The Virasoro structure and the scattering matrix for Liouville conformal field theory

TL;DR

This paper constructs a Virasoro algebra representation in Liouville conformal field theory, demonstrating the diagonalization of the Hamiltonian and the scattering matrix, and extending primary fields analytically.

Contribution

It introduces a new representation of the Virasoro algebra in Liouville theory and proves the diagonalization of the Hamiltonian and scattering matrix, confirming conjectures from physics.

Findings

Virasoro algebra representation constructed in Liouville theory

Hamiltonian diagonalization via Virasoro algebra

Scattering matrix shown to be diagonal and primary fields analytically extended

Abstract

In this work, we construct a representation of the Virasoro algebra in the canonical Hilbert space associated to Liouville conformal field theory. The study of the Virasoro operators is performed through the introduction of a new family of Markovian dynamics associated to holomorphic vector fields defined in the disk. As an output, we show that the Hamiltonian of Liouville conformal field theory can be diagonalized through the action of the Virasoro algebra. This enables to show that the scattering matrix of the theory is diagonal and that the family of the so-called primary fields (which are eigenvectors of the Hamiltonian) admits an analytic extension to the whole complex plane, as conjectured in the physics literature.