nLab
chiral differential operator

Contents

Idea

(…) see at chiral de Rham complex (…)

Properties

Relation to the Witten genus

For UU \subset \mathbb{C} an open subset of the complex plane then the space 𝒟 ch(U)\mathcal{D}^{ch}(U) of chiral differential operators on UU is naturally a super vertex operator algebra. For XX 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 𝒟 X ch()\mathcal{D}^{ch}_X(-) on XX. Its cochain cohomology H (𝒟 X ch)H^\bullet(\mathcal{D}^{ch}_X) is itself a super vertex operator algebra and its super-Kac-Weyl character is proportional to the Witten genus w(X)w(X) of XX:

charH (𝒟 X ch)w(X). char H^\bullet(\mathcal{D}^{ch}_X)\propto w(X) \,.

Physically this result is understood by observing that 𝒟 X ch\mathcal{D}^{ch}_Xis 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.)

Relation to formal loop space

With XX a suitable scheme, its formal loop space L infXL_inf X in the sense of (Kapranov-Vasserot I) has a Tate structure? and hence an associated determinantal gerbe Det L infXDet_{L_{inf} X} with band? 𝒪 L infX *\mathcal{O}^\ast_{L_{inf} X}. According to (Kapranov-Vasserot IV) this gerbe is essentially identified with the gerbe CDO XCDO_X of chiral differential operators on XX.

References

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 W 3W_3 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

Tentative aspects of a generalization to differential geometry are discussed in

Revised on May 13, 2014 05:08:32 by Urs Schreiber (109.144.240.2)