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

The cyclic open-closed map, u-connections and R-matrices