On the extensions of the left modules for a meromorphic open-string vertex algebra, I

We study the extensions of two left modules $W_1,W_2$ for a meromorphic open-string vertex algebra $V$. We show that the extensions satisfying some technical but natural convergence conditions are in bijective correspondence to the first cohomology classes associated to the $V$-bimodule ${\mathcal{H}}_N(W_1,W_2)$ constructed in \cite{HQ-Red}. When $V$ is grading-restricted and contains a nice vertex subalgebra $V_0$, those convergence conditions hold automatically. In addition, we show that the dimension of ${\text{Ext}}^1(W_1,W_2)$ is bounded above by the fusion rule $N\left(\genfrac{}{}{0px}{}{W_2}{V{W}_1}\right)$ in the category of $V_0$-modules. In particular, if the fusion rule is finite, then ${\text{Ext}}^1(W_1,W_2)$ is finite-dimensional. We also give an example of an abelian category consisting of certain modules of the Virasoro VOA that does not contain any nice subalgebras, while the convergence conditions hold for every object.