Cyclic quantum Teichm\"uller theory

TL;DR

This paper constructs explicit finite-dimensional projective representations of the dotted Ptolemy groupoid at roots of unity, linking quantum Teichmüller theory with quantum cluster algebras and providing geometric decompositions and intertwiners.

Contribution

It introduces a new explicit construction of quantum representations at roots of unity, reinterprets parameter relations for the cyclic quantum dilogarithm, and connects these to quantum cluster algebra structures.

Findings

Constructed finite-dimensional projective representations of the dotted Ptolemy groupoid.

Reinterpreted relations among cyclic quantum dilogarithm parameters to satisfy the pentagon identity.

Provided geometric decomposition of quantum state space into irreducible modules.

Abstract

Based on the pioneering ideas of Kashaev [Kas98,Kas00], we present a fully explicit construction of a finite-dimensional projective representation of the dotted Ptolemy groupoid when the quantum parameter $q$ is a root of unity, which reproduces the central charge of the $SU(2)$ Wess--Zumino--Witten model. A basic ingredient is the cyclic quantum dilogarithm [FK94]. A notable contribution of this work is a reinterpretation of the relations among the parameters in the cyclic quantum dilogarithm to ensure its pentagon identity in terms of the mutations of coefficients. In particular, we find the dual roles of these parameters: as coefficients in quantum cluster algebras and as the central characters of quantum cluster variables. We also provide a geometric method to decompose the space of quantum states into irreducible modules of the Chekhov--Fock algebra. We introduce two versions of quantum intertwiners associated with a mapping class: on the entire representation space and on each irreducible component, each being an explicit composite of cyclic quantum dilogarithm operators. We prove that the former gives an intertwiner of local representations of quantum Teichm\"uller space in the sense of Bai--Bonahon--Liu [BBL07], and also coincides with the transpose of the reduced quantum hyperbolic operator of Baseilhac--Benedetti [BB18]. The mutation relation of coefficients is equivalent to the quantum gluing equation. The irreducible intertwiner conjecturally coincides with the Bonahon--Liu intertwiner [BL07], and we give a partial evidence.