Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself? A well-known theorem of Atiyah and Bott states that given a finite dimensional oriented manifold $M$ with circle action, the $S^1$-equivariant cohomology of $M$ (with $\mathbb{Q}$ coefficients) is isomorphic to the $S^1$-equivariant cohomology of the fixed points after inverting the Euler class of the normal bundle $\nu$ of $i: Fix(M)\hookrightarrow M$ (and shifting degree by the rank of $\nu$). In particular, the isomorphisms are the equivariant pushforward $i_\ast$ in one direction and $i^\ast/eul(\nu)$ in the other. Notice that the $S^1$-equivariant cohomology of a space is a $\mathbb{Q}[u]$-module and $eul(\nu)$ is a multiple of a power of $u$. This means that we just need to localize at $u$ to obtain the theorem.
Now, if we let $X$ be a smooth manifold and $LX$ its free loop space, $S^1$ acts trivially on the former and by loop rotation on the latter. The inclusion map $i:X\hookrightarrow LX$ sending $X$ to the trivial loops is exactly the inclusion of the fixed points. We cannot apply the theorem above for dimensional reasons.
Question 1 Is it still true that ${{H_{S^1}}(LX)}_{(u)} \cong{H_{S^1}(X)}_{(u)}$?
The same could be rephrased as an isomorphism between the localized cyclic homology (see Jones) of $\Omega^\ast(X)$, namely $H\hat{C}_{-\ast}(\Omega^\ast (X))$ and $H^*(X)\otimes \mathbb{Q}[u,u^{-1}]$.
On one hand it seems natural enough for this isomorphism to hold, on the other hand, I have not seen anything stating any connection between them, not even in the paper of Jones in which he defines localized cyclic homology.
Question 2 Is the corresponding statement for localized cyclic homology true? That is, given a cochain complex $S$, is it true that
$H\hat{C}_{-*}(S)\cong H^n(S)\otimes\mathbb{Q}[u,u^{-1}]$?
Atiyah, Michael F.; Bott, Raoul, The moment map and equivariant cohomology, Topology 23, 1-28 (1984). ZBL0521.58025.
Jones, John D. S., Cyclic homology and equivariant homology, Invent. Math. 87, 403-423 (1987). ZBL0644.55005.
 A: I don't know what you would mean by the normal bundle of $M$ in $LM$, but the answer to one question is "no": When the $\mathbb Q[u]$-module $H^\ast_{S^1}(LM)$ is localized by inverting $u$, it does not become the same as the cohomology of $M$.
In fact, in my paper in Topology (1985) "Cyclic homology, derivations, and the free loopspace" I prove a very different and incompatible general statement: this localized cohomology depends only on the fundamental group of $M$. More precisely, for a $2$-connected map $X\to Y$ the relative equivariant cohomology of $LX\to LY$ is a torsion $\mathbb Q[u]$-module.
There are related statements involving cyclic homology of differential graded algebras.
A: The first question appears to have an affirmative answer in Corollary 1 in Exotic Twisted Equivariant Cohomology of Loop Spaces, Twisted Bismut–Chern Character and T-Duality by Fei Han and Varghese Mathai. However this seems to conflict with the answer by Tom Goodwillie, and I am not yet able to exactly see what is causing this discrepancy. My guess is that actually two different versions of $LM$ as a topological space/infinite dimensional manifold occur in the two articles so that both statements are true relative to the version of $LM$ they assume. But this is just a quick guess.
