超出刚性与有限性条件下，由顶点算子代数构造带奇点的模函子
Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness
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中文总结 AI 辅助
本文构造超出刚性与有限性条件下由顶点算子代数得到的带奇点开-闭模函子，证明其满足切除性质等，还将其应用于三重态模型的关联函数研究。
中文摘要 AI 辅助
对于顶点算子代数$V$及其合适的模范畴，我们提出一种构造带奇点的开-闭模函子形式的共形块空间的方法，该方法的核心是从一开始就贯彻全纯分解原则。更确切地说，我们利用Costello引入的模扩张策略，并结合我们前期工作的进展，对每个路径分支至少含一个边界分支、且标记区间或边界圆附有指定边界标签的曲面$\boldsymbol{\u03A3}$，构造其映射类群的表示$\boldsymbol{\u03A9}_V(\boldsymbol{\u03A3};-)$。该构造可在生成德恩扭转上显式描述，且本质上独立于其他基于代数几何或含手术等拓扑技术的构造，但我们也概述了现有可比较的结果。在$V$的模范畴为非半单模范畴$\boldsymbol{\u00A6}$的特殊情形下，空间$\boldsymbol{\u03A9}_V(\boldsymbol{\u03A3})$等价于$\boldsymbol{\u00A6}$的弦网空间，进而等价于Drinfeld中心$\boldsymbol{Z(\u00A6)}\backsimeq\bar{\boldsymbol{\u00A6}}\boxtimes\boldsymbol{\u00A6}$的模函子。然而，本文中$\boldsymbol{\u03A9}_V$的构造具有超出有理性、刚性、自对偶性与有限性条件的优势。此外，我们证明$\boldsymbol{\u03A9}_V$满足切除性质，在$C_2$-余有限情形下是有限维的，且能生成推广Brochier-Jordan结果的曲面辫群表示。我们还证明，对于融合积非正合的三重态$\boldsymbol{\u00A6}_{2,3}$，只要采用我们发展的带奇点模函子概念，Gaberdiel-Runkel-Wood引入的边界条件可产生关联函数。
英文摘要
For a vertex operator algebra $V$ and a certain category of its modules, we propose a construction for spaces of conformal blocks organized into an open-closed modular functor with singularities. This is inspired by the idea of implementing directly from the start the principle of holomorphic factorization. More precisely, using the strategy of modular extension introduced by Costello and developed further in our previous work, we build for each surface $Σ$ with at least one boundary component per path component and specified boundary labels attached to marked intervals or boundary circles a representation $Ω_V(Σ;-)$ of the mapping class group of $Σ$. The construction can be described explicitly on generating Dehn twists. This approach is a priori independent from other constructions based on algebraic geometry or topological techniques involving e.g. surgery, but we include an overview over the available comparisons. In the special case in which the module category of $V$ is a not necessarily semisimple modular category $\mathcal{A}$, the spaces $Ω_V(Σ)$ are equivalent to the string-net spaces for $\mathcal{A}$ and hence to the modular functor for the Drinfeld center $Z(\mathcal{A})\simeq \bar{\mathcal{A}}\boxtimes\mathcal{A}$. However, the construction of $Ω_V$ in this paper has the advantage of being available beyond rationality, rigidity, self-contragredience and finiteness. Moreover, we prove that $Ω_V$ satisfies excision, is finite-dimensional in the $C_2$-cofinite case and produces in genus one a generalization of the elliptic double of Brochier-Jordan. We prove for the triplet $\mathcal{W}_{2,3}$ with non-exact fusion product that the boundary conditions introduced by Gaberdiel-Runkel-Wood produce, as expected by these authors, correlation functions, provided that one uses the notion of a modular functor with singularities that we develop.
发表机构
- Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）
- Université Bourgogne Europe CNRS IMB UMR 5584（勃艮第欧洲大学法国国家科学研究中心IMB）
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