带尖点的三维流形的$\boldsymbol{\rm U}_{q\tilde q}\frak{sl}(2;\boldsymbol{\rm R})$ Turaev-Viro不变量
$\mathrm {U}_{q\tilde q}\mathfrak{sl}(2;\mathbb R)$ Turaev-Viro invariants for cusped $3$-manifolds
AI总结：
该研究拓展了双曲三维流形的Turaev-Viro不变量，构造出带尖点三维流形的$\boldsymbol{\rm U}_{q\tilde q}\frak{sl}(2;\boldsymbol{\rm R})$型不变量，并证明其衰减率由流形双曲体积决定。
AI中文摘要：
我们为带尖点端点的双曲三维流形定义了一族Turaev-Viro型不变量，拓展了文献[LMSWY]中针对带全测地边界的双曲三维流形引入的不变量。这些不变量由我们所称的理想$\boldsymbol{\rm U}_{q\tilde q}\frak{sl}(2;\boldsymbol{\rm R})$-$6j$符号构造，该符号是与$\boldsymbol{\rm U}_q\frak{sl}(2;\boldsymbol{\rm R})$模双的正表示相关的$6j$符号的变体。我们还证明这些不变量呈指数衰减，其指数衰减率由该流形的双曲体积决定。
英文摘要：
We define a family of Turaev-Viro type invariants for hyperbolic $3$-manifolds with cusp ends, extending the invariants introduced in \cite{LMSWY} for hyperbolic $3$-manifolds with totally geodesic boundary. These invariants are constructed from what we call the ideal $\mathrm{U}_{q\tilde q}\mathfrak{sl}(2;\mathbb R)$-$6j$ symbols, which are variants of the $6j$-symbols associated with the positive representations of the modular double of $\mathrm{U}_q\mathfrak{sl}(2;\mathbb R)$. We also prove that these invariants decay exponentially, with the exponential decay rate determined by the hyperbolic volume of the manifold.
