Categorical primitive forms of Calabi-Yau $A_{\infty }$-categories with semi-simple cohomology

Abstract

We study categorical primitive forms for Calabi--Yau $A_{\infty }$ categories with semi-simple Hochschild cohomology. We classify these primitive forms in terms of certain grading operators on the Hochschild homology. We use this result to prove that, if the Fukaya category $\mathsf{Fuk}(M)$ of a symplectic manifold $M$ has semi-simple Hochschild cohomology, then its genus zero Gromov--Witten invariants may be recovered from the $A_{\infty }$-category $\mathsf{Fuk}(M)$ together with the closed-open map. An immediate corollary of this is that in the semi-simple case, homological mirror symmetry implies enumerative mirror symmetry.

Comments welcome. v2: sign corrections and other improvements. v3: some improvements