# Deformations of Vertex Algebras

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.)

• Thank you, I was hoping you'd answer this! My understanding of your remark re. my precise question is that we shouldn't expect to be able to compute the vertex deformations of a holom vertex algebra $(A, \delta)$ purely in terms of the hochschild calculus of $A$, since this would only compute holomorphic deformations and we should expect to have non-holomorphic ones too? This seems very reasonable to me.