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Liouville central charge in quantum Teichmuller theory

R. M. Kashaev

TL;DR

The work addresses the projective representation of mapping class groups in quantum Teichmüller theory based on Penner coordinates and proves that the resulting projective factor for the genus-3, one-puncture surface is the exponential of the Liouville central charge. Using the non-compact quantum dilogarithm as a building block, the authors construct unitary representations from decorated ideal triangulations and compute the central extension by exploiting lantern and chain relations in the mapping class group. They provide explicit Dehn-twist operators for $\Sigma_{3,1}$ and show that, after removing Gaussian, homology-related degrees of freedom, the projective factor satisfies $\xi_D = -\xi_F = -\zeta^{-72} = \exp(i\pi c_L)$ with $c_L = 1+6(\lambda+\lambda^{-1})^2 \pmod{2}$, aligning with Virasoro conformal blocks in Liouville theory. This establishes a direct link between quantum Teichmüller theory, $SL(2,\mathbb R)$ Chern–Simons theory, and Liouville/CFT, reinforcing the interpretation of the quantum Teichmüller Hilbert space as a space of Virasoro conformal blocks.

Abstract

In the quantum Teichmuller theory, based on Penner coordinates, the mapping class groups of punctured surfaces are represented projectively. The case of a genus three surface with one puncture is worked out explicitly. The projective factor is calculated. It is given by the exponential of the Liouville central charge.

Liouville central charge in quantum Teichmuller theory

TL;DR

The work addresses the projective representation of mapping class groups in quantum Teichmüller theory based on Penner coordinates and proves that the resulting projective factor for the genus-3, one-puncture surface is the exponential of the Liouville central charge. Using the non-compact quantum dilogarithm as a building block, the authors construct unitary representations from decorated ideal triangulations and compute the central extension by exploiting lantern and chain relations in the mapping class group. They provide explicit Dehn-twist operators for and show that, after removing Gaussian, homology-related degrees of freedom, the projective factor satisfies with , aligning with Virasoro conformal blocks in Liouville theory. This establishes a direct link between quantum Teichmüller theory, Chern–Simons theory, and Liouville/CFT, reinforcing the interpretation of the quantum Teichmüller Hilbert space as a space of Virasoro conformal blocks.

Abstract

In the quantum Teichmuller theory, based on Penner coordinates, the mapping class groups of punctured surfaces are represented projectively. The case of a genus three surface with one puncture is worked out explicitly. The projective factor is calculated. It is given by the exponential of the Liouville central charge.

Paper Structure

This paper contains 10 sections, 1 theorem, 56 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The non-compact quantum dilogarithm
  3. Notation
  4. Some algebraic relations
  5. Projective representations of mapping class groups
  6. The case of $\Sigma_{3,1}$
  7. Realization of the Dehn twists
  8. Relation to the Liouville central charge
  9. Summary
  10. Acknowledgements

Key Result

Theorem 1

For any compact oriented surface $\Sigma$ of genus $g>1$, the mapping class group $M_\Sigma$ has the following presentation:

Figures (8)

  • Figure 1: Elementary transformation $\rho^t$ consisting of changing the marked corner of triangle $t$.
  • Figure 2: Elementary transformation $\omega_e$ consisting of replacing edge $e$ by the complementary edge $e'$.
  • Figure 3: Resolution from $a$ to $b$ at $p$.
  • Figure 4: The Dehn twists along these curves satisfy the chain relation.
  • Figure 5: The Dehn twist $\tau'\equiv D_\alpha(\tau)$ of d.i.t. $\tau$ of an annulus along contour $\alpha$ is mapped by the elementary transformation $\omega_e$ back to $\tau=\omega_e(\tau')$.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1: Gervais Gervais