Fukaya category and (open) Gromov-Witten invariants
This thesis contributes to the problem of obtaining both open and closed Gromov-Witten invariants from the Fukaya category. For closed invariants, the main ingredient is the cyclic open-closed map which maps the cyclic homology of the Fukaya category of a symplectic manifold to the S¹-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 the cyclic open-closed map 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 the cyclic open-closed map in the semisimple case; we also consider the non-semisimple case. To obtain open Gromov-Witten invariants from the Fukaya category, we associate to an object in an A∞-category an extension of the negative cyclic homology, called relative cyclic homology. We extend the Getzler-Gauss-Manin connection to relative cyclic homology. Then, we construct (under simplifying technical assumptions) a relative cyclic open-closed map, which maps the relative cyclic homology of a Lagrangian L in the Fukaya category of a symplectic manifold X to the S¹-equivariant relative quantum homology of (X,L). Relative quantum homology is the dual to the relative quantum cohomology constructed by Solomon-Tukachinsky. This is an extension of quantum cohomology, and comes equipped with a connection extending the quantum connection. We prove that the relative open-closed map respects connections. As an application of this framework, we show, assuming a construction of the relative cyclic open-closed map in a broader technical setup, that the Fukaya category of a Calabi-Yau variety determines the open Gromov-Witten invariants with one interior marked point for any null-homologous Lagrangian brane with vanishing Maslov class.