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 $\Omega^{ ch}_X$, 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 $(\Omega_X, d_{ DR})\hra (\Omega_X^{ ch}, d_{ DR}^{ ch})$ which is a quasiisomorphism. If $X$ is Calabi-Yau then this sheaf admits an N=2 supersymmetry. For some $X$ (for example, for curves or for the flag spaces $G/B$), one can construct also a purely even analogue of this sheaf, a { \it chiral structure sheaf} $\CO^{ ch}_X$. For the projective line, the space of global sections of the last sheaf is the irreducible vacuum $\hsl(2)$-module on the critical level.
Categories: math.AG