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

Abstract This paper considers the (negative) cyclic open–closed map OC^- O C -, which maps the cyclic homology of the Fukaya category of a symplectic manifold to its S¹ 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 OC^- O 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 OC^- O C - in the semisimple case; we also consider the non-semisimple case.