I gave *Jones’ theorem* (long requested at *Hochschild homology*) a quick statement and references. Copied this also to the entries *free loop space*, *cyclic loop space* and *cyclic homology* and *Sullivan models of free loop spaces*:

Let $X$ be a simply connected topological space.

The ordinary cohomology $H^\bullet$ of its free loop space is the Hochschild homology $HH_\bullet$ of its singular chains $C^\bullet(X)$:

$H^\bullet(\mathcal{L}X) \simeq HH_\bullet( C^\bullet(X) ) \,.$Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space $\mathcal{L}X/^h S^1$ is the cyclic homology $HC_\bullet$ of the singular chains:

$H^\bullet(\mathcal{L}X/^h S^1) \simeq HC_\bullet( C^\bullet(X) )$(Loday 11)

If the coefficients are rational, and $X$ is of finite type then this may be computed by the *Sullivan model for free loop spaces*, see there the section on *Relation to Hochschild homology*.

In the special case that the topological space $X$ carries the structure of a smooth manifold, then the singular cochains on $X$ are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that

$H^\bullet(\mathcal{L}X) \simeq HH_\bullet( \Omega^\bullet(X) ) \,.$ $H^\bullet(\mathcal{L}X/^h S^1) \simeq HC_\bullet( \Omega^\bullet(X) ) \,.$This is known as *Jones’ theorem* (Jones 87)

An infinity-category theoretic proof of this fact is indicated at *Hochschild cohomology – Jones’ theorem*.