Quantum Liouville theory versus quantized Teichmüller spaces

TL;DR

The paper proves that Liouville conformal blocks and the Hilbert spaces from quantized Teichmüller spaces carry equivalent mapping class group representations, establishing a geometric bridge between quantum Liouville theory and quantum geometry of Riemann surfaces. It constructs the quantum Teichmüller framework with length operators and a Moore-Seiberg-compatible automorphism structure, showing that the fusion and braiding data match Liouville theory under a precise identification of weights. This yields a chiral-geometric realization of Liouville theory that extends to higher genus and clarifies how geodesic data encode Liouville representations. The result provides a foundational link for holographic interpretations and for understanding quantum gravity in low dimensions through moduli-space quantization.

Abstract

This note announces the proof of a conjecture of H. Verlinde, according to which the spaces of Liouville conformal blocks and the Hilbert spaces from the quantization of the Teichmüller spaces of Riemann surfaces carry equivalent representations of the mapping class group. This provides a basis for the geometrical interpretation of quantum Liouville theory in its relation to quantized spaces of Riemann surfaces.

Figures (5)

• Figure 1: Triangulation of the once-punctured torus.
• Figure 2: The elementary move between two triangulations
• Figure 3: The generators of the Moore-Seiberg groupoid
• Figure 4: Fat graph for the sphere with four holes (shaded) with numbered edges.
• Figure 5: Neighborhood $N$ and the part of the fat graph $\Gamma$ contained in $N$