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arXiv 1901.02249hep-thcond-mat.str-elquant-ph
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拓扑相的2型规范模型

On 2-form gauge models of topological phases

Clement Delcamp, Apoorv Tiwari

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78 pages
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摘要：

我们探讨了（3+1）d中2-形式拓扑规范理论的各个方面。这些理论可以构造成目标空间为对称群$G$的第二分类空间$B^2G$的西格玛模型，并按$B^2G$的上同调类进行分类.离散拓扑规范理论可以典型地嵌入到连续量子场论中。在2-形式的情况下，连续理论被证明是严格的2-群规范理论。通过使用Deligne-Beilinson上同调技术仔细构造$q$ -形式联络的空间来研究这种嵌入。同样的技术可以用来研究从Postnikov塔建造的更一般的模型。对于有限对称群，2-形式拓扑理论有一个自然的格解释，我们用它来构造一个在（3+1）d中精确可解的格哈密顿模型。这种构造依赖于代数上环的上同调（称为2-形式上同调）的引入，这些代数上环与$B^2G$的单纯上环相同，如Eilenberg-MacLane空间的所谓$W$ -构造所提供的。本文从代数和几何上证明了2-形式4-上圈如何约化为$G$ -分次向量空间的准模范畴的结合子和辫子同构。这是用来显示我们的2形式规范模型和沃克-王模型之间的对应关系。

英文摘要：

We explore various aspects of 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space $B^2G$ of the symmetry group $G$, and they are classified by cohomology classes of $B^2G$. Discrete topological gauge theories can typically be embedded into continuous quantum field theories. In the 2-form case, the continuous theory is shown to be a strict 2-group gauge theory. This embedding is studied by carefully constructing the space of $q$-form connections using the technology of Deligne-Beilinson cohomology. The same techniques can then be used to study more general models built from Postnikov towers. For finite symmetry groups, 2-form topological theories have a natural lattice interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d that is exactly solvable. This construction relies on the introduction of a cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified with the simplicial cocycles of $B^2G$ as provided by the so-called $W$-construction of Eilenberg-MacLane spaces. We show algebraically and geometrically how a 2-form 4-cocycle reduces to the associator and the braiding isomorphisms of a premodular category of $G$-graded vector spaces. This is used to show the correspondence between our 2-form gauge model and the Walker-Wang model.

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