Quantum invariants of closed framed $3$-manifolds based on ideal triangulations
Quantum invariants of closed framed $3$-manifolds based on ideal triangulations
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发表时间：
2022-09
影响因子：
8.6
通讯作者：
S. Mihalache;Sakie Suzuki;Yuji Terashima
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文献类型：
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作者：
S. Mihalache;Sakie Suzuki;Yuji Terashima
我们构建了一种新型封闭框架 3 美元流形的量子不变量，其第一个贝蒂数消失。该不变量是为任何有限维 Hopf 代数（例如小量子群）定义的，并且基于理想三角剖分。我们使用海森堡双元的规范元素，它满足五边形方程，以及 R. Benedetti 和 C. Petronio 引入的 $3$ 流形的图形表示。结构简单，易于直观理解；五边形方程反映了理想三角剖分的 Pachner $(2,3)$ 移动，而 Hopf 代数的非对合性反映了框架。对于合合 Hopf 代数，不变量简化为闭合梳状 $3$ 流形的不变量。对于对合幺模共模 Hopf 代数，不变量简化为我们之前论文中介绍的闭合 $3$-流形的拓扑不变量。在本文中，我们在对称关键类别中使用更普遍的 Hopf 幺半群来形式化构造，并使用张量网络进行计算。
We construct a new type of quantum invariant of closed framed $3$-manifolds with the vanishing first Betti number. The invariant is defined for any finite dimensional Hopf algebra, such as small quantum groups, and is based on ideal triangulations. We use the canonical element of the Heisenberg double, which satisfies a pentagon equation, and graphical representations of $3$-manifolds introduced by R. Benedetti and C. Petronio. The construction is simple and easy to be understood intuitively; the pentagon equation reflects the Pachner $(2,3)$ move of ideal triangulations and the non-involutiveness of the Hopf algebra reflects framings. For an involutory Hopf algebra, the invariant reduces to an invariant of closed combed $3$-manifolds. For an involutory unimodular counimodular Hopf algebra, the invariant reduces to the topological invariant of closed $3$-manifolds which is introduced in our previous paper. In this paper we formalize the construction using more generally a Hopf monoid in a symmetric pivotal category and use tensor networks for calculations.