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

This paper considers the (negative) cyclic open–closed map \({\mathcal{O}\mathcal{C}}^{-}\), 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 \({\mathcal{O}\mathcal{C}}^{-}\) 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 \({\mathcal{O}\mathcal{C}}^{-}\) in the semisimple case; we also consider the non-semisimple case.