  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.
