nLab formal loop space

Idea

The space of infinitesimal loops in a given space $X$ – hence the formal neighbourhood of the constant loops $X \hookrightarrow L X$ – is often called the formal loop space (as in formal geometry).

Formal loop spaces have application for instance in the theory of elliptic genera, since it was shown that the Witten genus is the $S^1$-equivariant index of a Dirac-Ramond operator which is like a Dirac operator on smooth loop space, only that due to $S^1$-equivariance (or else due to a large volume limit) the index is fully determined by the restriction of that operator to formal loops.

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 $Det_{L_{inf} X}$ 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

• loop space, loop space object

• smooth loop space

• 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)