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.

arXiv commentComments welcome. v2: sign corrections and other improvements. v3: some improvements

References in corpus (16)

• The structure of 2D semi-simple field theories
• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Quantum periods - I. Semi-infinite variations of Hodge structures
• The monotone wrapped Fukaya category and the open-closed string map
• On the Fukaya category of a Fano hypersurface in projective space
• Products of Floer cohomology of torus fibers in toric Fano manifolds
• Semi-infinite Hodge structures and mirror symmetry for projective spaces
• Primitive forms via polyvector fields
• Mirror symmetry: from categories to curve counts
• Hirzebruch-Riemann-Roch theorem for DG algebras
• Lagrangian Floer theory and mirror symmetry on compact toric manifolds
• Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures
• Formulae in noncommutative Hodge theory
• Automatically generating Fukaya categories and computing quantum cohomology
• Categorical primitive forms and Gromov-Witten invariants of $A_n$ singularities
• Categorical Saito theory, I: A comparison result

Cited by in corpus (3)

• Categorical Enumerative Invariants, II: Givental formula
• The cyclic open-closed map, u-connections and R-matrices
• Open Gromov-Witten invariants from the Fukaya category