(…) see at chiral de Rham complex (…)
For an open subset of the complex plane then the space of chiral differential operators on is naturally a super vertex operator algebra. For a complex manifold such that its first Chern class and second Chern class vanish over the rational numbers, then this assignment gives a sheaf of vertex operator algebras on . Its cochain cohomology is itself a super vertex operator algebra and its super-Kac-Weyl character is proportional to the Witten genus of :
Physically this result is understood by observing that is the sheaf of quantum observables of the topologically twisted 2d (2,0)-superconformal QFT (see there for more on this) of which the Witten genus is (the large volume limit of) the partition function.
See (Cheung 10) for a brief review (where the problem of generalizing of this construction to sheaves of vertex operator algebras over more general string structure manifolds is addressed.)
With a suitable scheme, its formal loop space in the sense of (Kapranov-Vasserot I) has a Tate structure? and hence an associated determinantal gerbe with band . According to (Kapranov-Vasserot IV) this gerbe is essentially identified with the gerbe of chiral differential operators on .
The original articles are
Surveys and further developments include
Andrew R. Linshaw, Introduction to invariant chiral differential operators, in: Vertex operator algebras and related areas, 157–168, Contemp. Math. 497, Amer. Math. Soc. 2009; Invariant chiral differential operators and the algebra, J. Pure Appl. Algebra 213 (2009), 632-648, arxiv/0710.0194, MR2010b:17035, doi
Pokman Cheung, Chiral differential operators and topology, arxiv/1009.5479
The relation to formal loop space geometry is discussed in
Mikhail Kapranov, E. Vasserot, Vertex algebras and the formal loop space Publications Mathématiques de l’IHÉS,
100 (2004), 209–269. (arXiv:math/0107143)
Mikhail Kapranov, E. Vasserot, Formal Loops II : the local Riemann-Roch theorem for determinantal gerbes, Ann. Sci. ENS, (arXiv:math/0509646)
Mikhail Kapranov, E. Vasserot, Formal loops III: Factorizing functions and the Radon transform (arXiv:math/0510476)
Mikhail Kapranov, E. Vasserot, Formal loops IV: Chiral differential operators (arXiv:math/0612371)
Tentative aspects of a generalization to differential geometry are discussed in
Last revised on May 13, 2014 at 05:08:32. See the history of this page for a list of all contributions to it.