  This paper considers the (negative) cyclic open-closed map
$\mathcal{OC}^{-}$, which maps the cyclic homology of the Fukaya category of a
symplectic manifold to its $S^1$-equivariant quantum cohomology. We prove
(under simplifying technical hypotheses) that this map respects the respective
natural connections in the direction of the equivariant parameter. In the
monotone setting this allows us to conclude that $\mathcal{OC}^{-}$ intertwines
the decomposition of the Fukaya category by eigenvalues of quantum cup product
with the first Chern class, with the Hukuhara-Levelt-Turrittin decomposition of
the quantum cohomology. We also explain how our results relate to the
Givental-Teleman classification of semisimple cohomological field theories: in
particular, how the R-matrix is related to $\mathcal{OC}^{-}$ in the semisimple
case; we also consider the non-semisimple case.
