As the title suggests, I'm interested in deformation theory of vertex algebras and their representations.

In the paper https://arxiv.org/abs/1806.08754, the authors construct, for a vertex algebra $V$ with a module $M$, a family of cohomology spaces $H^{i}_{VA}(V, M)$. Naturally, they in fact construct a complex computing the above and call it vertex algebra cohomology (of $V$ with coefficients in $M$). Further they prove that low degree cohomology groups can be interpreted in the usual way (a version of singular vectors, derivations, extensions etc). In the case of the adjoint representation I believe we obtain a complex controlling the deformation theory of the vertex algebra, as one would hope.

I find the calculus of vertex algebras somewhat daunting at times and so I find the construction hard to follow. I'd like to understand it in a simple case, hopefully not so simple as to be completely degenerate.

Let $V$ then he a holomorphic vertex algebra, so that for all $v\in V$ the field $v(z)$ is an element of $End(V)[[z]]$. It is not hard to show that such a $V$ is equivalent to the data of a commutative algebra with a derivation. Switching to this language I'll write $(A, \delta)$ for such an object. What is the vertex algebra cohomology of $(A, \delta)$ with cohomolgy in the adjoint representation?

If I'm not mistaken, square zero deformations of $(A, \delta) $ are Hochschild cohomology classes $\gamma\in HH^{2}(A)$ such that $Lie_{\delta} (\gamma) =0$. Perhaps this generalizes in the obvious way to other cohomology groups, whatever they are. (Note that the answer should have the structure of a dgla and $Ker(Lie_{\delta})$ indeed has such a structure, in fact that of a Gerstenhaber algebra I believe.)