##
**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).

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 |