Suppose that a finite group $G$ acts on a smooth complex variety $X$. Then this action lifts to the Chiral de Rham Complex of $X$ and to its cohomology by automorphisms of the vertex algebra structure. We define twisted sectors for the Chiral de Rham Complex (and their cohomologies) as sheaves of twisted vertex algebra modules supported on the components of the fixed-point sets $X^{g}, g \in G$. Each twisted sector sheaf carries a BRST differential and is quasi-isomorphic to the de Rham complex of $X^{g}$. Putting the twisted sectors together with the vacuum sector and taking $G$--invariants, we recover the additive and graded structures of Chen-Ruan orbifold cohomology. Finally, we show that the orbifold elliptic genus is the partition function of the direct sum of the cohomologies of the twisted sectors.
Given a compact complex algebraic variety with an effective action of a finite group $G$, and a class $α\in H^2(G,U(1))$, we introduce an orbifold elliptic genus with discrete torsion $α$, denoted $Ell^α_{orb}(X,G, q, y)$. We give an interpretation of this genus in terms of the chiral de Rham complex attached to the orbifold $[X/G]$. If $X$ is Calabi-Yau and $G$ preserves the volume form, $Ell^α_{orb}(X,G, q, y)$ is a weak Jacobi form. We also obtain a formula for the generating function of the elliptic genera of symmetric products with discrete torsion.
In this paper, we study the perturbative aspects of the half-twisted variant of Witten's topological A-model on a complex orbifold X/G, where G is an isometry group of X. The objective is to furnish a purely physical interpretation of the mathematical theory of the Chiral de Rham complex on orbifolds recently constructed by Frenkel and Szczesny in [Chiral de Rham complex and orbifolds, arXiv: math.AG/0307181]. In turn, one can obtain a novel understanding of the holomorphic (twisted) N=2 superconformal structure underlying the untwisted and twisted sectors of the quantum sigma model, purely in…
We present a superfield formulation of the chiral de Rham complex (CDR) of Malikov-Schechtman-Vaintrob in the setting of a general smooth manifold, and use it to endow CDR with superconformal structures of geometric origin. Given a Riemannian metric, we construct an N=1 structure on CDR (action of the N=1 super--Virasoro, or Neveu--Schwarz, algebra). If the metric is K"ahler, and the manifold Ricci-flat, this is augmented to an N=2 structure. Finally, if the manifold is hyperk"ahler, we obtain an N=4 structure. The superconformal structures are constructed directly from the Levi-Civita connect…
On any Calabi-Yau manifold X one can define a certain sheaf of chiral N=2 superconformal field theories, known as the chiral de Rham complex of X. It depends only on the complex structure of X, and its local structure is described by a simple free field theory. We show that the cohomology of this sheaf can be identified with the infinite-volume limit of the half-twisted sigma-model defined by E. Witten more than a decade ago. We also show that the correlators of the half-twisted model are independent of the Kahler moduli to all orders in worldsheet perturbation theory, and that the relation to…
We define a version of a derived chiral De Rham complex over a locally complete intersection, thereby "chiralizing" a result by Illusie and Bhatt. A similar construction attaches to a graded ring a dg vertex algebra, which we prove to be Morita equivalent to a dg algebra of differential operators. For example, the dg vertex algebra associated to a fat point, which also arises in the Landau-Ginzburg model, is shown to be derived rational.
The chiral de Rham complex is a sheaf of vertex algebras Ω^ch_M on any nonsingular algebraic variety or complex manifold M, which contains the ordinary de Rham complex as the weight zero subspace. We show that when M is a Kummer surface, the algebra of global sections is isomorphic to the N = 4 superconformal vertex algebra with central charge 6. Previously, CP^n was the only manifold where a complete description of the global section algebra was known.
The aim of this note is to define certain sheaves of vertex algebras on smooth manifolds. For each smooth complex algebraic (or analytic) manifold $X$, we construct a sheaf $Ω^{ch}_X$, called the {\bf chiral de Rham complex} of $X$. It is a sheaf of vertex algebras in the Zarisky (or classical) topology, It comes equipped with a $\BZ$-grading by {\it fermionic charge}, and the {\it chiral de Rham differential} $d_{DR}^{ch}$, which is an endomorphism of degree 1 such that $(d_{DR}^{ch})^2=0$. One has a canonical embedding of the usual de Rham complex $(Ω_X, d_{DR})\hra (Ω_X^{ch}, d_{DR}^{ch})$ …
This paper surveys the authors recent work on two variable elliptic genus of singular varieties. The last section calculates a generating function for the elliptic genera of symmetric products. This generalizes the classical results of Macdonald and Zagier.
In this paper, we study the perturbative aspects of a "B-twisted" 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 as 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 $X$. In addition, one can also obtain …
This paper is a sequel to math.AG/9803041. It consists of three parts. In the first part we give certain construction of vertex algebras which includes in particular the ones appearing in op. cit. In the second part we show how the cohomology ring $H^*(X)$ of a smooth complex variety $X$ could be restored from the correlation functions of the vertex algebra $RΓ(X;Ω^{ch}_X)$. In the third part, we prove first a useful general statement that the sheaf of loop algebras over the tangent sheaf $\Cal{T}_X$ acts naturally on $Ω^{ch}_X$ for every smooth $X$ (see §1). The Z-graded vertex algebra $H^*(X…
The data from collider experiments and cosmic observatories indicates the existence of three light matter generations. In some classes of string compactifications the number of generations is related to a topological quantity, the Euler characteristic. However, these do not explain the existence of three generations. In a class of free fermionic string models, related to the Z2 X Z2 orbifold compactification, the existence of three generations is correlated with the existence of three twisted sectors in this class of compactifications. However, the three generation models are constructed in th…
We construct a spectral sequence that converges to the cohomology of the chiral de Rham complex over a Calabi-Yau hypersurface and whose first term is a vertex algebra closely related to the Landau-Ginburg orbifold. As an application, we prove an explicit orbifold formula for the elliptic genus of Calabi-Yau hypersurfaces.
We reproduce the quantum cohomology of toric varieties (and of some hypersurfaces in projective spaces) as the cohomology of certain vertex algebras with differential. The deformation technique allows us to compute the cohomology of the chiral de Rham complex over the projective space.
For any algebraic super-manifold M we define the super-ind-scheme LM of formal loops and study the transgression map (Radon transform) on differential forms in this context. Applying this to the super-manifold M=SX, the spectrum of the de Rham complex of a manifold X, we obtain, in particular, that the transgression map for X is a quasi-isomorphism between the [2,3)-truncated de Rham complex of X and the additive part of the [1,2)-truncated de Rham complex of LX. The proof uses the super-manifold SSX and the action of the Lie superalgebra sl(1|2) on this manifold. This quasi-isomorphism result…
We conjecture a description of the vertex (chiral) algebras of the (0,2) nonlinear sigma models on smooth quintic threefolds. We provide evidence in favor of the conjecture by connecting our algebras to the cohomology of a twisted chiral de Rham sheaf. We discuss CY/LG correspondence in this setting.
