Turaev-Viro invariant from the modular double of ${\mathrm{U}}_q\mathfrak{sl}(2;\mathbb{R})$

Abstract

We define a family of Turaev-Viro type invariants of hyperbolic $3$-manifolds with totally geodesic boundary from the $6j$-symbols of the modular double of ${\mathrm{U}}_q\mathfrak{sl}(2;\mathbb{R})$, and prove that these invariants decay exponentially with the rate the hyperbolic volume of the manifolds and with the $1$-loop term the adjoint twisted Reidemeister torsion of the double of the manifolds.

111 pages, 26 figures. We add the proof of the absolute convergence of the invariant for all $b$ in $(0,1)$, where previously we only had the convergence for sufficiently small $b.$

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