Crossing Symmetry in the $H_3^+$ WZNW model
J. Teschner
TL;DR
The paper addresses crossing symmetry of the four-point function in the $H_3^+$ WZNW model and derives it from known properties of Liouville theory's five-point correlators. It constructs a factorization identity that links the $H_3^+$ four-point function to a Liouville correlator via an auxiliary factor, relying on a precise map between $j$-parameters and Liouville $\alpha$-parameters. A central technical ingredient, drawn from FZ, shows that Liouville decoupling equations correspond to Knizhnik–Zamolodchikov equations for the WZNW blocks when related by a prefactor, enabling a term-by-term matching of conformal blocks and structure constants through $b$-Racah–Wigner data. The verification uses asymptotic matching of initial terms and functional identities of the $\Upsilon$-functions and Gamma functions, producing a genus-zero consistent construction of the $H_3^+$ model via Liouville theory. Overall, the work demonstrates a concrete Liouville–WZNW correspondence at the level of correlation functions and strengthens the noncompact CFT framework for $AdS_3$-related string theory.
Abstract
We show that crossing symmetry of four point functions in the $H_3^+$ WZNW model follows from similar properties of certain five point correlation functions in Liouville theory that have already been proven previously.