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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 S1-action on the free loop space. For Sullivan models (∧•(V⊕sV,dℒX) such that dX is simple enough, then it is easy to guess a Sullivan model for the S1-homotopy quotient. Namely add a generator ω2 of degre 2 and then modify the differential on the unshifted generators by adding a term proportional to the corresponding shifted generator wedge ω2.
Is there any published statement about Sullivan models for the homotopy quotients ℒX/S1?
I have found a source for the proof of the Sullivan model for ℒX/S1. 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 ℒS4//S1 and makes an observation of how this relates to a kind of caloron correspondence.
added pointer to:
also this one:
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