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From Liouville Theory to the Quantum Geometry of Riemann Surfaces

J. Teschner

TL;DR

This work proposes a geometric interpretation of the full non-chiral Liouville correlation functions through the quantum geometry of Teichmüller spaces. By connecting Liouville conformal blocks to the quantized Teichmüller framework, it establishes a coherent-state-like picture in which length operators correspond to classical accessory parameters and Liouville blocks become eigenfunctions of the quantum geometry. The manuscript develops the Liouville fusion and Moore-Seiberg structures, then extends them to the quantum Teichmüller theory via Penner coordinates, yielding a modular functor with explicit A-, B-, and S-move kernels that reproduce Liouville data in the semi-classical limit. The proposed program provides a pathway to understanding 2D quantum gravity and hints at insights into 3D quantum gravity through a unified Liouville–Teichmüller formalism, with concrete predictions in degeneration limits and a set of rigorous exercises to consolidate the framework.

Abstract

The aim of this note is to propose an interpretation for the full (non-chiral) correlation functions of the Liouville conformal field theory within the context of the quantization of spaces of Riemann surfaces.

From Liouville Theory to the Quantum Geometry of Riemann Surfaces

TL;DR

This work proposes a geometric interpretation of the full non-chiral Liouville correlation functions through the quantum geometry of Teichmüller spaces. By connecting Liouville conformal blocks to the quantized Teichmüller framework, it establishes a coherent-state-like picture in which length operators correspond to classical accessory parameters and Liouville blocks become eigenfunctions of the quantum geometry. The manuscript develops the Liouville fusion and Moore-Seiberg structures, then extends them to the quantum Teichmüller theory via Penner coordinates, yielding a modular functor with explicit A-, B-, and S-move kernels that reproduce Liouville data in the semi-classical limit. The proposed program provides a pathway to understanding 2D quantum gravity and hints at insights into 3D quantum gravity through a unified Liouville–Teichmüller formalism, with concrete predictions in degeneration limits and a set of rigorous exercises to consolidate the framework.

Abstract

The aim of this note is to propose an interpretation for the full (non-chiral) correlation functions of the Liouville conformal field theory within the context of the quantization of spaces of Riemann surfaces.

Paper Structure

This paper contains 19 sections, 4 theorems, 60 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Liouville conformal field theory
  3. The conformal Ward identities
  4. Sewing of conformal blocks
  5. Feynman rules for the construction of conformal blocks
  6. The Moore-Seiberg groupoid
  7. Representation of the Moore-Seiberg groupoid on ${\mathcal{H}}^{\rm L}(\Sigma)$
  8. Relations between classical Liouville theory and Teichmüller theory
  9. The semi-classical limit
  10. Energy-momentum tensor and accessory parameters
  11. Geometrical interpretation
  12. Classical and quantized Teichmüller spaces
  13. The Fenchel-Nielsen coordinates
  14. The Moore-Seiberg groupoid
  15. Quantization of the Teichmüller spaces
  16. ...and 4 more sections

Key Result

Theorem 3.1

(Takhtajan-Zograf)TZ1

Figures (4)

  • Figure 1: The A- and B-moves
  • Figure 2: A marked pair of pants
  • Figure 3: The S-move
  • Figure 4: Action of a Dehn-twist on an annulus

Theorems & Definitions (7)

  • Remark 3.1
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Conjecture 5.1
  • Remark 1