Softcover ISBN:  9782856299203 
Product Code:  AST/419 
List Price:  $68.00 
AMS Member Price:  $54.40 
Softcover ISBN:  9782856299203 
Product Code:  AST/419 
List Price:  $68.00 
AMS Member Price:  $54.40 

Book DetailsAstérisqueVolume: 419; 2020; 210 ppMSC: Primary 81; 17; 55
The curved \(\beta\gamma\) system is a nonlinear \(\sigma\)model 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 \(X\). The authors 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, the authors find that the factorization algebra of quantum observables of the curved \(\beta\gamma\) 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.
The authors 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 \(W_n\) 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 \(W_n\) by \(\hat{\Omega}^{2}_{{\mathrm{cl}}}\), the closed 2forms on the disk. By machinery mentioned above, the authors 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, the authors introduce a version of GelfandKazhdan formal geometry suitable for factorization algebras, and verify that for a complex manifold \(X\) with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs of \(X\).
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
ReadershipGraduate students and research mathematicians.

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The curved \(\beta\gamma\) system is a nonlinear \(\sigma\)model 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 \(X\). The authors 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, the authors find that the factorization algebra of quantum observables of the curved \(\beta\gamma\) 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.
The authors 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 \(W_n\) 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 \(W_n\) by \(\hat{\Omega}^{2}_{{\mathrm{cl}}}\), the closed 2forms on the disk. By machinery mentioned above, the authors 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, the authors introduce a version of GelfandKazhdan formal geometry suitable for factorization algebras, and verify that for a complex manifold \(X\) with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs of \(X\).
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and research mathematicians.