Vertex operator algebra and parenthesized braid operad

Abstract: 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.