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Given a simply connected topological space $X$, it is well known that its free loop space $LX$ has cohomology being the Hochschild homology of the singular cochains [1]:

$$HH_\bullet(S^\star X) \simeq H^\bullet(LX)$$

The circle $S^1 = SO(2)$ naturally acts on $LX$, so I suppose given any $\theta \in S^1$, there should be an induced action

$$ \tilde{\theta}: HH_\bullet(S^\star X) \to HH_\bullet(S^\star X).$$

Unfortunately, the isomorphism is given in the context of the simplicial setting, which I'm not familiar with. In particular, it isn't clear how I should let $\theta$ act on the simplicial model of $LX$.

Question

How to describe $\tilde{\theta}$ explicity? Can I even lift that action to an explicit endo chain map of the Hochschild complex?

EDIT The action is homotopic to the identity, so is trivial on the Hochschild homology. It remains to investigate the lifted action to the Hochschild complex.

Reference

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    $\begingroup$ Isn’t the action of a point of the circle homotopic to the identity, and thus induces the identity on homology? $\endgroup$ Commented Aug 30, 2021 at 23:51
  • $\begingroup$ Maybe the OP meant to describe the action of the fundamental class of the circle? In which case it is precisely the degree one map given by Connes' boundary operator. $\endgroup$ Commented Aug 31, 2021 at 10:52
  • $\begingroup$ @ManuelRivera would you elaborate on Connes' boundary operator? (I hope to see the action on the chain level.) $\endgroup$ Commented Aug 31, 2021 at 12:21

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