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

TL;DR

This paper reviews the connection between quantum Liouville theory and quantized Teichmüller spaces, exploring their Hilbert space constructions and the conjectured equivalence of their mapping class group representations.

Contribution

It discusses the construction of conformal blocks and the quantization of Teichmüller spaces, providing insights into the verification of Verlinde's conjecture on their equivalence.

Findings

Hilbert spaces assigned to Riemann surfaces in both theories

Representation of the mapping class group in each theory

Partial verification of Verlinde's conjecture

Abstract

We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichm\"uller 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.