nLab chiral differential operator

Idea

(…) see at chiral de Rham complex (…)

Properties

Relation to the Witten genus

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

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

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

Relation to formal loop space

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

Related concepts

• chiral de Rham complex, vertex operator algebra, chiral algebra

• formal loop space

References

The original articles are

• 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

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 $W_3$ 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

• 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

• Pokman Cheung, Chiral differential operators: formal loop group actions and associated modules (arXiv:1205.7089)