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

TL;DR

This paper introduces a new family of invariants for hyperbolic 3-manifolds with geodesic boundary, derived from the modular double of quantum groups, which decay exponentially with volume and relate to Reidemeister torsion.

Contribution

It defines Turaev-Viro type invariants from the modular double of U_q(sl(2,R)) and proves their exponential decay related to hyperbolic volume and Reidemeister torsion.

Findings

Invariants decay exponentially with hyperbolic volume

The invariants relate to the 1-loop term and Reidemeister torsion

Provides a new quantum invariant framework for hyperbolic 3-manifolds

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.