Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions
Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions
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发表时间：
2014-05
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通讯作者：
Liang Kong;X. Wen
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作者：
Liang Kong;X. Wen
引力反常可以在一维拓扑有序态的边界上实现，并且可以用一维拓扑有序来描述。在这篇文章中，我们试图发展一个关于任意维的拓扑序和引力反常的一般理论。(1)我们引入了BF范畴的概念来描述拓扑激发的编织和融合性质，它可以是点状、弦状等。BF范畴的一个子集--闭合BF范畴--对任何维度的拓扑序进行分类，而一般的BF范畴则对在一维有间隙的量子液体的边界上出现的(潜在的)反常拓扑序进行分类。(2)引入基于张量网络的拓扑路积分来实现这些拓扑序。(3)玻色拓扑序有一个重要的拓扑不变量：具有不同度规的闭空间的模空间上的简并基态向量丛。它们可以完全刻画拓扑序。(4)我们猜想拓扑序有可开边界当且仅当上述向量丛是平坦的。(5)我们发现了一个全息现象，即每一个具有可开边界的拓扑序都可以由边界的知识唯一地确定。因此，不同维度的BF范畴形成了一个(么半群)余链复合体，揭示了不同维度的拓扑序和引力反常的结构和关系。我们还研究了最简单的一类没有非平凡拓扑激发的玻色拓扑序。我们发现，这种拓扑序在2+1D(无间隙边)中形成$\mathbb{Z}$类，在4+1D(具有可开边界)中形成$\mathbb{Z}_2$类，在6+1D(无间隙边界)中形成$\mathbb{Z}\Oplus\mathbb{Z}$类。
Gravitational anomalies can be realized on the boundary of topologically ordered states in one higher dimension and are described by topological orders in one higher dimension. In this paper, we try to develop a general theory for both topological order and gravitational anomaly in any dimensions. (1) We introduce the notion of BF category to describe the braiding and fusion properties of topological excitations that can be point-like, string-like, etc. A subset of BF categories -- closed BF categories -- classify topological orders in any dimensions, while generic BF categories classify (potentially) anomalous topological orders that can appear at a boundary of a gapped quantum liquid in one higher dimension. (2) We introduce topological path integral based on tensor network to realize those topological orders. (3) Bosonic topological orders have an important topological invariant: the vector bundles of the degenerate ground states over the moduli spaces of closed spaces with different metrics. They may fully characterize topological orders. (4) We conjecture that a topological order has a gappable boundary iff the above mentioned vector bundles are flat. (5) We find a holographic phenomenon that every topological order with a gappable boundary can be uniquely determined by the knowledge of the boundary. As a consequence, BF categories in different dimensions form a (monoid) cochain complex, that reveals the structure and relation of topological orders and gravitational anomalies in different dimensions. We also studied the simplest kind of bosonic topological orders that have no non-trivial topological excitations. We find that this kind of topological orders form a $\mathbb{Z}$ class in 2+1D (with gapless edge), a $\mathbb{Z}_2$ class in 4+1D (with gappable boundary), and a $\mathbb{Z}\oplus \mathbb{Z}$ class in 6+1D (with gapless boundary).