Asymptotics of quantum $6j$-symbols and generalized hyperbolic tetrahedra

TL;DR

This paper links quantum 6j-symbols to the geometry of various tetrahedra, revealing their asymptotic behavior and introducing the novel phenomenon of exponential decay linked to negative volumes of generalized hyperbolic tetrahedra.

Contribution

It classifies the geometric interpretation of quantum 6j-symbols and establishes their asymptotics in relation to generalized hyperbolic tetrahedra volumes.

Findings

Quantum 6j-symbols correspond to dihedral angles of specific tetrahedra.

The exponential growth rate of 6j-symbols equals the volume of a generalized hyperbolic tetrahedron.

Negative volumes can cause exponential decay of the 6j-symbols.

Abstract

We establish the geometry behind the quantum $6j$-symbols under only the admissibility conditions as in the definition of the Turaev-Viro invariants of $3$-manifolds. As a classification, we show that the $6$-tuples in the quantum $6j$-symbols give in a precise way to the dihedral angles of (1) a spherical tetrahedron, (2) a generalized Euclidean tetrahedron, (3) a generalized hyperbolic tetrahedron or (4) in the degenerate case the angles between four oriented straight lines in the Euclidean plane. We also show that for a large proportion of the cases, the $6$-tuples always give the dihedral angles of a generalized hyperbolic tetrahedron and the exponential growth rate of the corresponding quantum $6j$-symbols equals the suitably defined volume of this generalized hyperbolic tetrahedron. It is worth mentioning that the volume of a generalized hyperbolic tetrahedron can be negative, hence the corresponding sequence of the quantum $6j$-symbols could decay exponentially. This is a phenomenon that has never been aware of before.