The Braiding of Chiral Vertex Operators with Continuous Spins in 2D Gravity

TL;DR

This paper introduces generalized chiral vertex operators for continuous spins in 2D gravity, demonstrating their braiding relations and extending Liouville exponentials to continuous powers while maintaining locality.

Contribution

It develops a free-field realization of chiral vertex operators for continuous spins and extends braiding relations and Liouville exponentials to this new setting.

Findings

Generalized chiral vertex operators satisfy closed braiding relations.

Braiding matrices are extended using orthogonal polynomials.

Liouville exponentials are extended to continuous powers, preserving locality.

Abstract

Chiral vertex-operators are defined for continuous quantum-group spins $J$ from free-field realizations of the Coulomb-gas type. It is shown that these generalized chiral vertex operators satisfy closed braiding relations on the unit circle, which are given by an extension in terms of orthogonal polynomials of the braiding matrix recently derived by Cremmer, Gervais and Roussel. This leads to a natural extension of the Liouville exponentials to continuous powers that remain local.