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I have created Sullivan model of free loop space with the formula and pointers to the literature.
There is a canonical $S^1$-action on the free loop space. For Sullivan models $(\wedge^\bullet (V \oplus s V, d_{\mathcal{L}X} )$ such that $d_X$ is simple enough, then it is easy to guess a Sullivan model for the $S^1$-homotopy quotient. Namely add a generator $\omega_2$ of degre 2 and then modify the differential on the unshifted generators by adding a term proportional to the corresponding shifted generator wedge $\omega_2$.
Is there any published statement about Sullivan models for the homotopy quotients $\mathcal{L}X/S^1$?
I have found a source for the proof of the Sullivan model for $\mathcal{L}X/S^1$. It is theorem A in Vigué-Burghelea 85. I have added the statement here.
I have added an Examples-section The 4-sphere and twisted de Rham cohomology which spells out the Sullivan model for $\mathcal{L}S^4 // S^1$ and makes an observation of how this relates to a kind of caloron correspondence.
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