We push forward the investigation of holographic dualities in 3D quantum gravity formulated as a topological quantum field theory, studying the correspondence between boundary and bulk structures. Working with the Ponzano–Regge topological state-sum model defining an exact discretization of 3D quantum gravity, we analyse how the partition function for a solid twisted torus depends on the boundary quantum state. This configuration is relevant to the AdS $$_{3}$$ 3 /CFT $$_{2}$$ 2 correspondence. We introduce boundary spin network states with coherent superposition of spins on a square lattice on the boundary surface. This allows for the first exact analytical calculation of Ponzano–Regge amplitudes with extended 2D boundary (beyond the single tetrahedron). We get a regularized finite truncation of the BMS character formula obtained from the one-loop perturbative quantization of 3D gravity. This hints towards the existence of an underlying symmetry and the integrability of the theory for finite boundary at the quantum level for coherent boundary spin network states.

中文翻译：

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非微扰 3D 量子引力：量子边界态和精确配分函数

我们推进了 3D 量子引力中全息二元性的研究，该研究被制定为拓扑量子场论，研究边界和体结构之间的对应关系。使用定义 3D 量子引力精确离散化的 Ponzano-Regge 拓扑状态和模型，我们分析了固体扭曲环面的配分函数如何取决于边界量子态。此配置与 AdS $$_{3}$$ 3 /CFT $$_{2}$$ 2 对应关系相关。我们引入了边界自旋网络状态，在边界面上的方形晶格上具有自旋的相干叠加。这允许对具有扩展 2D 边界（超出单个四面体）的 Ponzano-Regge 振幅进行第一次精确分析计算。我们得到了从 3D 重力的单循环微扰量化中获得的 BMS 字符公式的正则化有限截断。这暗示了潜在对称性的存在以及相干边界自旋网络状态量子级有限边界理论的可积性。