Asymptotics of $b$-$6j$ symbols and anti-de Sitter tetrahedra

TL;DR

This work establishes a complete semiclassical picture for the $b$-deformed $6j$ symbols of the modular double of $ ext{U}_q ext{sl}(2,oldsymbol{R})$, showing that all asymptotic regimes correspond to geometric realizations as edge-length data of truncated hyperideal tetrahedra in hyperbolic or anti-de Sitter space. The authors classify sextuples of parameters into three geometric types and derive explicit volume/co-volume linked asymptotics in both edge-length and dihedral-angle scales, using saddle-point analysis and Gram-matrix criteria. A central novelty is the appearance of anti-de Sitter geometry in quantum invariants, alongside a precise CFT interpretation via the Virasoro fusion kernel and a 3D/2D holographic correspondence tying Liouville theory to AdS/Teichmüller data. The results connect fusion kernels, Fenchel-Nielsen coordinates, and geometric invariants, offering new insights into the interplay between quantum groups, hyperbolic/AdS geometry, and holography in three dimensions.

Abstract

In this paper, we study the asymptotics of the $6j$-symbols for the principal series of the modular double of $\mathrm U_q\mathfrak{sl}(2;\mathbb R)$, and of their analytic extension -- what we call the $b$-$6j$ symbols, relating them in various cases to the volume of truncated hyperideal tetrahedra in the hyperbolic and the anti-de Sitter geometry. To the best of our knowledge, this is the first time that the anti-de Sitter geometry appears in the asymptotics of quantum invariants. In addition, based on the connection to conformal field theory, we reveal a correspondence between the edge lengths and the dihedral angles of truncated hyperideal anti-de Sitter tetrahedra and the Fenchel-Nielsen coordinates of hyperbolic four-holed spheres. We also provide a concrete instance of $3D/2D$ holography, in the spirit of the AdS/CFT correspondence.

Figures (11)

• Figure 1: Labeling for the edge lengths: in which way the edge lengths appearing in row $i$ and in column $i$ of the Gram matrix are adjacent to the triangle of truncation $T_i$ of $\Delta$.
• Figure 2: Labeling for the dihedral angles: in which way the dihedral angles appearing in row $i$ and in column $i$ of the Gram matrix are adjacent to the face $F_i$ of $\Delta$.
• Figure 3: On the left we have the hyperbolic four-hole sphere $S,$ where the shaded regions are bounded by $\gamma_i$'s and the shortest geodesic arcs between them, hence are twisted right-angled hyperbolic hexagons as in RY. On the right we have the double $D(\Delta)$ of the truncated hyperideal tetrahedron $\Delta$ in $\mathbb A\mathrm d\mathbb S^3$ obtained by gluing $\Delta$ and its mirror image $\overline\Delta$ together along the shaded triangles of truncation $T_1,\dots, T_4$, which is topologically a handlebody whose boundary is a closed surface of genus $3$.
• Figure 4: Contours $\Gamma$, $\Gamma^*$ and possible zeros and poles (located in the white rays) of the integrand in \ref{['b6j']}.
• Figure 5: Contour $\Gamma$ and possible zeros and poles (located in the white rays) of the integrand in \ref{['b-6j']}, where $\{i_1,i_2,i_3,i_4\}=\{1,2,3,4\}$ with $\mathrm{Im}t_{i_1}\leqslant \mathrm{Im}t_{i_2}\leqslant \mathrm{Im}t_{i_3}\leqslant \mathrm{Im}t_{i_4},$ and $\{j_1,j_2,j_3\}=\{1,2,3\}$ with $0=\mathrm{Im}q_4<\mathrm{Im}q_{j_1}\leqslant \mathrm{Im}q_{j_2}\leqslant \mathrm{Im}q_{j_3}.$

Theorems & Definitions (88)

• Definition 1.1: $b$-$6j$ symbols
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 2.1
• Remark 2.2
• Definition 2.3: Truncated hyperideal tetrahedra in $\mathbb A\mathrm d\mathbb S^3$
• Remark 2.4