拓扑序、辫状统计与五维两类扭曲$BF$理论的混合
Topological Orders, Braiding Statistics, and Mixture of Two Types of Twisted $BF$ Theories in Five Dimensions
摘要：
拓扑序是描述量子多体系统的一个重要范例，而不需要破坏拓扑序。利用拓扑量子场论方法研究了五维时空中的拓扑序，其中\textit{topological excitations}不仅包括点状的\textit{particles}，还包括两类空间扩展的物体：封闭的弦状\textit{loops}和二维封闭的\textit{membranes} .特别是在拓扑序的文献中，膜很少被研究。通过引入高阶规范场，我们构造了包含两种不同类型的$BF$拓扑项和许多扭曲拓扑项的奇异TQFT作用量。规范变换被适当地定义并用于计算TQFT的水平量化和分类。在所有的TQFT中，有些不属于Dijkgraaf-Witten上同调分类。为了表征拓扑序，我们具体地构建了拓扑激发之间的所有编织过程，这导致了由粒子、环和膜的闭合时空轨迹形成的非常奇异的链接。对于每个编织过程，我们构造规范不变的威尔逊算子，并计算相关的编织统计相位。结果，我们得到了链接不变量的表达式，所有这些表达式都具有明显的几何解释。按照温的定义，拓扑序的边界理论表现出引力反常。我们期望本文中的5维拓扑序的表征和分类能够编码4维重力异常信息。进一步考虑，例如，将TQFT放置在具有边界的5D流形上，留给未来的工作。
英文摘要：
Topological orders are a prominent paradigm for describing quantum many-body systems without symmetry-breaking orders. We present a topological quantum field theoretical (TQFT) study on topological orders in five-dimensional spacetime ($5$D) in which \textit{topological excitations} include not only point-like \textit{particles}, but also two types of spatially extended objects: closed string-like \textit{loops} and two-dimensional closed \textit{membranes}. Especially, membranes have been rarely explored in the literature of topological orders. By introducing higher-form gauge fields, we construct exotic TQFT actions that include mixture of two distinct types of $BF$ topological terms and many twisted topological terms. The gauge transformations are properly defined and utilized to compute level quantization and classification of TQFTs. Among all TQFTs, some are not in Dijkgraaf-Witten cohomological classification. To characterize topological orders, we concretely construct all braiding processes among topological excitations, which leads to very exotic links formed by closed spacetime trajectories of particles, loops, and membranes. For each braiding process, we construct gauge-invariant Wilson operators and calculate the associated braiding statistical phases. As a result, we obtain expressions of link invariants all of which have manifest geometric interpretation. Following Wen's definition, the boundary theory of a topological order exhibits gravitational anomaly. We expect that the characterization and classification of 5D topological orders in this paper encode information of 4D gravitational anomaly. Further consideration, e.g., putting TQFTs on 5D manifolds with boundaries, is left to future work.
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