Cohomology of vertex algebras

TL;DR

This paper develops a cohomology theory for vertex algebras, linking second cohomology to extensions and deformations, thus advancing the understanding of their algebraic structure.

Contribution

It introduces the first and second cohomology groups for vertex algebras and establishes their correspondence with extensions and deformations.

Findings

Second cohomology classifies square-zero extensions.

When coefficients are the algebra itself, cohomology classifies first order deformations.

Provides a cohomological framework for vertex algebra structure analysis.

Abstract

Let $V$ be a vertex algebra and $M$ a $V$-module. We define the first and second cohomology of $V$ with coefficients in $M$, and we show that the second cohomology $H^{2}(V, M)$ corresponds bijectively to the set of equivalence classes of square-zero extensions of $V$ by $M$. In the case that $M=V$, we show that the second cohomology $H^{2}(V, V)$ corresponds bijectively to the set of equivalence classes of first order deformations of $V$.