nLab chiral Dolbeault complex

Idea

The chiral de Rham complex sheaf of vertex operator algebras of a complex manifold admits a resolution by a fine sheaf which in degree-0 is the Dolbeault-resolution of the structure sheaf. This is the chiral Dolbeault complex (Cheung 10).

Where the construction of the chiral de Rham complex as a sheaf of vertex operator algebras requires the complex analog of a string structure in that the second Chern class vanishes, the construction of the chiral Dolbeault complex requires an analog of a differential string structure where a differential 3-form is chosen such that $\mathbf{d}H \propto \langle R \wedge R\rangle$.

Related concepts

• Witten genus

References

• Pokman Cheung, The Witten genus and vertex algebras (arXiv:0811.1418)