Abstract
The curved βγ system is a nonlinear σmodel with a Riemann surface as the source and a complex manifold X as the target. Its classical solutions pick
out the holomorphic maps from the Riemann surface into X. Physical arguments
identify its algebra of operators with a vertex algebra known as the chiral differential operators (CDO) of X. We verify these claims mathematically by constructing and quantizing rigorously this system using machinery developed by Kevin Costello and the second author, which combine renormalization, the BatalinVilkovisky formalism, and factorization algebras. Furthermore, we find that the factorization algebra of quantum observables of the curved βγ system encodes the sheaf of chiral differential operators. In this sense our approach provides deformation quantization for vertex algebras. As in many approaches to deformation quantization, a key role is played by GelfandKazhdan formal geometry. We begin by constructing a quantization of the βγ 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 Wn 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 Wn by Ωb2cl, the closed 2forms on the disk. By machinery mentioned above, 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 GelfandKazhdan 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.
out the holomorphic maps from the Riemann surface into X. Physical arguments
identify its algebra of operators with a vertex algebra known as the chiral differential operators (CDO) of X. We verify these claims mathematically by constructing and quantizing rigorously this system using machinery developed by Kevin Costello and the second author, which combine renormalization, the BatalinVilkovisky formalism, and factorization algebras. Furthermore, we find that the factorization algebra of quantum observables of the curved βγ system encodes the sheaf of chiral differential operators. In this sense our approach provides deformation quantization for vertex algebras. As in many approaches to deformation quantization, a key role is played by GelfandKazhdan formal geometry. We begin by constructing a quantization of the βγ 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 Wn 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 Wn by Ωb2cl, the closed 2forms on the disk. By machinery mentioned above, 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 GelfandKazhdan 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.
Original language  English 

Pages (fromto)  viii+210 
Number of pages  211 
Journal  Astérisque 
Volume  419 
DOIs  
Publication status  Published  14 Jun 2020 
Keywords
 GanGrossPrasad conjecture
 Local trace formula
 Padic Lie groups
 Representations of real
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Vasily Gorbunov
 School of Natural & Computing Sciences, Mathematical Science  Emeritus Professor
Person: Honorary