Analytic continuations of 3-point functions of the conformal field theory

Vladimir S Dotsenko

doi:10.1016/j.nuclphysb.2016.03.037

Abstract

It is shown that the general 3-point function 〈ΦcΦbΦa〉, with continuous values of charges a,b,c of a statistical model operators, and the 3-point function of the Liouville model, could all be obtained by successive analytical continuations starting from the 3-point function of the minimal model.

Highlights

Recent interest in the 3-point functions c b a with continuous values of charges a, b, c, which do not satisfy the neutrality conditions of the Coulomb gas minimal models, is, principally, due to recently found realisations of these correlation functions in the context of statistical models, on the lattice: Potts model 3 spin correlation function [1], loop models [2].On the other side, the interest in the Liouville model correlation function was always present, since 1981 [3].The Liouville 3-point function was defined in [4,5]
We summarise that the formula (3.45) has all been obtained by the analytic continuation from general 3-point function for degenerate operators of the minimal model
We observe that all the ρ-factors get cancelled and we find, the normalised 3-point function for the normalised operators in the form: c (∞)b (1)a (0)

Summary

Introduction

Recent interest in the 3-point functions c b a with continuous values of charges a, b, c, which do not satisfy the neutrality conditions of the Coulomb gas minimal models, is, principally, due to recently found realisations of these correlation functions in the context of statistical models, on the lattice: Potts model 3 spin correlation function [1], loop models [2]. We summarise that the formula (3.45) has all been obtained by the analytic continuation from general 3-point function for degenerate operators of the minimal model. The “naive” norms of vertex operators Va , Va+ have been defined in [10, Section 9.1] They differ from the actual norms in (4.34) by the absence of the partition function Z, of the Coulomb gas, in the norm of Va+ , N (Va+ )naive = 1/Na. By the formula (4.9) we obtain:. We observe that all the ρ-factors get cancelled and we find, the normalised 3-point function for the normalised operators in the form: ΥM (a + b + c − 2α0 )ΥM (−a + b + c)ΥM (a − b + c)ΥM (a + b − c) ΥM (−2α0 ) In this formula everything is expressed, analytically, in terms of charges a, b, c, so that we can continue the formula to the general, continuous values of charges. The fact that log ΥM (−i x, −i h), in (5.1.18), is well defined, by its integral, as was discussed above, is seen by the first figure in Fig. 6: the poles stay away from the initial integration line, the real axes of t

Analytic continuation of the function c b a

Discussions

One also finds that