Cyclic quantum Teichmüller theory

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 \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 prove that it 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]. We provide a geometric method to decompose the space of quantum states into irreducible modules of the Chekhov--Fock algebra. The reduced version of quantum intertwiner conjecturally coincides with the Bonahon--Liu intertwiner [BL07].