F4在level 6的一个特殊例外模不变量的量子对称性与nimreps
Quantum symmetries and nimreps for an exotic exceptional modular invariant of F4 at level 6
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中文总结 AI 辅助
本文研究F4在level 6的例外模不变量，通过模范畴和étale代数构造量子对称性，给出Ocneanu图、nimreps及融合多项式等关键数据。
中文摘要 AI 辅助
我们重新审视了一个1989年发现的F4在level 6的非对角模不变量。在它被发现时，人们注意到它不能通过仿射李代数的共形嵌入得到。后来，研究表明它可以通过一个非李型顶点算子代数的扩张得到。对于这个模不变量，可以构造一个完整的CFT，它被定义（或表述）为融合范畴A = C(F4,6)的模范畴M。最近，人们认识到M与A的一个étale代数相关。换言之，所得的量子模是一个量子子群（也称为具有自融合、I型或平坦）。我们研究A关于M的对偶O。特别地，我们描述了量子对称性代数（我们获得了手征生成元并展示了A的基本不可约表示的Ocneanu图）。手征图不是连通的，每个图包含两对同构的分支。这些分支的顶点集被等同于两个共享同一模不变量的模范畴M和M′的简单对象（在等价意义下）。M具有自融合，而M′不具有。Verlinde代数A对相关阿贝尔群M和M′的作用的限制定义了两个具有非负整数项矩阵族（nimreps）及其邻接图。作为副产品，我们获得了描述理论缺陷线的矩阵、与A及其模相关的整数和有理融合多项式，以及相关的数值不变量、量子维数、量子质量和诱导-限制规则。
英文摘要
We revisit an old non-diagonal modular invariant of $F_4$ at level $6$, discovered in 1989. At the time of its discovery, it was noticed that it could not be obtained from a conformal embedding of affine Lie algebras. Later, it was shown that it can be obtained from an extension of a vertex operator algebra which is not of Lie type. To this modular invariant, one can attach a full CFT defined (or formulated) as a module category $\mathcal M$ for the fusion category $\mathcal A = \mathcal C(F_4,6)$. More recently, it was recognized that $\mathcal M$ is associated with an étale algebra of $\mathcal A$. In other words, the resulting quantum module is a quantum subgroup (it is also said to have self-fusion, to be of type I, or to be flat). We study the dual $\mathcal O$ of $\mathcal A$ with respect to $\mathcal M$. In particular, we describe the algebra of quantum symmetries (we obtain the chiral generators and display the Ocneanu graphs for the fundamental irreps of $\mathcal A$). The chiral graphs are not connected, each one containing two pairs of isomorphic components. The vertex sets of these components are identified with the simple objects, up to equivalence, of two module categories $\mathcal M$ and $\mathcal M^\prime$ sharing the same modular invariant. $\mathcal M$ enjoys self-fusion, whereas $\mathcal M^\prime$ does not. The restriction of the action of the Verlinde algebra $A$ to the associated abelian groups $M$ and $M^\prime$ defines two families of matrices with non-negative integer entries (nimreps) and their adjacency graphs. As a by-product, we obtain the matrices describing the defect lines of the theory, the integer and rational fusion polynomials attached to $A$ and its modules, and the relevant numerical invariants, quantum dimensions, quantum masses, and induction-restriction rules.
发表机构
- Aix Marseille Univ, CNRS, I2M, Marseille, France（艾克斯-马赛大学，法国国家科学研究中心，I2M）
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