nLab chiral de Rham complex

Contents

Contents

Idea

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.

Properties

Fine resolution

The chiral de Rham complex sheaf of vertex operator algebras as a resolution to a fine sheaf by the chiral Dolbeault complex (Cheung 10).

Relation to 2d (2,0)-superconformal QFT

The chiral de Rham complex of XX arises as the quantum observables of the topologically twisted 2d (2,0)-superconformal QFT sigma-model with target space XX.

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.

References

The resolution by the chiral Dolbeault complex is due to

We construct a certain algebro-geometric version (X)\mathcal{L}(X) of the free loop space for a complex algebraic variety XX. This is an ind-scheme containing the scheme 0(X)\mathcal{L}_0(X) of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on (X)\mathcal{L}(X) supported in 0(X)\mathcal{L}_0(X). We also show that (X)\mathcal{L}(X) possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that “all” linear constructions applied to the free loop space produce vertex algebras.

Last revised on October 27, 2023 at 13:00:14. See the history of this page for a list of all contributions to it.