On modules for meromorphic open-string vertex algebras

We study representations of the meromorphic open-string vertex algebra (MOSVAs hereafter) defined in [H3], a noncommutative generalization of vertex (operator) algebra. We start by recalling the definition of a MOSVA $V$ and left $V$-modules in [H3]. Then we define right $V$-modules and $V$-bimodules that reflect the noncommutative nature of $V$. When $V$ satisfies a condition on the order of poles of the correlation function (which we call pole-order condition), we prove that the rationality of products of two vertex operators implies the rationality of products of any numbers of vertex operators. Also, the rationality of iterates of any numbers of vertex operators is established, and is used to construct the opposite MOSVA $V^{op}$ of $V$. It is proved here that right (resp. left) $V$-modules are equivalent to left (resp. right) $V^{op}$-modules. Using this equivalence, we prove that if $V$ and a grading-restricted left $V$-module $W$ is endowed with a M\"obius structure, then the graded dual $W'$ of $W$ is a right $V$-module. This proof is the only place in this paper that needs the grading-restriction condition. Also, this result is generalized to not-grading-restricted modules under a strong pole-order condition that is satisfied by all existing examples of MOSVAs and modules.