Chiral differential operators via BatalinVilkovisky quantization
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 BatalinVilkovisky formalism, and factorization algebras. Our approach is in the spirit of deformation quantization via GelfandKazhdan formal geometry. We begin by constructing a quantization of the beta gamma system with an ndimensional 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 GelfandFuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of formal vector fields by the closed 2forms 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 GelfandKazhdan 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2016
 arXiv:
 arXiv:1610.09657
 Bibcode:
 2016arXiv161009657G
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Algebraic Geometry
 EPrint:
 Some minor errors and typos fixed. Added section on the formal construction of the Witten genus. Added section on the conformal anomaly and realization of the Virasoro vertex algebra in the case of a curved target. Added a further comparisons to and interpretations of physical aspects of the nonlinear sigma model. This is the final, published version