Let $(Y, \omega)$ be a compact symplectic manifold and let $Fuk(X,\omega)$ be its Fukaya category. The Hochschild cohomology of this category should be given by $HH^\bullet(Fuk(Y,\omega))=H^\bullet(Y, \mathbb{C})$, at least at the level of vector spaces (the product structure should be the quantum cohomology on $H^\bullet(Y,\mathbb{C})$: see for example Hochschild (co)homology of Fukaya categories and (quantum) (co)homology ). As $Fuk(Y,\omega)$ should be a Calabi-Yau category (see for example Are Fukaya categories Calabi-Yau categories? ), the Hochschild homology $HH_\bullet(Fuk(Y,\omega))$ coincides with the Hochschild cohomology $HH^\bullet(Fuk(Y,\omega))$ up to a shift of the grading.
Let me deform the preceding setting: let me assume that $(Y,\omega)$ is a Kähler manifold and that $W \colon Y \rightarrow \mathbb{C}$ is an holomorphic function on $Y$ (in particular $Y$ is noncompact if one wants $W$ non-constant). One should be able to define the Fukaya-Seidel category $FS((Y,\omega),W)$ (I probably need to assume that $W$ has only isolated non-degenerate critical points given the current technology). The Hochschild homology of this category should be given by:
$HH_\bullet(FS((Y,\omega),W)=H^\bullet(Y,Y_{-\infty},\mathbb{C})$,
where we take the cohomology relative to the fiber $Y_{-\infty}=W^{-1}(z)$ of a point $z\in\mathbb{C}$ with $Re(z)<<0$. But now $FS((Y,\omega),W)$ is no longer Calabi-Yau and so the Hochschild cohomology is no longer simply related to the Hochschild homology. So my question is:
What is (or should be) the Hochschild cohomology $HH^\bullet(FS((Y,\omega),W))$ of the Fukaya-Seidel category $FS((Y,\omega),W)$?
If $(Y,\omega),W)$ is the mirror of a smooth projective Fano variety $X$ then
$FS((Y,\omega),W)=D^b Coh(X)$
and one knows by the Hochschild-Kostant-Rosenberg theorem that
$HH_\bullet(D^bCoh(X))=\oplus_{p-q=\bullet}H^p(X,\Omega_X^q)$
and
$HH^\bullet(D^bCoh(X))=\oplus_{p+q=\bullet}H^p(X,\wedge^q T_X)$.
In particular, Hochschild homology and cohomology are indeed very different in general.
This question is motivated by the general principle saying that the deformations of a category are governed by $HH^2$. In particular, a subquestion could be: what are the deformations of the Fukaya-Seidel category?, and this should be equivalent to understand what $HH^2$ is.
Disclaimer: when I write that something "should be" true, it is what I think is the general expectation but as I am not an expert in symplectic geometry, I don't know the technical state of the art of the rigorous proofs. As my question is itself of the "should be" form, I hope it is fine.
