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

TL;DR

This paper classifies primitive forms for Calabi-Yau $A_ $-categories with semi-simple Hochschild cohomology and shows how they relate to Gromov--Witten invariants and mirror symmetry.

Contribution

It provides a classification of categorical primitive forms via grading operators and links semi-simple Hochschild cohomology to enumerative mirror symmetry.

Findings

Primitive forms classified by grading operators.

Gromov--Witten invariants recovered from Fukaya categories.

Homological mirror symmetry implies enumerative mirror symmetry in the semi-simple case.

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 ${{\sf Fuk}}(M)$ 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 ${{\sf Fuk}}(M)$ 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.