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 a grading-restricted vertex algebra containing a nice vertex subalgebra $V_0$, those convergence conditions hold automatically. In addition, we show that $\text{Ext}^1(W_1, W_2)$ is finite dimensional, with its dimension bounded above by the fusion rule $N\binom{W_2}{VW_1}$ of $V_0$-modules. When $V$ is the Virasoro VOA that does not contain any nice subalgebras, we also give an example of an abelian category consisting of certain modules where the convergence conditions hold for every object.Comment: 51 pages. Comments are welcome