nLab derived loop space

Idea

A derived loop space is a free loop space object in derived geometry.

Definition

Let $T$ be an (∞,1)-algebraic theory and $C \subset T Alg_\infty^{op}$ an (∞,1)-site of formal duals of $\infty$-algebras over $T$. Then the (∞,1)-topos $\mathbf{H} = (\infty,1) Sh(C)$ encodes derived geometry modeled on $T$.

A derived loop space is a free loop space object in such $\mathbf{H}$.

More specifically, if $T$ is an ordinary Lawvere theory, regarded as a 1-truncated $(\infty,1)$-theory, then $T Alg_{\infty}$ are its simplicial algebras. There is a canonical embedding $T Alg^{op} \hookrightarrow T Alg_\infty^{op}$ of the ordinary algebras into the $\infty$-algebras, so that we may regard $X \in T Alg^{op}$ as an object of $\mathbf{H}$. Then the derived loop space of $X$ is its free loop space object computed in $\mathbf{H}$.

The point is that the derived loop space of an ordinary $X \in T Alg^{op}$ in general is a significantly richer object than the free loop space object of $X$ as computed just in the underived $(\infty,1)$-topos $(\infty,1)Sh(T Alg^{op})$. In fact, since $X$ is 0-truncated in $(\infty,1)Sh(T Alg^{op})$, it coincides with its free loop space object there, but the derived loop space does not.

Function complexes on derived loop spaces: Hochschild homology

The function complex on the derived loop space $\mathcal{L}X$ is the Hochschild homology complex of $C(X)$. 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.

Related concepts

• loop space object, free loop space object,

  • delooping

  • loop space, free loop space,

  • formal loop space, derived loop space

  • derived microlocalization

  • inertia orbifold

• suspension object

  • suspension

References

• David Ben-Zvi, David Nadler:

  • Loop Spaces and Langlands Parameters (arXiv:0706.0322)

  • Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry (arXiv:0805.0157)

  • Loop Spaces and Representations (arXiv:1004.5120)

Using Toën’s theory of function algebras on ∞-stacks for showing that the function complex on a derived loop space $\mathcal{L}X$ is, under mild conditions, the Hochschild homology complex of $X$ hence by Hochschild-Kostant-Rosenberg theorem the collection of Kähler differential forms on $X$, and that the functions on $\mathcal{L}X$ that are invariant under the canonical $S^1$-action on $\mathcal{L}X$ 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):

• Bertrand Toën, Gabriele Vezzosi: $S^1$-Equivariant simplicial algebras and de Rham theory [arXiv:0904.3256]

• David Ben-Zvi, David Nadler: Loop Spaces and Connections, Journal of Topology 5 2 (2012) [arXiv:1002.3636, doi:10.1112/jtopol/jts007]