标题： 解析共形块的$C_2$-有限顶点算子代数 III：缝合-分解定理 显示英文标题

标题： Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras III: The Sewing-Factorization Theorems

摘要： 设$\mathbb V=\bigoplus_{n\in\mathbb N}\mathbb V(n)$是一个$C_2$-有限的 VOA，不一定有理或自对偶。在本文中，我们为与$\mathbb V^{\otimes N}$的分次限制广义模相关的共形块建立了各种版本的缝合-分解（SF）定理（其中$N\in\mathbb N$）。 除了在[GZ23]引言中宣布的版本外，我们证明了SF定理的下列共端版本： 令$\mathfrak F$为一个具有$N$个输入和$R$个输出标记点的紧致黎曼曲面，并令$\mathfrak G$为另一个具有$K$个输入和$R$个输出标记点的紧致黎曼曲面。 将$\mathbb W\in\mathrm{Mod}(\mathbb V^{\otimes N})$和$\mathbb X\in\mathrm{Mod}(\mathbb V^{\otimes K})$分别分配给$\mathfrak F$和$\mathfrak G$的输入标记点。 对于每个$\mathbb{M} \in \mathrm{Mod}(\mathbb{V}^{\otimes R})$，将$\mathbb{M}$及其反不变量$\mathbb M'$分配给$\mathfrak F$和$\mathfrak G$的出射标记点分别。 将相应的共形块空间表示为$\mathscr T_{\mathfrak F}^*(\mathbb M\otimes\mathbb W)$和$\mathscr T_{\mathfrak{G}}^*(\mathbb M'\otimes\mathbb X)$。 设 $\mathfrak X$ 是通过在它们的出向标记点上缝合 $\mathfrak F$ 和 $\mathfrak G$ 所得到的 $(N+K)$点曲面。 然后，通过共形块证明在 [GZ24] 中收敛的缝合，给出了向量空间 $$\int^{\mathbb{M}\in\mathrm{Mod}(\mathbb V^{\otimes R})}\mathscr T_{\mathfrak F}^*(\mathbb M\otimes\mathbb{W})\otimes_{\mathbb C} \mathscr T_{\mathfrak G}^*(\mathbb M'\otimes \mathbb X)\simeq\mathscr T_{\mathfrak X}^*(\mathbb W\otimes \mathbb X)$$的同构。 我们还讨论了当 $\mathbb V$ 是强有限且刚性时，共形块与使用 Lyubashenko 的余端定义的模 functor 之间的关系。 显示英文摘要

摘要： Let $\mathbb V=\bigoplus_{n\in\mathbb N}\mathbb V(n)$ be a $C_2$-cofinite VOA, not necessarily rational or self-dual. In this paper, we establish various versions of the sewing-factorization (SF) theorems for conformal blocks associated to grading-restricted generalized modules of $\mathbb V^{\otimes N}$ (where $N\in\mathbb N$). In addition to the versions announced in the Introduction of [GZ23], we prove the following coend version of the SF theorem: Let $\mathfrak F$ be a compact Riemann surface with $N$ incoming and $R$ outgoing marked points, and let $\mathfrak G$ be another compact Riemann surface with $K$ incoming and $R$ outgoing marked points. Assign $\mathbb W\in\mathrm{Mod}(\mathbb V^{\otimes N})$ and $\mathbb X\in\mathrm{Mod}(\mathbb V^{\otimes K})$ to the incoming marked points of $\mathfrak F$ and $\mathfrak G$ respectively. For each $\mathbb{M} \in \mathrm{Mod}(\mathbb{V}^{\otimes R})$, assign $\mathbb{M}$ and its contragredient $\mathbb M'$ to the outgoing marked points of $\mathfrak F$ and $\mathfrak G$ respectively. Denote the corresponding spaces of conformal blocks by $\mathscr T_{\mathfrak F}^*(\mathbb M\otimes\mathbb W)$ and $\mathscr T_{\mathfrak{G}}^*(\mathbb M'\otimes\mathbb X)$. Let the $\mathfrak X$ be the $(N+K)$-pointed surface obtained by sewing $\mathfrak F$, $\mathfrak G$ along their outgoing marked points. Then the sewing of conformal blocks-proved to be convergent in [GZ24]-yields an isomorphism of vector spaces $$\int^{\mathbb{M}\in\mathrm{Mod}(\mathbb V^{\otimes R})}\mathscr T_{\mathfrak F}^*(\mathbb M\otimes\mathbb{W})\otimes_{\mathbb C} \mathscr T_{\mathfrak G}^*(\mathbb M'\otimes \mathbb X)\simeq\mathscr T_{\mathfrak X}^*(\mathbb W\otimes \mathbb X)$$ We also discuss the relation between conformal blocks and the modular functors defined using Lyubashenko's coend in the case where $\mathbb V$ is strongly finite and rigid.