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Abstract

We show that the local observables of the curved beta gamma system encode the
sheaf of chiral differential operators using the machinery of the book
"Factorization algebras in quantum field theory", by Kevin Costello and the
second author, which combines 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 beta gamma 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 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 formal vector fields
by the closed 2-forms on the disk. By results in the book listed above, we then
naturally obtain a factorization algebra of quantum observables, which has an
associated vertex algebra easily identified with the formal beta gamma vertex
algebra. Next, we introduce a version of Gelfand-Kazhdan formal geometry
suitable for factorization algebras, and we verify that for a complex manifold
with trivialized first Pontryagin class, the associated factorization algebra
recovers the vertex algebra of CDOs on the complex manifold.