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Liouville theory revisited

J. Teschner

Abstract

We try to develop a coherent picture on Liouville theory as a two-dimensional conformal field theory that takes into account the perspectives of path-integral approach, bootstrap, canonical quantization and operator approach. To do this, we need to develop further some of these approaches. This includes in particular a construction of general exponential field operators from a set of covariant chiral operators. The latter are shown to satisfy braid relations that allow one to prove the locality of the former.

Liouville theory revisited

Abstract

We try to develop a coherent picture on Liouville theory as a two-dimensional conformal field theory that takes into account the perspectives of path-integral approach, bootstrap, canonical quantization and operator approach. To do this, we need to develop further some of these approaches. This includes in particular a construction of general exponential field operators from a set of covariant chiral operators. The latter are shown to satisfy braid relations that allow one to prove the locality of the former.

Paper Structure

This paper contains 107 sections, 1 theorem, 284 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Aims and scope
  4. Overview
  5. Notational conventions
  6. The DOZZ proposal
  7. Conformal symmetry and three point function
  8. Descendants
  9. The DOZZ-formula
  10. Path integral approach
  11. Path integral on the Riemann sphere
  12. Scaling behavior
  13. Relation to free field?
  14. Evaluation of the residues
  15. Continuation in $s$
  16. ...and 92 more sections

Key Result

Theorem 1

Null vector decoupling: Let $i,j,k\in\{1,2,3\}$ be chosen such that $j\neq i$, $k\neq i$, $j\neq k$. Assume that (i) $\alpha_i=\alpha_{r,s}$, and (ii) $\xi_i$ lies in the singular subspace ${\mathcal{S}}_{r,s}$. One then finds that $\rho^{\alpha_3,\alpha_2,\alpha_1}_{z_3,z_2,z_1} (\xi^{}_3,\xi^{}_2,

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Remark 6
  • Definition 1
  • Remark 7
  • Remark 8
  • ...and 2 more