Chiral de Rham complex and orbifolds

Suppose that a finite group $G$ acts on a smooth complex variety $X$. Then this action lifts to the Chiral de Rham complex $\Omega ^{\operatorname {ch}}_{X}$ of $X$ and to its cohomology by automorphisms of the vertex algebra structure. We define twisted sectors for $\Omega ^{\operatorname {ch}}_{X}$ (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.