Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC $\mathcal{A}$ representing the Witt class of an anomaly, the article [10] gave a commuting projector model associated to an $\mathcal{A}$-enriched unitary fusion category $\mathcal{X}$ on a 2D boundary of the 3D Walker-Wang model associated to $\mathcal{A}$. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer $Z^\mathcal{A}(\mathcal{X})$ of $\mathcal{A}$ in $Z(\mathcal{X})$.
In this article, we give a rigorous treatment of this 2D boundary model, and we verify this assertion using topological quantum field theory (TQFT) techniques, including skein modules and a certain semisimple algebra whose representation category describes boundary excitations. We also use TQFT techniques to show the 3D bulk point excitations of the Walker-Wang bulk are given by the Müger center $Z_2(\mathcal{A})$, and we construct bulk-to-boundary hopping operators $Z_2(\mathcal{A})\to Z^{\mathcal{A}}(\mathcal{X})$ reflecting how the UMTC of boundary excitations $Z^{\mathcal{A}}(\mathcal{X})$ is symmetric-braided enriched in $Z_2(\mathcal{A})$.
This article also includes a self-contained comprehensive review of the Levin-Wen string net model from a unitary tensor category viewpoint, as opposed to the skeletal $6j$ symbol viewpoint.

中文翻译：

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丰富的弦网模型及其激励

Walker-Wang 模型的边界已用于构建通勤投影模型，该模型实现手征酉模张量类别 (UMTC) 作为边界激励。给定一个表示异常 Witt 类的 UMTC $\mathcal{A}$，文章 [10] 给出了与 $\mathcal{A}$ 丰富的单一融合类别 $\mathcal{X}$ 相关的通勤投影模型在与 $\mathcal{A}$ 关联的 3D Walker-Wang 模型的 2D 边界上。该文章声称边界激励是由 $Z(\mathcal{X}) 中 $\mathcal{A}$ 的富集中心/Müger 集中器 $Z^\mathcal{A}(\mathcal{X})$ 给出的$。
在本文中，我们对这个二维边界模型进行了严格的处理，并使用拓扑量子场论（TQFT）技术验证了这个断言，包括绞纱模块和某种半简单代数，其表示类别描述了边界激励。我们还使用 TQFT 技术来显示 Walker-Wang 体的 3D 体点激励由 Müger 中心 $Z_2(\mathcal{A})$ 给出，并且我们构造体到边界跳跃算子 $Z_2(\mathcal {A})\to Z^{\mathcal{A}}(\mathcal{X})$ 反映边界激励 $Z^{\mathcal{A}}(\mathcal{X})$ 的 UMTC 如何对称-编织丰富$Z_2(\mathcal{A})$。
本文还从酉张量类别的角度（而不是骨架 $6j$ 符号的角度）对 Levin-Wen 弦网模型进行了独立的全面回顾。