源自半单 Hopf 代数交叉模的 2+1 维拓扑相精确可解模型
Exactly solvable models for 2+1D topological phases derived from crossed modules of semisimple Hopf algebras
AI总结：
本文提出源自半单 Hopf 代数交叉模的 Hopf 代数高维 Kitaev 模型，构造四有限群交叉模族，并在两类情形证明基态空间与三角剖分无关且可由 TQFT 描述。
摘要：
我们在三角剖分曲面上定义了一个关于物质 2+1 维拓扑相的精确可解模型，该模型源自半单有限维 Hopf 代数的交叉模，即“Hopf 代数高维 Kitaev 模型”。此模型推广了半单 Hopf 代数的 Kitaev 量子双模型，以及源自 2-群的完整高维 Kitaev 模型，因此可被解释为一种 Hopf 代数离散高阶规范理论。我们构造了一族半单 Hopf 代数的交叉模 $(\mathscr{F}_{\mathbb{C}}(X) \otimes \mathbb{C}E \xrightarrow{\partial} \mathscr{F}_{\mathbb{C}}(Y) \rtimes \mathbb{C} G, \triangleright)$，它依赖于四个有限群 $E,G,X$ 和 $Y$。我们分别在 $G=E=\{1\}$ 以及 $Y=\{1\}$ 时计算了所得模型在三角剖分曲面上的基态空间，证明这些基态空间以典范方式独立于三角剖分，因而只依赖于底层曲面；此外，我们找到了一个 2+1 维 TQFT，其在曲面上的状态空间给出这些基态空间。这些 TQFT 是 Quinn 有限全同伦 TQFT 的特例，因此赋予曲面的状态空间是从曲面到同伦有限空间的映射同伦类集合上的自由向量空间；在本文情形中，这些同伦有限空间由有限广群以及有限广群交叉模的分类空间得到。对于一般的半单 Hopf 代数交叉模，Hopf 代数高维 Kitaev 模型在三角剖分曲面上的基态空间是否独立于三角剖分、是否总存在一个 TQFT 使其在曲面上的状态空间给出该模型的基态空间，以及由 $E,G,X,Y$ 得到的模型基态空间是否总能给出同伦解释，我们将这些留作开放问题。
英文摘要：
We define an exactly solvable model for 2+1D topological phases of matter on a triangulated surface derived from a crossed module of semisimple finite-dimensional Hopf algebras, the `Hopf-algebraic higher Kitaev model'. This model generalizes both the Kitaev quantum double model for a semisimple Hopf algebra and the full higher Kitaev model derived from a 2-group, and can hence be interpreted as a Hopf-algebraic discrete higher gauge theory. We construct a family of crossed modules of semisimple Hopf algebras, $(\mathscr{F}_{\mathbb{C}}(X) \otimes \mathbb{C}E \xrightarrow{\partial} \mathscr{F}_{\mathbb{C}}(Y) \rtimes \mathbb{C} G, \triangleright)$, that depends on four finite groups, $E,G,X$ and $Y$. We calculate the ground-state spaces of the resulting model on a triangulated surface when $G=E=\{1\}$ and when $Y=\{1\}$, prove that those ground-state spaces are canonically independent of the triangulations, and so depend only on the underlying surface; and moreover we find a 2+1D TQFT whose state spaces on surfaces give the ground-state spaces. These TQFTs are particular cases of Quinn's finite total homotopy TQFT and hence the state spaces assigned to surfaces are free vector spaces on sets of homotopy classes of maps from a surface to homotopy finite spaces, in this case obtained as classifying spaces of finite groupoids and finite crossed modules of groupoids. We leave it as an open problem whether the ground-state space of the Hopf-algebraic higher Kitaev model on a triangulated surface is independent of the triangulation for general crossed modules of semisimple Hopf algebras, whether a TQFT always exists whose state space on a surface gives the ground-state space of the model, and whether the ground-state space of the model obtained from $E,G,X,Y$ can always be given a homotopical explanation.
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