Categorical Enumerative Invariants and Homological Mirror Symmetry for Log Calabi-Yau Manifolds

We investigate the structural correspondence between categorical enumerative invariants (CEI) arising from the cyclic homology of A-categories and the generative functions of open Gromov-Witten invariants in the context of Homological Mirror Symmetry (HMS) for log Calabi-Yau manifolds. By analyzing the wrapped Fukaya category W(X D) of a log Calabi-Yau pair (X, D), we construct a trace map on the Hochschild chain complex that identifies Costello's categorical invariants with the period integrals of the mirror Landau-Ginzburg model (Y, W). We utilize the Gross-Siebert program's intrinsic mirror symmetry construction to define the mirror geometry via wall-crossing structures and scattering diagrams. Furthermore, we demonstrate that the cyclic open-closed map provides an isomorphism of variations of semi-infinite Hodge structures, effectively reconstructing the Frobenius manifold structure of the B-model from the A-structure of the wrapped Fukaya category. This framework provides a rigorous categorical derivation of the enumerative predictions of mirror symmetry, extending the results of Ganatra, Perutz, and Sheridan to the log smooth setting.