A sheaf of vertex operator algebras (in fact a vertex operator algebroid) which is naturally associated with a complex manifold whose first and second Chern class vanishes.
The chiral de Rham complex sheaf of vertex operator algebras as a resolution to a fine sheaf by the chiral Dolbeault complex (Cheung 10).
The chiral de Rham complex of $X$ arises as the quantum observables of the topologically twisted 2d (2,0)-superconformal QFT sigma-model with target space $X$.
Under suitable geometric conditions (a version of string structure) the local chiral de Rham complexes glue together to a sheaf of vertex operator algebras and serves to compute the Witten genus.
The resolution by the chiral Dolbeault complex is due to
Pokman Cheung, The Witten genus and vertex algebras (arXiv:0811.1418)
Urs Schreiber at string cafe:
Yuly Billig, Vyacheslav Futorny, Representations of Lie algebra of vector fields on a torus and chiral de Rham complex, arxiv/1108.6092