# Describing the THH of function spectra?

Are there any results describing the $$THH$$ of spectra of the form $$F(X, E)$$ where $$X$$ is a space (say, finite CW) and $$E$$ is a (nice enough) ring spectrum?

I'm happy to put various (further, or other) restrictions on $$X$$ and $$E$$. I'm also happy to consider specific examples for $$E$$ (e.g. $$H\mathbb{Z}$$, $$KU$$, $$MU$$, $$\mathbb{S}$$ etc).

I'm hoping for something similar-ish to the result that $$HH_*S^*X$$, the Hochschild homology of the singular cochains on $$X$$, is equivalent to $$H^*LX$$, the cohomology of loop space. That already has a $$THH$$ analog in the Blumberg-Cohen-Schlichtkrull theorem, which describes the $$THH$$ of certain Thom spectra in terms of Thom spectra over loop space.

I would also be interested in "cyclic" results, i.e. some kind of topological analogs of the fact that the cyclic homology of $$S^*X$$ is the $$S^1$$-equivariant cohomology of $$LX$$.

• If $X$ is finite CW, then it is dualizable in spectra, so that $F(X, E)\cong E\tensor DX$, and then $THH(F(X,E))\cong THH(E)\tensor B^{cyc}(DX)$, where $D(X)\tensor D(X)\to D(X)$ is the dual of the diagonal. I don't know what is known about the $B^{cyc}(DX)$, but it seems approachable at least. – Noah Riggenbach Mar 30 at 2:19
• This might also give a connection to the Blumberg-Cohen-Schlichtkrull result, since Atiyah duality gives that $\Sigma_+^\infty M$ and $Th(NM)$ are dual. – Noah Riggenbach Mar 30 at 2:26