Chiral differential operators via quantization of the holomorphic $\sigma$-model
We show that the local observables of the curved βγ system encode the sheaf of chiral differential operators using the machinery of [CG17, CG], which combine renormalization, the Batalin-Vilkovisky formalism, and factorization algebras. Our approach is in the spirit of deformation quantization via Gelfand- Kazhdan formal geometry. We begin by constructing a quantization of the βγ system with an n-dimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields Wn on the target disk, and it is naturally identified with the first Pontryagin class in Gelfand-Fuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of Wn by the closed 2-forms on the disk. By results in [CG17], we then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal βγ vertex algebra. Next, we introduce a version of Gelfand-Kazhdan formal geometry suitable for factorization algebras, and we verify that for a complex manifold X with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs of X.