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arXiv · 2511.20953

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

Tianyue Liu·Shuang Ming·Xin Sun·Baojun Wu·Tian Yang
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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.

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Tianyue Liu, Shuang Ming, Xin Sun, Baojun Wu, Tian Yang. 2025-11-26. Asymptotics of $b$-$6j$ symbols and anti-de Sitter tetrahedra. https://arxiv.org/abs/2511.20953

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