Gerbes of chiral differential operators. II

Chiral differential operators on abelian varieties

The paper consists of two parts. In the first, we describe a way of getting from an algebra of chiral differential operators (cdo) on an abelian variety a cdo on the dual variety. The second is an introduction to the sigma-model on a torus and to the Wess-Zumino-Witten model from the cdo perspective.

Differential operators and the loop group via chiral algebras

Let $G$ be an algebraic group and let $\widetilde{\mathfrak g}$ be the corresponding affine algebra on some level. Consider the induced module $V:=Ind^{\widetilde{\mathfrak g}}_{{\mathfrak g}[[t]](O_{G[[t]]})$, where $O_{G[[t]]}$ is the ring of regular functions on the group $G[[t]]$. In this paper we show that $V$ is naturally a vertex operator algebra, which is "responsible" for D-modules on the loop group $G((t))$. Using the techiques of VOA we show that $V$ is in fact a bimodule over the affine algebra. In addition, we show that $V$ possesses a remarkable property related to its BRST reduc…

W-algebras and chiral differential operators at the critical Level

Let $\mathcal{A}_{crit}$ be the chiral algebra corresponding to the affine Kac-Moody algebra at the critical level $\hat{\mathfrak{g}}_{crit}$. Let $\mathfrak{Z}_{crit}$ be the center of $\mathcal{A}_{crit}$. The commutative chiral algebra $\mathfrak{Z}_{crit}$ admits a canonical deformation into a non-commutative chiral algebra $\mathcl{W}_{h}$. In this paper we will express the resulting first order deformation via the chiral algebra $\mathcal{D}_{crit}$ of chiral differential operators of $G((t))$ at the critical level.

Algebras of twisted chiral differential operators and affine localization of $\frak{g}$-modules

We propose a notion of algebra of {\it twisted} chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of "smallest" such modules are irreducible $\ghat$-modules and all …

One example of a chiral Lie group

We quantize the Khesin-Zakharevich Poisson-Lie group of pseudo-differential symbols.

Two-Dimensional Twisted Sigma Models And The Theory of Chiral Differential Operators

In this paper, we study the perturbative aspects of a twisted version of the two-dimensional $(0,2)$ heterotic sigma model on a holomorphic gauge bundle $\mathcal E$ over a complex, hermitian manifold $X$. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". In particular, the physical anomalies of the sigma model can be reinterpreted in terms of an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on $X$. One can also …

手征临界轨迹与拓扑结构

The chiral critical locus and topological structures

该文研究光滑有向簇上函数导出临界轨迹对应的微分分次VOA，推广Witten的物理构造证明齐次函数下该VOA具拓扑结构，算出其秩的表达式，并讨论了BV量子化与特征标计算。

Chiral differential operators via Batalin-Vilkovisky quantization

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 …