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Any free loop space $\mathcal{L}X$ has a canonical action (infinity-action) of the circle group $S^1$. The homotopy quotient $\mathcal{L}(X)/S^1$ of this action might be called the cyclic loop space of $X$.
If $X = Spec(A)$ is an affine variety regarded in derived algebraic geometry, then $\mathcal{O}(\mathcal{L}Spec(A))$ is the Hochschild homology of $A$ and $\mathcal{O}((\mathcal{L}Spec(A))/S^1)$ the corresponding cyclic homology, see the discussion at Hochschild cohomology.
If $X = Y//G$ is the homotopy quotient of a topological space by a topological group action, regarded as a locally constant $\infty$-stack, so that the $S^1$-action on $\mathcal{L}(X//G)$ is an $B \mathbb{Z}$-action, then the restriction of the cyclic loop space to the constant loops $\mathcal{L}_{const}Y//G \to \mathcal{L}(Y//G)$ has been called the twisted loop space in (Witten 88). This terminology has been widely adopted, for example in the context of the transchromatic character map (Stapleton 11)
The cyclic loop space $\mathcal{L}X \sslash S^1$ is equivalently the right base change/dependent product along the canonical point inclusion $\ast \to B S^1$ (this prop.) into the delooping of $S^1$ (the classifying space of the circle group when realized in the homotopy theory of topological spaces). See also at double dimensional reduction (BMSS 19, Sec. 2.2, following FSS 18, Sec. 3).
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 $C^\bullet(X)$:
Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space $\mathcal{L}X \sslash S^1$ is the cyclic homology $HC_\bullet$ of the singular chains:
(Jones 87, Thm. A, review in Loday 92, Cor. 7.3.14, Loday 11, Sec. 4)
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
This is known as Jones' theorem (Jones 87)
An infinity-category theoretic proof of this fact is indicated at Hochschild cohomology – Jones’ theorem.
For $X \in TopSpaces^{\ast/}$ a pointed topological space with reduced suspension denoted $\Sigma X$, the stabilization of its cyclic loop spaces splits as a direct sum of suspension spectra, hence the underlying infinite loop space of that as a product space, of the following form:
In terms of the suspension spectra themselves:
(Carlsson & Cohen 1987, Cor. C, announced in Cohen 1985, p. 194) See also at stable splitting of mapping spaces.
Here
$\wedge$ denotes the smash product,
$\mathbb{Z}/k$ a cyclic group,
$B G$ the classifying space of a Lie group $G$ and $E G$ its universal principal bundle,
$(-)_+$ the adjoining of a base point
and $(-)\wedge_{G} (-) \;\coloneqq\; \big((-) \wedge (-)\big)/G$.
For $X = S^n$ the n-sphere, we have $\Sigma X \,=\, S^{n+1}$ and hence
For the rational homotopy type of cyclic loop spaces see at Sullivan model for free loop space.
The notion of the cyclic loop space of a topological space appears as:
Ralph Cohen, p. 194 of: A Model for the Free Loop Space of a Suspension, in: Haynes Miller, Douglas Ravenel (eds.), Algebraic Topology, Lecture Notes in Mathematics 1286, Springer 1985 (doi:10.1007/BFb0078743)
John D.S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87, 403-423 (1987) (pdf, doi:10.1007/BF01389424)
Gunnar Carlsson, Ralph Cohen, The cyclic groups and the free loop space, Commentarii Mathematici Helvetici 62 (1987) 423–449 (doi:10.1007/BF02564455, dml:140092)
Edward Witten, The index of the Dirac operator in loop space. In Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), volume 1326 of Lecture Notes in Math., pages 161–181. Springer, Berlin, 1988 (doi:10.1007/BFb0078045)
Jean-Louis Loday, Cyclic Spaces and $S^1$-Equivariant Homology (doi:10.1007/978-3-662-21739-9_7)
Chapter 7 in: Cyclic Homology, Grundlehren 301, Springer 1992 (doi:10.1007/978-3-662-21739-9)
Jean-Louis Loday, Section 4 of: Free loop space and homology, Chapter 4 in: Janko Latchev, Alexandru Oancea (eds.): Free Loop Spaces in Geometry and Topology, IRMA Lectures in Mathematics and Theoretical Physics 24, EMS 2015 (arXiv:1110.0405, ISBN:978-3-03719-153-8)
Nathaniel Stapleton, Transchromatic generalized character maps, Algebr. Geom. Topol. 13 (2013) 171-203 (arXiv:1110.3346)
Specifically on cyclic loop spaces of n-spheres:
See also:
A version of the cyclic loop stack of orbifolds, or at least its restriction to constant loops, namely Huan's inertia orbifold, is discussed in the context of equivariant elliptic cohomology via Tate K-theory in:
following
and recalled/expanded on in several followup articles, such as in
The above formulation of cyclic loop spaces, in the generality of ∞-stacks, as right base change to the delooping of the circle group, and its relation to double dimensional reduction in brane-physics, is due to:
following the analogous discussion in rational homotopy theory in
with exposition in
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