## Chiral de Rham complex.(English)Zbl 0952.14013

From the introduction: The aim of this note is to define sheaves of vertex algebras on smooth manifolds. In this note, “vertex algebra” will have the same meaning as in V. Kac’s book: “Vertex algebras for beginners” (1996; Zbl 0861.17017; 2nd ed. 1998; Zbl 0924.17023). Recall that these algebras are by definition $$\mathbb{Z}/ (2)$$-graded. “Smooth manifold” will mean a smooth scheme of finite type over $$\mathbb{C}$$.
For each smooth manifold $$X$$, we construct a sheaf $$\Omega_X^{\text{ch}}$$, called the chiral de Rham complex of $$X$$. It is a sheaf of vertex algebras in the Zariski topology. It comes equipped with a $$\mathbb{Z}$$-grading by fermionic charge, and the chiral de Rham differential $$d_{\text{DR}}^{\text{ch}}$$, which is an endomorphism of degree 1 such that $$(d_{\text{DR}}^{\text{ch}})^2= 0$$. One has a canonical embedding of the usual de Rham complex $(\Omega_X, d_{\text{DR}}) \hookrightarrow (\Omega_X^{\text{ch}}, d_{\text{DR}}^{\text{ch}}). \tag{1}$ The sheaf $$\Omega_X^{\text{ch}}$$ has also another $$\mathbb{Z}_{\geq 0}$$-grading, by conformal weight, compatible with fermionic charge one. The differential $$d_{\text{DR}}^{\text{ch}}$$ respects conformal weight, and the subcomplex $$\Omega_X$$ coincides with the conformal weight zero component of $$\Omega_X^{\text{ch}}$$. The wedge multiplication on $$\Omega_X$$ may be restored from the operator product $$\Omega_X^{\text{ch}}$$. The map (1) is a quasi-isomorphism. Each component of $$\Omega_X^{\text{ch}}$$ of fixed conformal weight admits a canonical finite filtration whose graded factors are symmetric and exterior powers of the tangent bundle $${\mathcal T}_X$$ and of the bundle of 1-forms $$\Omega_X^1$$. Similar sheaves exist in complex-analytic and $$C^\infty$$ settings. – If $$X$$ is Calabi-Yau, then the sheaf $$\Omega_X^{\text{ch}}$$ has a structure of a topological vertex algebra (i.e. it admits $$N=2$$ supersymmetry).
The intuitive geometric picture behind our construction is as follows. Let $$LX$$ be the space of “formal loops” on $$X$$, i.e. of the maps of the punctured formal disk to $$X$$. Let $$L^+ X\subset LX$$ be the subspace of loops regular at 0. Note that we have a natural projection $$L^+X\to X$$ (value at 0). We have a functor $p: (\text{Sheaves on }LX) \longrightarrow (\text{Sheaves on }X),$ namely, if $${\mathcal F}$$ is a sheaf on $$LX$$, then $$\Gamma(U; p({\mathcal F}))= \Gamma (LU;F)$$ for an open $$U \subset X$$. Now, the sheaf $$\Omega_X^{\text{ch}}$$ is the image under $$p$$ of the semi-infinite de Rham complex of the $${\mathcal D}$$-module of $$\delta$$-functions along $$L^+ X$$. – This sheaf is a particular case of a more general construction which associates with every $${\mathcal D}$$-module $${\mathcal M}$$ over $$X$$ its “chiral de Rham complex” $$\Omega_X^{\text{ch}} ({\mathcal M})$$ which is a sheaf of vertex modules over the vertex algebra $$\Omega_X^{\text{ch}}$$. Its construction is sketched in §6.
One can also try to define a purely even sheaf $${\mathcal O}_X^{\text{ch}}$$ of vertex algebras, which could be called chiral structure sheaf. Here the situation is more subtle than in the case of $$\Omega_X^{\text{ch}}$$ , where “fermions cancel the anomaly”. One can define this sheaf for curves. If $$\dim (X)> 1$$, then there exists a non-trivial obstruction of cohomological nature to the construction of $${\mathcal O}_X^{\text{ch}}$$. This obstruction can be expressed in terms of a certain homotopy Lie algebra. – However, one can define $${\mathcal O}_X^{\text{ch}}$$ for the flag manifolds $$X= G/B$$ ($$G$$ being a simple algebraic group and $$B$$ a Borel subgroup). The space of global sections $$\Gamma (X;{\mathcal O}_X^{\text{ch}})$$ is the irreducible vaccum $$\widehat{\mathfrak g}$$-module for $${\mathfrak g}= sl(2)$$.
More generally, if we start from an arbitrary $${\mathcal D}$$-module $${\mathcal M}$$ on $$X= G/B$$ corresponding to some $${\mathfrak g}$$-module $$M$$, then we can define its “chiralization” $${\mathcal M}^{\text{ch}}$$ which is an $${\mathcal O}^{\text{ch}}$$-module. It seems plausible that the space of global sections $$\Gamma(X;{\mathcal M}^{\text{ch}})$$ coincides with the Weyl module over $$\widehat{\mathfrak g}$$ corresponding to $$M$$ (on the critical level).

### MSC:

 14F40 de Rham cohomology and algebraic geometry 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 17B68 Virasoro and related algebras 14M15 Grassmannians, Schubert varieties, flag manifolds 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14H81 Relationships between algebraic curves and physics 14J81 Relationships between surfaces, higher-dimensional varieties, and physics 17B69 Vertex operators; vertex operator algebras and related structures

### Citations:

Zbl 0861.17017; Zbl 0924.17023
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