Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A lecture on the Liouville vertex operators

J. Teschner

TL;DR

This work develops a rigorous free-field construction of Liouville vertex operators by introducing a continuous family of chiral vertex operators built from a chiral free field and screening charges. It derives the fusion and braid relations of these chiral operators, proving locality and crossing symmetry for the Liouville vertex operators and providing a constructive route to the DOZZ three-point function. The approach centers on explicit matrix elements, conformal blocks, and functional equations, tying these to Weyl-type algebras, quantum dilogarithms, and b-Racah–Wigner data to realize a complete operator algebra for quantum Liouville theory. The resulting framework not only clarifies the operator product expansions and monodromies of Liouville conformal blocks but also furnishes a concrete determination of Liouville structure constants, with implications for the exact solution of the theory and its boundary variants.

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.

A lecture on the Liouville vertex operators

TL;DR

This work develops a rigorous free-field construction of Liouville vertex operators by introducing a continuous family of chiral vertex operators built from a chiral free field and screening charges. It derives the fusion and braid relations of these chiral operators, proving locality and crossing symmetry for the Liouville vertex operators and providing a constructive route to the DOZZ three-point function. The approach centers on explicit matrix elements, conformal blocks, and functional equations, tying these to Weyl-type algebras, quantum dilogarithms, and b-Racah–Wigner data to realize a complete operator algebra for quantum Liouville theory. The resulting framework not only clarifies the operator product expansions and monodromies of Liouville conformal blocks but also furnishes a concrete determination of Liouville structure constants, with implications for the exact solution of the theory and its boundary variants.

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.

Paper Structure

This paper contains 32 sections, 1 theorem, 115 equations.

Table of Contents

  1. Introduction
  2. Definition and main properties of chiral vertex operators
  3. The chiral free field
  4. Conformal symmetry
  5. Building blocks
  6. Chiral vertex operators
  7. Conformal covariance
  8. More advanced properties of ${\mathsf h}_{p_{\mathfrak 2}p_{\mathfrak 1}}^{\alpha}(w)$
  9. Matrix elements
  10. Auxiliary fields
  11. Strategy
  12. Calculation of the conformal blocks
  13. Functional equations
  14. Braiding
  15. Operator ordering, I
  16. ...and 17 more sections

Key Result

Theorem 1

PT2 The following two sets of distributions form bases for $L^2({\mathbb R})$ in the sense of generalized functions. We have the relations

Theorems & Definitions (1)

  • Theorem 1