W-algebras and chiral differential operators at the critical Level

Let ${\mathcal{A}}_{crit}$ be the chiral algebra corresponding to the affine Kac-Moody algebra at the critical level ${\widehat{\mathfrak{g}}}_{crit}$. Let ${\mathfrak{Z}}_{crit}$ be the center of ${\mathcal{A}}_{crit}$. The commutative chiral algebra ${\mathfrak{Z}}_{crit}$ admits a canonical deformation into a non-commutative chiral algebra $\text{mathcl}{W}_h$. In this paper we will express the resulting first order deformation via the chiral algebra ${\mathcal{D}}_{crit}$ of chiral differential operators of $G((t))$ at the critical level.