Vertex operator algebra and parenthesized braid operad

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 $V={⨁}_{n\ge 0}{V}_n$, $\dim V_n<\mathrm{\infty }$, $V_0=\mathbb{C}\mathbf{1}$ and $V{\text{-mod}}_{\mathrm{f}\mathrm{.}\mathrm{g}}$ 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 $V{\text{-mod}}_{\mathrm{f}\mathrm{.}\mathrm{g}}$, and consequently $V{\text{-mod}}_{\mathrm{f}\mathrm{.}\mathrm{g}}$ has a structure of the (unital) pseudo-braided category. Moreover, if $V$ is rational and $C_2$-cofinite, then $V{\text{-mod}}_{\mathrm{f}\mathrm{.}\mathrm{g}}$ is a balanced braided tensor category, which gives an alternative proof of a result of Huang and Lepowsky.