Cyclic quantum Teichm\"uller theory

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 \emph{mutations of coefficients}. In particular, we elucidate the dual roles of these parameters: as coefficients in quantum cluster algebras and as the central characters of quantum cluster variables. We introduce the quantum intertwiner associated with a mapping class as a composite of cyclic quantum dilogarithm operators, whose trace defines a quantum invariant. We provide a canonical method to reduce the entire theory to an irreducible representation of the Chekhov--Fock algebra. The reduced intertwiner corresponds to the one studied by Bonahon--Liu [BL07] and Bonahon--Wong--Yang [BWY21,BWY22], which is now expressed using the cyclic quantum dilogarithm operators. Finally, we describe the connection to the Kashaev's $6j$-symbol [Kas94,Kas95] within our framework, which elucidates the fundamental relationship between our quantum invariant and the Kashaev's link invariant.