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\begin{document}

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\section*{cyclic loop space}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_right_base_change_along_}{As right base change along $\ast \to \mathbf{B} S^1$}\dotfill \pageref*{as_right_base_change_along_} \linebreak
\noindent\hyperlink{ordinary_cohomology_of__on_cyclic_cohomology_of_}{Ordinary cohomology of $\mathcal{L}X/S^1$ on cyclic cohomology of $X$}\dotfill \pageref*{ordinary_cohomology_of__on_cyclic_cohomology_of_} \linebreak
\noindent\hyperlink{rational_sullivan_model}{Rational Sullivan model}\dotfill \pageref*{rational_sullivan_model} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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 \emph{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 \emph{[[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 \emph{twisted loop space} in (\hyperlink{Witten88}{Witten 88}). This terminology has been widely adopted, for example in the context of the [[transchromatic character]] map (\hyperlink{Stapleton11}{Stapleton 11})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{as_right_base_change_along_}{}\subsubsection*{{As right base change along $\ast \to \mathbf{B} S^1$}}\label{as_right_base_change_along_}

The cyclic loop space $\mathcal{L}X/S^1$ is equivalently the right [[base change]]/[[dependent product]] along the canonical point inclusion $\ast \to B S^1$ (\href{base+change#CyclicLoopSpace}{this prop.}). See also at \emph{[[double dimensional reduction]]}.

\hypertarget{ordinary_cohomology_of__on_cyclic_cohomology_of_}{}\subsubsection*{{Ordinary cohomology of $\mathcal{L}X/S^1$ on cyclic cohomology of $X$}}\label{ordinary_cohomology_of__on_cyclic_cohomology_of_}

Let $X$ be a [[simply connected topological space|simply connected]] [[topological space]].

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

\begin{displaymath}
H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( C^\bullet(X) )
  \,.
\end{displaymath}
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:

\begin{displaymath}
H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( C^\bullet(X) )
\end{displaymath}
(\hyperlink{Loday11}{Loday 11})

If the [[coefficients]] are [[rational numbers|rational]], and $X$ is of [[finite type]] then this may be computed by the \emph{[[Sullivan model for free loop spaces]]}, see there the section on \emph{\href{Sullivan+model+of+free+loop+space#RelationToHochschildHomology}{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

\begin{displaymath}
H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( \Omega^\bullet(X) )
  \,.
\end{displaymath}
\begin{displaymath}
H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( \Omega^\bullet(X) )
  \,.
\end{displaymath}
This is known as \emph{[[Jones' theorem]]} (\hyperlink{Jones87}{Jones 87})

An [[(infinity,1)-category theory|infinity-category theoretic]] proof of this fact is indicated at \emph{\href{Hochschild+cohomology#JonesTheorem}{Hochschild cohomology -- Jones' theorem}}.

\hypertarget{rational_sullivan_model}{}\subsubsection*{{Rational Sullivan model}}\label{rational_sullivan_model}

See at \emph{[[Sullivan model for free loop space]]}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[double dimensional reduction]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[John D.S. Jones]], \emph{Cyclic homology and equivariant homology}, Invent. Math. \textbf{87}, 403-423 (1987) (\href{https://math.berkeley.edu/~nadler/jones.pdf}{pdf})


\item [[Edward Witten]], \emph{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.


\item [[Jean-Louis Loday]], \emph{Free loop space and homology} (\href{https://arxiv.org/abs/1110.0405}{arXiv:1110.0405})


\item [[Nathaniel Stapleton]], \emph{Transchromatic generalized character maps}, Algebr. Geom. Topol. 13 (2013) 171-203 (\href{https://arxiv.org/abs/1110.3346}{arXiv:1110.3346})



\end{itemize}
[[!redirects cyclic loop spaces]]

[[!redirects twisted loop space]] [[!redirects twisted loop spaces]]



\end{document}