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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.

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