Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman
In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by A.Vaintrob and two of the authors. Hopefully our result clarifies to some extent the constructions of the above work.
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This note is a sequel to "Gerbes of chiral differential operators. II", math.AG/0003170. We study gerbes of chiral differential operators acting on the exterior algebra $ΛE$ of a vector bundle over a smooth algebraic variety $X$. When $E=Ω^1_X$ this gerbe admits a canonical global section which coincides with the chiral de Rham complex of $X$.
Results of our previous note, "Gerbes of chiral differential operators" (Math. Res. Letters, 7(2000), 55-66), are discussed in the algebraic category.
We relate the gerbe of sheaves of chiral differential operators (CDO) on a algebraic variety X, studied by Gorbounov, Malikov and Schechtman, to the determinantal gerbe of the formal loop space LX introduced in our earlier paper. The liens of the two gerbes are related by a version of the symplectic action homomorphism. The determinantal gerbe of LX has a factorization structure in the spirit of Beilinson and Drinfeld. In our identification, sheaves of CDO correspond to factorizing objects of this factorization gerbe.
We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes, due to W. Messing and the author (arXiv:math.AG/0106083), and give a more direct derivation of the associated cocycle equations. This leads us to a more restrictive definition of the corresponding coboundary relations. We also show that the diagrammatic proofs of certain local curving and curvature equations may be replaced by computations with differential forms.
We define in a global manner the notion of a connective structure for a gerbe on a space X. When the gerbe is endowed with trivializing data with respect to an open cover of X, we describe this connective structure in two separate ways, which extend from abelian to general gerbes the corresponding descriptions due to J.- L. Brylinski and N. Hitchin. We give a global definition of the 3-curvature of this connective structure as a 3-form on X with values in the Lie stack of the gauge stack of the gerbe. We also study this notion locally in terms of more traditional Lie algebra-valued 3-forms. Th…
In this note we study algebras of chiral differential operators over an algebraic group and over homogeneous spaces $G/G'$ where $G$ is simple and $G'$ is unipotent or parabolic.
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.
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…
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.
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 …
We quantize the Khesin-Zakharevich Poisson-Lie group of pseudo-differential symbols.
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 …
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。
We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. This is applied to (1) finding the character of $G$-integrable irreducible highest weight modules over the affine Lie algebra at the critical level, and (2) computing a certain elliptic genus of the flag manifold. The main tool is a result that interprets the Drinfeld-Sokolov reduction as a derived functor.
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 …
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