# Item

ITEM ACTIONSEXPORT

Released

Book

#### Chiral differential operators via Batalin-Vilkovisky quantization of the holomorphic σ-model

##### External Resource

##### Fulltext (restricted access)

There are currently no full texts shared for your IP range.

##### Fulltext (public)

1610.09657.pdf

(Preprint), 2MB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Gorbounov, V., Gwilliam, O., & Williams, B. R. (2020). *Chiral
differential operators via Batalin-Vilkovisky quantization of the holomorphic σ-model*. Paris: Société Mathématique
France.

Cite as: https://hdl.handle.net/21.11116/0000-0007-4730-3

##### 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.

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.