论文标题
顶点操作员代数和括号的辫子
Vertex operator algebra and parenthesized braid operad
论文作者
论文摘要
保形块，手性2D共形场理论的物理量是复杂平面的配置空间上的或骨，它们是根据顶点操作员代数，其模块和相关的D模型在数学上表达的。我们表明，配置空间的基本组素（括号的编织式手术）作用于单肌代表作用于保形块上。 更准确地说，令$ v $为$ v = \ bigoplus_ {n \ geq 0} v_n $，$ \ dim v_n <\ infty $，$ v_0 = \ v_0 = \ mathbb {c} \ bf {c} \ bf {1} $ v \ v \ text f.-mod and} f. $ V $ -MODULES $ M $的类别是$ M $是$ C_1 $ -COFINITE，双模块$ M^\ VEE $是有限生成的$ V $ -MODULE。我们表明，括号的辫子弱在$ V \ text {-mod} _ {\ mathrm {f.g}} $上作用于$ V \ text {-mod} _ {\ mathrm {f.g}} $，因此$ v \ text {-mod} _ {\ mathrm {f.g}} $具有（Unital Iital）pseudo-brad，此外，如果$ v $是理性的，而$ c_2 $ -cofinite，则$ v \ text {-mod} _ {\ mathrm {f.g}} $是平衡的编织张量类别，它提供了Huang和Lepowsky的替代证明。
Conformal blocks, physical quantities of chiral 2d conformal field theory, are sheaves on the configuration spaces of the complex plane, which are mathematically formulated in terms of a vertex operator algebra, its modules and associated D-modules. We show that the operad of fundamental groupoids of the configuration spaces, the parenthesized braid operad, acts on the conformal blocks by the monodromy representation. More precisely, let $V$ be a vertex operator algebra with $V=\bigoplus_{n\geq 0} V_n$, $\dim V_n <\infty$, $V_0=\mathbb{C}\bf{1}$ and $V\text{-mod}_{\mathrm{f.g}}$ the category of $V$-modules $M$ such that $M$ is $C_1$-cofinite and the dual module $M^\vee$ is a finitely generated $V$-module. We show that the parenthesized braid operad weakly 2-categorically acts on $V\text{-mod}_{\mathrm{f.g}}$, and consequently $V\text{-mod}_{\mathrm{f.g}}$ has a structure of the (unital) pseudo-braided category. Moreover, if $V$ is rational and $C_2$-cofinite, then $V\text{-mod}_{\mathrm{f.g}}$ is a balanced braided tensor category, which gives an alternative proof of a result of Huang and Lepowsky.
