${\mathrm{U}}_{q\tilde{q}}\mathfrak{sl}(2;\mathbb{R})$ Turaev-Viro invariants for cusped $3$-manifolds

Abstract

We define a family of Turaev-Viro type invariants for hyperbolic $3$-manifolds with cusp ends, extending the invariants introduced in \cite{LMSWY} for hyperbolic $3$-manifolds with totally geodesic boundary. These invariants are constructed from what we call the ideal ${\mathrm{U}}_{q\tilde{q}}\mathfrak{sl}(2;\mathbb{R})$-$6j$ symbols, which are variants of the $6j$-symbols associated with the positive representations of the modular double of ${\mathrm{U}}_q\mathfrak{sl}(2;\mathbb{R})$. We also prove that these invariants decay exponentially, with the exponential decay rate determined by the hyperbolic volume of the manifold.

37 pages, 1 figure