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On the relation between quantum Liouville theory and the quantized Teichm"uller spaces

J. Teschner

TL;DR

This work surveys the deep link between quantum Liouville theory and the quantization of Teichmüller spaces, framing both as Hilbert-space assignments with mapping class group actions and proposing a precise Verlinde-type equivalence. It develops the Moore-Seiberg formalism for Liouville blocks and the Penner–Fock quantization of Teichmüller spaces, introducing length operators and their spectra as a bridge between the two pictures. By constructing representations of the Moore-Seiberg groupoid on both sides and showing a matching kernel structure when a Liouville parameter a matches a Teichmüller length via a Q–dependent relation, it provides a pathway to proving equivalence of the two quantizations. The final perspective hints at a coherent-state viewpoint where Liouville blocks implement a basis change to length-diagonal states, offering a unified geometric interpretation of the quantum theory of 2D gravity and moduli spaces.

Abstract

We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichmüller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verification of this conjecture.

On the relation between quantum Liouville theory and the quantized Teichm"uller spaces

TL;DR

This work surveys the deep link between quantum Liouville theory and the quantization of Teichmüller spaces, framing both as Hilbert-space assignments with mapping class group actions and proposing a precise Verlinde-type equivalence. It develops the Moore-Seiberg formalism for Liouville blocks and the Penner–Fock quantization of Teichmüller spaces, introducing length operators and their spectra as a bridge between the two pictures. By constructing representations of the Moore-Seiberg groupoid on both sides and showing a matching kernel structure when a Liouville parameter a matches a Teichmüller length via a Q–dependent relation, it provides a pathway to proving equivalence of the two quantizations. The final perspective hints at a coherent-state viewpoint where Liouville blocks implement a basis change to length-diagonal states, offering a unified geometric interpretation of the quantum theory of 2D gravity and moduli spaces.

Abstract

We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichmüller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verification of this conjecture.

Paper Structure

This paper contains 26 sections, 2 theorems, 49 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Teichmüller spaces
  3. Penner coordinates
  4. Symplectic structure
  5. Changing the triangulation
  6. The representation of the mapping class group
  7. Quantization of Teichmüller spaces
  8. Algebra of observables and Hilbert space
  9. Representation of the mapping class group on ${\mathcal{A}}(\Sigma)$
  10. Representation of the mapping class group on ${\mathcal{H}}(\Sigma)$
  11. Liouville theory in the Moore-Seiberg formalism
  12. Holomorphic factorization
  13. The conformal Ward identities
  14. Sewing of conformal blocks
  15. Feynman rules for the construction of conformal blocks
  16. ...and 11 more sections

Key Result

Theorem 1

Ka4 The following set of distributions forms a basis for $L^2({\mathbb R})$ in the sense of generalized functions. We have the relations where the measure $m(p)$ is defined as $m(p)=4\sinh 2\pi bp\,\sinh 2\pi b^{-1}p$.

Figures (12)

  • Figure 1: Triangulation of the once-punctured torus.
  • Figure 2: The labeling of the edges
  • Figure 3: The elementary move between two triangulations
  • Figure 4: The pentagon relation.
  • Figure 5: Action of a Dehn-twist on an annulus.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2