constants one obtains a proposal for the fusion rules from canonical quantization. Taken together these pieces of information allow an unambigous definition of any genus zero n-point function.