On the Liouville coupling constants

TL;DR

This paper explores the structure of Liouville coupling constants, linking them to quantum group 6j symbols and path-independent lattice products, and extends their definition to continuous screening numbers.

Contribution

It demonstrates that the Dorn-Otto and Zamolodchikov-Zamolodchikov three-point function ansatz naturally generalizes coupling constants to continuous screening numbers.

Findings

Coupling constants relate to quantum group 6j symbols.

Path independent lattice products define these constants.

Extension to continuous screening numbers is achieved through lattice connection analysis.

Abstract

For the general operator product algebra coefficients derived by Cremmer Roussel Schnittger and the present author with (positive integer) screening numbers, the coupling constants determine the factors additional to the quantum group 6j symbols. They are given by path independent products over a two dimensional lattice in the zero mode space. It is shown that the ansatz for the three point function of Dorn-Otto and Zamolodchikov-Zamolodchikov precisely defines the corresponding flat lattice connection, so that it does give a natural generalization of these coupling constants to continuous screening numbers. The consistency of the restriction to integer screening charges is reviewed, and shown to be linked with the orthogonality of the (generalized) 6j symbols. Thus extending this last relation is the key to general screening numbers.