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

TL;DR

This paper explores the relationship between module extensions and cohomology in meromorphic open-string vertex algebras, establishing bounds on extension dimensions and providing examples in the Virasoro VOA context.

Contribution

It establishes a bijective correspondence between module extensions and first cohomology classes, and bounds the extension dimension by fusion rules, with automatic convergence under certain conditions.

Findings

Extensions correspond to first cohomology classes.

Extension dimension is bounded by fusion rules.

An example in the Virasoro VOA context shows convergence conditions hold without nice subalgebras.

Abstract

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 $N\binom{W_2}{VW_1}$ 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.