nLab chiral de Rham complex

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 $X$ 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.

Related concepts

• chiral differential operator, vertex operator algebra, chiral algebra

• Berkovits superstring

• formal loop space

References

• 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:

  • Nekrasov on pure spinor superstring

  • Sheaves of CDOs

• Yuly Billig, Vyacheslav Futorny, Representations of Lie algebra of vector fields on a torus and chiral de Rham complex, arxiv/1108.6092

• A. Linshaw, Varghese Mathai, Twisted chiral de Rham complex, generalized geometry, and T-duality, Commun. Math. Phys. 339 (2015) 663–697 doi

• Mikhail Kapranov, Eric Vasserot, Vertex algebras and formal loop space, Publ. Math., Inst. Hautes Étud. Sci. 100 (2004) 209–269 (2004) doi arXiv:math/0107143

We construct a certain algebro-geometric version $\mathcal{L}(X)$ of the free loop space for a complex algebraic variety $X$. This is an ind-scheme containing the scheme $\mathcal{L}_0(X)$ 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 $\mathcal{L}(X)$ supported in $\mathcal{L}_0(X)$. We also show that $\mathcal{L}(X)$ 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.