A lecture on the Liouville vertex operators

TL;DR

This paper revisits the construction of exponential fields in quantum Liouville theory, providing a simplified derivation of key properties and confirming a conjectured formula through explicit calculations.

Contribution

It introduces a free-field approach to construct chiral vertex operators and derives their fusion and braid relations, simplifying the verification of locality and crossing symmetry.

Findings

Derived fusion and braid relations for chiral vertex operators

Simplified proof of locality and crossing symmetry

Constructive derivation of Dorn/Otto and Zamolodchikov formula

Abstract

We reconsider the construction of exponential fields in the quantized Liouville theory. It is based on a free-field construction of a continuous family or chiral vertex operators. We derive the fusion and braid relations of the chiral vertex operators. This allows us to simplify the verification of locality and crossing symmetry of the exponential fields considerably. The calculation of the matrix elements of the exponential fields leads to a constructive derivation of the formula proposed by Dorn/Otto and the brothers Zamolodchikov.