Recursive relations for the S-matrix of Liouville theory

TL;DR

This paper derives recursive relations for the Liouville theory S-matrix using vertex operator relations, expressing solutions as contour integrals, and confirming previous functional integral proposals.

Contribution

It introduces a recursive relation for the S-matrix of Liouville theory and expresses solutions as contour integrals, advancing the understanding of its scattering properties.

Findings

Recursive relation for the S-matrix derived

Solutions expressed as contour integrals

Agreement with previous functional integral proposals

Abstract

We analyze the relation between the vertex operators of the in and out fields in Liouville theory. This is used to derive equations for the S-matrix, from which a recursive relation for the normal symbol of the S-matrix for discrete center-of-mass momenta is obtained. Its solution is expressed as multiple contour-integrals of a generalized Dotsenko-Fateev type. This agrees with the functional integral representation of the scattering matrix of Liouville theory which we had proposed previously.