(…) see at chiral de Rham complex (…)
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 $X$:
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.)
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 $X$.
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 $W_3$ 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