Chiral de rham complex and orbifolds

Abstract

Suppose that a finite group G acts on a smooth complex variety X. Then this action lifts to the Chiral de Rham complex Ωchx of X and to its cohomology by automorphisms of the vertex algebra structure. We define twisted sectors for Ωxch ( and their cohomologies) as sheaves of twisted vertex algebra modules supported on the components of the fixed-point sets Xg,g ∈ G. Each twisted sector sheaf carries a BRST differential and is quasi-isomorphic to the de Rham complex of Xg . Putting the twisted sectors together with the vacuum sector and taking G-invariants, we recover the additive and graded structures of ChenRuan 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.