Liouville central charge in quantum Teichmuller theory

TL;DR

This paper explores the quantum Teichmuller theory, explicitly computing the projective factor related to the Liouville central charge for a genus three surface with one puncture, revealing deep connections between geometry and quantum algebra.

Contribution

It explicitly calculates the projective factor in quantum Teichmuller theory for a genus three surface, linking it to the Liouville central charge.

Findings

Projective representation of mapping class groups is established.

Explicit calculation of the projective factor as exponential of Liouville central charge.

Deepens understanding of quantum geometry of punctured surfaces.

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.