derived smooth geometry
A derived loop space is a free loop space object in such .
More specifically, if is an ordinary Lawvere theory, regarded as a 1-truncated -theory, then are its simplicial algebras. There is a canonical embedding of the ordinary algebras into the -algebras, so that we may regard as an object of . Then the derived loop space of is its free loop space object computed in .
The point is that the derived loop space of an ordinary in general is a significantly richer object than the free loop space object of as computed just in the underived -topos . In fact, since is 0-truncated in , it coincides with its free loop space object there, but the derived loop space does not.
The function complex on the derived loop space is the Hochschild homology complex of . See there for further details. In particular see the section Hochschild cohomology – As function algebra on the derived loop space.
Also see free loop space object for more information.
Loop Spaces and Langlands Parameters (arXiv:0706.0322)
Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry (arXiv:0805.0157)
Loop Spaces and Connections (arXiv:1002.3636)
This article uses Toën’s theory of function algebras on ∞-stacks for showing that the function complex on a derived loop space is under mild conditions the Hochschild homology complex of hence by Hochschild-Kostant-Rosenberg theorem the collection of Kähler differential forms on , and that the functions on that are invariant under the canonical -action on are the closed forms. This also gives a geometric interpretation of the old observation by Maxim Kontsevich and others, that the differential and grading on the de Rham complex may be understood as induced from automorphisms of the odd line.
Loop Spaces and Representations (arXiv:1004.5120)