标题： 任意子链 -- $α$-归纳 -- CFT -- 缺陷 -- 子因子 显示英文标题

标题： Anyonic Chains -- $α$-Induction -- CFT -- Defects -- Subfactors

摘要： 给定一个单态融合范畴，可以定义所谓的“任意子自旋链”的希尔伯特空间以及提供实时演化的最近邻哈密顿量。 有大量证据表明，这类系统的适当尺度极限可能导致$1+1$维共形场理论（CFTs），实际上，它们可能被用来构建新型的CFT类。 除了哈密顿量及其密度外，自旋链还携带与哈密顿量对易的对称算子代数，这些算子可以有趣地表示为矩阵乘积算子（MPOs）。 另一方面，融合范畴众所周知地来源于冯·诺依曼代数-子因子对。 在本工作中，我们研究了此类结构对于相应任意子自旋链模型的一些有趣后果。 我们的主要结果之一是构造了一个作用于双部分任意子链上的新型MPO代数。 我们证明该代数精确地同构于Fröhlich等人和Bischoff等人构造的$1+1$CFTs 的缺陷代数，尽管该模型是在有限格点上定义的。 因此，我们猜想其中心投影与模型尺度极限中的不可见（透明）垂直缺陷有关。 我们的结果部分依赖于这样一个观察，即MPOs与子因子理论中出现的所谓“双三角代数”密切相关。 在后续的构造中，我们利用了Böckenhauer等人对双三角代数结构的见解，这些见解基于范畴的辫结构以及$\alpha$-诱导。 本文关于子因子和融合范畴的引言部分具有综述性质。 显示英文摘要

摘要： Given a unitary fusion category, one can define the Hilbert space of a so-called ``anyonic spin-chain'' and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to $1+1$-dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bi-partite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of $1+1$ CFTs as constructed by Fr\" ohlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called ``double triangle algebra'' arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by B\" ockenhauer et al. based on the braided structure of the categories and on $\alpha$-induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.