nLab
formal loop space

Idea

The space of infinitesimal loops in a given space.

Properties

Relation to chiral differential operators

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 ${\mathrm{Det}}_{L_{\inf }X}$ with band? ${𝒪}_{L_{\inf }X}^{*}$. According to (Kapranov-Vasserot IV) this gerbe is essentially identified with the gerbe ${\mathrm{CDO}}_X$ of chiral differential operators on $X$.

Related concepts

• loop space, loop space object

• free loop space, free loop space object,

• derived loop space

References

In the context of algebraic geometry formal loop spaces have been introduced and studied 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)