chiral de Rham complex
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).
Relation to 2d (2,0)-superconformal QFT
The chiral de Rham complex of arises as the quantum observables of the topologically twisted 2d (2,0)-superconformal QFT sigma-model with target space .
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.
- Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman, Gerbes of chiral differential operators Math. Res. Lett. 7 (2000), no. 1, 55–66, MR2002c:17040, math.AG/9906117; Gerbes of chiral differential operators. II. Vertex algebroids, Invent. Math. 155 (2004), no. 3, 605–680, MR2005e:17047, math.AG/0003170, doi; Gerbes of chiral differential operators. III, in: The orbit method in geometry and physics (Marseille, 2000), 73–100, Progr. Math. 213, Birkhäuser 2003, MR2005a:17028, math.AG/0005201, On chiral differential operators over homogeneous spaces, Int. J. Math. Math. Sci. 26 (2001), no. 2, 83–106, MR2002g:14020, math.AG/0008154, doi
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
Revised on March 21, 2014 09:05:17
by Urs Schreiber