Quantum Liouville theory versus quantized Teichm\"uller spaces

TL;DR

This paper proves a conjecture linking quantum Liouville theory and the quantization of Teichmüller spaces, showing they produce equivalent mapping class group representations, thus clarifying their geometric relationship.

Contribution

It establishes the equivalence of representations of the mapping class group from quantum Liouville theory and quantized Teichmüller spaces, confirming a conjecture by H. Verlinde.

Findings

Spaces of Liouville conformal blocks and quantized Teichmüller spaces have equivalent mapping class group representations.

Provides a geometric interpretation of quantum Liouville theory.

Supports the connection between conformal field theory and quantum geometry.

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 Teichm\"uller 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.