# Free almost commutative vertex algebras

Given a commutative $$k$$-algebra $$A$$, we can freely generate a commutative vertex algebra by formally adjoining a derivation. We obtain a functor $$CAlg_{k} \rightarrow CVAlg_{k}$$, which I'll denote $$\mathcal{J}$$. Geometrically this corresponds to taking jets.

We could sensibly consider deformations of this functor. We could also (possibly non-sensibly) consider to what extend deformations of $$A$$ as an associative algebra give deformations of $$\mathcal{J}A$$ as a vertex algebra. Up to order $$\hbar^{2}$$ such deformations of $$A$$ are controlled by Poisson structures $$\pi$$. On the vertex side we can use $$\pi$$ to define a Poisson vertex structure on $$\mathcal{J} A$$ so we're off to a decent start.

My question then, given the data of a flat deformation of $$A$$ to an algebra $$\mathcal{A}$$ over $$k[[\hbar]]$$, can we produce a vertex algebra over $$k[[\hbar]]$$ with special fibre the Poisson vertex algebra $$\mathcal{J}A$$?

A remark, $$\mathcal{A}$$ corresponds to a star product and thus a sequence of bi-differential operators. One should then attempt to see what structure this sequence of operators gives on $$\mathcal{J}A[[\hbar]]$$. Mimicking the construction of the $$\lambda$$ bracket corresponding to $$\pi$$ at each stage it doesn't seem impossible that such operators suffice to define a product $$\mathcal{J}A\otimes\mathcal{J} A\rightarrow\mathcal{J} A[[\hbar]] ((z))$$ defining a vertex algebra.