arXiv 1502.016902015-02cond-mat.str-elmath.CTmath.QA

拓扑序作为高范畴到中心映射函子的边界-体关系

Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers

Liang Kong, Xiao-Gang Wen, Hao Zheng

备注：: 48 pages, 28 figures. a major revision, including a reformulation of the categorical classification (or the macroscopic definition) of local topological orders, the correction of a mistake in Section 6.1 and a removal of some mathematical proofs, which can be found in [arXiv:1507.00503]

摘要：

本文研究了拓扑序与它们的有隙边界之间的关系。我们建议，对于一个给定的间隙边界理论的体积是唯一的。它实际上是局部拓扑序的微观定义的结果，局部拓扑序是定义在开放圆盘上的（潜在异常的）拓扑序。利用这个唯一性，我们证明了“体积”的概念等价于数学中的中心概念。我们实现这一点，首先引入的概念之间的一个态射的两个局部拓扑秩序相同的尺寸，然后证明，满足相同的普遍性质的中心在数学中的散装。我们提出了$n+$一维局部拓扑序的酉多融合$n$ -范畴的分类（表述为宏观定义），并解释了两个局部拓扑序之间的态射的概念在一些低维情况下与酉monoidal $n$ -函子的概念是相容的.我们还解释了在某些低维情况下，这种分类是兼容的“散装=中心”的结果。最后，我们解释了上述边界-体关系只是一个层次结构的第一层，这个层次结构可以用体（或中心）的函性来概括。这种函性也提供了一些著名的数学结果的物理意义的融合1-范畴。这一工作也可以被看作是系统研究局部拓扑序范畴的第一步，而边界-体关系实际上为这一研究提供了一个有用的工具。

英文摘要：

In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local topological order, which is a (potentially anomalous) topological order defined on an open disk. Using this uniqueness, we show that the notion of "bulk" is equivalent to the notion of center in mathematics. We achieve this by first introducing the notion of a morphism between two local topological orders of the same dimension, then proving that the bulk satisfying the same universal property as that of the center in mathematics. We propose a classification (formulated as a macroscopic definition) of $n+$1D local topological orders by unitary multi-fusion $n$-categories, and explain that the notion of a morphism between two local topological orders is compatible with that of a unitary monoidal $n$-functor in a few low dimensional cases. We also explain in some low dimensional cases that this classification is compatible with the result of "bulk = center". In the end, we explain that above boundary-bulk relation is only the first layer of a hierarchical structure which can be summarized by the functoriality of the bulk (or center). This functoriality also provides the physical meanings of some well-known mathematical results on fusion 1-categories. This work can also be viewed as the first step towards a systematic study of the category of local topological orders, and the boundary-bulk relation actually provides a useful tool for this study.

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