On the quantum Teichmüller invariants of fibred cusped 3-manifolds
This paper describes a relation between the Quantum Hyperbolic Invariants (QHI) defined by the authors [Topology 43, No. 6, 1373--1423 (2004; Zbl 1065.57008); Geom. Topol. 9, 493--569 (2005; Zbl 1093.57005); Algebr. Geom. Topol. 15, No. 4, 1983--2063 (2015; Zbl 1335.57017)] and the local representations of the quantum Teichmüller spaces defined by \textit{H. Bai}, \textit{F. Bonahon} and \textit{X. Liu} [``Local representations of the quantum Teichmüller spaces'', Preprint, \url{arXiv:0707.2151}]. Given \(q\) a root of unity of odd order, \(N>1\) the QHI \(\mathcal{H}_N(M, \tau, \kappa)\) are invariants of cusped \(3\)-manifolds \(M\) equipped with an augmented character \(\tau\) of a representation \(\pi_1(M)\to \mathrm{PSL}_2(\mathbb{\mathbb{C}})\) and a class \(\kappa \in \mathrm{H}^1(\partial \overline{M}; \mathbb{C}^*)\) such that \(\kappa^N\) corresponds to the augmentation. The pair \((\tau, \kappa)\) can be encoded combinatorially by an ideal triangulation \(T\), a pre-branching \(b\) and a set \(\mathbf{w}\) of weight associated to the tetrahedron and the QHI is defined as a product \(\mathcal{H}_N(M, \tau, \kappa):= \alpha_N(T,b,\mathbf{w}) \mathcal{H}_N^{red}(T, b, \mathbf{w})\) where \(\alpha_N\) is a normalization factor and \(\mathcal{H}_N^{red}\) is the contraction of some Kashaev \(6j\)-symbols. The state sum \(\mathcal{H}_N^{red}\) is an invariant of \((M, \tau , \kappa)\) equipped with the additional datum of a non-ambiguous structure [the authors, Quantum Topol. 8, No. 4, 749--846 (2017; Zbl 1402.57014)]. When \(M\) is the mapping torus of a pseudo Anosov mapping class \(\phi\) acting on an open surface \(S\), and \(T\) is induced by a triangulation \(T_{C_{\phi}}\) of the associated mapping cylinder \(C_{\phi}\), one can associate an operator \(\mathcal{H}_N^{red}(T_{C_{\phi}}, \widetilde{b}, \mathbf{w})\) to the decorated mapping cylinder such that \(\mathcal{H}_N^{red}(T, b, \mathbf{w})= \mathrm{Tr} \left( \mathcal{H}_N^{red}(T_{C_{\phi}}, \widetilde{b}, \mathbf{w}) \right)\). On the other hand, given a triangulation \(\lambda\) of \(S\), the Chekhov-Fock algebra \(\mathcal{T}_S^q\) admits a family of finite dimensional representations \(\rho\) named \textit{local}. For two pairs \((\lambda, \rho)\) and \((\lambda', \rho')\), the change of coordinates defines a set \(\mathcal{L}_{\lambda \lambda'}^{\rho \rho'}\) of intertwiners between the two representations which is a \(\mathrm{H}_1(S; \mathbb{Z}/N\mathbb{Z})\) torsor. Let \(\lambda, \lambda'\) be the triangulations of \(S\times \{0\}\) and \(S\times \{1\}\) obtained by restriction of a triangulation \(T_{C_{\phi}}\) of the mapping cylinder. The authors associate to the datum \((T_{C_{\phi}}, b, \mathbf{w})\) two local representations \(\rho\) and \(\rho'\) of the Chekhov-Fock algebras of \(\lambda\) and \( \lambda'\). The main result of the paper is that the transposed \(\left( \mathcal{H}_N^{red}(T_{C_{\phi}}, \widetilde{b}, \mathbf{w})\right)^T \in\mathcal{L}_{\lambda \lambda'}^{\rho \rho'} \) is an intertwiner and that the choice of the weight \(\kappa\) is related explicitly to the free transitive action of \(\mathrm{H}_1(S; \mathbb{Z}/N\mathbb{Z})\).
scientific article; zbMATH DE number 6982133
scientific article