Negative Screenings in Conformal Field Theory and 2D Gravity: The Braiding Matrix

TL;DR

This paper extends the Coulomb gas framework in conformal field theory to include negative screening powers, deriving explicit braiding matrices using q-hypergeometric functions and verifying them through Moore-Seiberg equations.

Contribution

It introduces a novel extension of the Coulomb gas approach with negative screenings and provides explicit braiding matrices in terms of 4F3 q-hypergeometric functions.

Findings

Derived explicit R-matrices for negative screening numbers

Verified braiding matrices using Moore-Seiberg equations

Discovered new relations for q-hypergeometric functions

Abstract

We consider an extension of the Coulomb gas picture which is motivated by Liouville theory and contains negative powers of screening operators on the same footing as positive ones. The braiding problem for chiral vertex operators in this extended framework is analyzed. We propose explicit expressions for the R-matrix with general integer screening numbers, which are given in terms of 4F3 q-hypergeometric functions through natural analytic continuations of the well-known expression for positive integer screenings. These proposals are subsequently verified using a subset of the Moore-Seiberg equations that is obtained by simple manipulations in the operator approach. Interesting new relations for q-hypergeometric functions (particularly of type 4F3) arise on the way.