发表机构
Università di Roma “Tor Vergata”（罗马托尔维加塔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过直极限完备化的代数对象分类，结合A₅-轨道模型的强合理性假设，完成了中心电荷c=1的强合理顶点算子代数的完整分类。
AI 中文摘要
设G为具有有限维有理表示的对称张量范畴𝒞_G的复约化仿射代数群。我们证明，𝒞_G的直极限完备化Ind(𝒞_G)中每个至多可数维的简单交换代数对象，都同构于齐性空间G/H上的正则函数代数𝒪(G/H)，其中H是G的某个约化代数子群。随后我们将该结果应用于顶点算子代数扩张理论。特别地，在假设与秩1根格𝒪₂:=√2ℤ关联的顶点算子代数V_{𝒪₂}的A₅-轨道模型具有强合理性的前提下，我们对满足特定谱条件的简单酉Virasoro顶点算子代数L(1,0)的所有（不一定有理的）简单CFT型预酉顶点算子代数扩张进行了分类，其中心电荷c=1。该结果是冯旭的共形网结果在顶点算子代数领域的对应。由于其特征标的同余子群模性性质，每个强合理的预酉顶点算子代数L(1,0)扩张都满足上述谱条件。因此，在V_{𝒪₂}^{A₅}的强合理性假设下，我们得到了强合理c=1顶点算子代数的完整分类结果。
英文摘要
Let $G$ be a complex reductive affine algebraic group with its symmetric tensor category $\mathcal{C}_G$ of finite-dimensional rational representations. We prove that every simple commutative algebra object with at most countable dimension in the direct limit completion $\operatorname{Ind}(\mathcal{C}_G)$ of $\mathcal{C}_G$ is isomorphic to the algebra $\mathcal{O}(G/H)$ of regular functions on the homogeneous space $G/H$, for some reductive algebraic subgroup $H$ of $G$. Then we apply this result to the theory of vertex operator algebra extensions. In particular, assuming the strong rationality of the $ \operatorname{A}_5$-orbifold of the vertex operator algebra $V_{\mathcal{L}_2}$ associated with the rank-one root lattice $\mathcal{L}_2:= \sqrt{2}\mathbb{Z}$, we classify all the not necessarily rational simple CFT type preunitary vertex operator algebra extensions of the simple unitary Virasoro vertex operator algebra $L(1,0)$ with central charge $c=1$ satisfying a certain spectrum condition. This result is the vertex operator algebra analogue of a conformal net result by Feng Xu. Every strongly rational preunitary vertex operator algebra extension of $L(1,0)$ satisfies the above spectrum condition because of the congruence subgroup modularity property of its characters. As a consequence, we get a complete classification result for strongly rational $c=1$ vertex operator algebras, up to the strong rationality of $V_{\mathcal{L}_2}^{\operatorname{A}_5}$.
