标题： 伪$q$-迹与(余)端有何关系？ 显示英文标题

标题： How are pseudo-$q$-traces related to (co)ends?

摘要： 设$\mathbb V$是一个$\mathbb N$-分次的$C_2$-有限顶点算子代数（VOA），不一定有理或自对偶。 利用[GZ25a]中缝合-分解定理的一个特例，我们证明了在$\mathrm{Mod}(\mathbb{V}^{\otimes2})$中的端点$\mathbb E=\int_{\mathbb M\in\mathrm{Mod}(\mathbb V)}\mathbb M\otimes_{\mathbb C}\mathbb M'$（其中$\mathbb{M}'$是$\mathbb{M}$的对偶模）具有与它的$\mathbb{V}^{\otimes2}$-模结构相容的结合$\mathbb C$-代数的自然结构。 Moreover, we show that a suitable category $\mathrm{Coh}_{\mathrm{L}}(\mathbb E)$ of left $\mathbb E$-modules is isomorphic, as a linear category, to $\mathrm{Mod}(\mathbb V)$, and that the space of vacuum torus conformal blocks is isomorphic to the space $\mathrm{SLF}(\mathbb E)$ of symmetric linear functionals on $\mathbb E$. 结合这些结果与[GZ25b]中的主定理，我们证明了Gainutdinov-Runkel的一个猜想：对于$\mathbb G$中的任何投射生成器，在$\mathrm{Mod}(\mathbb V)$中，伪$q$-迹构造从$\mathrm{SLF}(\mathrm{End}_{\mathbb V}(\mathbb{G})^{\mathrm{opp}})$映射到$\mathbb V$的真空环面共形块空间的线性同构。 特别是，如果$A$是一个单位有限维$\mathbb C$-代数，使得有限维左$A$-模的范畴与$\mathrm{Mod}(\mathbb V)$等价，那么$\mathrm{SLF}(A)$与$\mathbb V$的真空环面共形块空间线性同构。这证实了Arike-Nagatomo的一个猜想。 显示英文摘要

摘要： Let $\mathbb V$ be an $\mathbb N$-graded $C_2$-cofinite vertex operator algebra (VOA), not necessarily rational or self-dual. Using a special case of the sewing-factorization theorem from [GZ25a], we show that the end $\mathbb E=\int_{\mathbb M\in\mathrm{Mod}(\mathbb V)}\mathbb M\otimes_{\mathbb C}\mathbb M'$ in $\mathrm{Mod}(\mathbb{V}^{\otimes2})$ (where $\mathbb{M}'$ is the contragredient module of $\mathbb{M}$) admits a natural structure of associative $\mathbb C$-algebra compatible with its $\mathbb{V}^{\otimes2}$-module structure. Moreover, we show that a suitable category $\mathrm{Coh}_{\mathrm{L}}(\mathbb E)$ of left $\mathbb E$-modules is isomorphic, as a linear category, to $\mathrm{Mod}(\mathbb V)$, and that the space of vacuum torus conformal blocks is isomorphic to the space $\mathrm{SLF}(\mathbb E)$ of symmetric linear functionals on $\mathbb E$. Combining these results with the main theorem of [GZ25b], we prove a conjecture of Gainutdinov-Runkel: For any projective generator $\mathbb G$ in $\mathrm{Mod}(\mathbb V)$, the pseudo-$q$-trace construction yields a linear isomorphism from $\mathrm{SLF}(\mathrm{End}_{\mathbb V}(\mathbb{G})^{\mathrm{opp}})$ to the space of vacuum torus conformal blocks of $\mathbb V$. In particular, if $A$ is a unital finite-dimensional $\mathbb C$-algebra such that the category of finite-dimensional left $A$-modules is equivalent to $\mathrm{Mod}(\mathbb V)$, then $\mathrm{SLF}(A)$ is linearly isomorphic to the space of vacuum torus conformal blocks of $\mathbb V$. This confirms a conjecture of Arike-Nagatomo.