higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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 Dirac operator on smooth loop space, only that due to $S^1$-equivariance (orelse due to a large volume limit) the index is fully determined by the restriction of that operator to formal loops.
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 $X$.
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