Quantized Geodesic Lengths for Teichm\"uller Spaces: Algebraic Aspects

TL;DR

This paper advances the algebraic understanding of quantized geodesic lengths in Teichmüller spaces, building on quantum trace maps and recursion relations to support the modular functor conjecture in quantum Teichmüller theory.

Contribution

It constructs and analyzes quantized trace-of-monodromy operators using Chekhov-Fock-Goncharov theory, providing new algebraic insights into quantum geodesic length operators.

Findings

Quantized trace-of-monodromy satisfies Teschner's recursion relation.

Quantized trace-of-monodromy for disjoint loops commute strongly.

Supports the algebraic framework for the modular functor conjecture.

Abstract

In 1980's H. Verlinde suggested to construct and use a quantization of Teichm\"uller spaces to construct spaces of conformal blocks for the Liouville conformal field theory. This suggestion led to a mathematical formulation by Fock in 1990's and later by Fock, Goncharov and Shen, called the modular functor conjecture, based on the Chekhov-Fock quantum Teichm\"uller theory. In 2000's, Teschner combined the Chekhov-Fock version and the Kashaev version of quantum Teichm\"uller theory to construct a solution to a modified form of the conjecture. We embark on a direct approach to the conjecture based on the Chekhov-Fock(-Goncharov) theory. We construct quantized trace-of-monodromy along simple loops via Bonahon and Wong's quantum trace maps developed in 2010's, and investigate algebraic structures of them, which will eventually lead to construction and properties of quantized geodesic length operators. We show that a special recursion relation used by Teschner is satisfied by the quantized trace-of-monodromy, and that the quantized trace-of-monodromy for disjoint loops commute in a certain strong sense.