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