Notes on the SL(2,R) CFT

TL;DR

This paper reviews the SL(2,R) conformal field theory, focusing on vertex operators, correlation functions, and fusion rules, highlighting issues with OPE closure and consistency with four-point correlator factorizations.

Contribution

It provides a detailed analysis of the correlation functions, fusion rules, and spectral flow in SL(2,R) CFT, addressing the closure problem of the OPE and its implications.

Findings

Derived selection rules and reduction formulas for correlators.

Identified issues with the closure of the SL(2,R) OPE.

Discussed the consistency of fusion rules with four-point correlator factorizations.

Abstract

This is a set of notes which reviews and addresses issues in the SL(2,R) conformal field theory, while working primarily in a basis of vertex operators of definite weight under the affine algebra. Following a review of the H3 coset model and the spectrum of the SL(2,R) CFT, derivations are given for the selection rules and the reduction of descendant to primary correlators involving arbitrary spectral flowed modules. Together with a description of the equivalence of correlation functions of fixed total spectral flow, this leads to an enumeration of the minimal set of three-point amplitudes required to determine the fusion rules. The corresponding fusion rules for elements of the spectrum are compared to those which arise through analytic continuation of the OPE of primary fields in the H3 model, and the apparent failure of the closure of the associated SL(2,R) OPE is described in detail. A discussion of the consistency of the respective fusion rules with the conjectured structure of equivalent factorizations of SL(2,R) four-point correlators is presented.