Open Gromov-Witten invariants from the Fukaya category

Abstract

This paper proposes a framework to show that the Fukaya category of a symplectic manifold $X$ determines the open Gromov-Witten invariants of Lagrangians $L\subset X$. We associate to an object in an $A_{\infty }$-category an extension of the negative cyclic homology, called \emph{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 $S^1$-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.

arXiv comment35 pages. Minor edits taking into account referee suggestions. Updated references. Numbering agrees with journal version. arXiv admin note: text overlap with arXiv:2205.13436

References in corpus (11)

• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Symplectic cohomology and duality for the wrapped Fukaya category
• Opening Mirror Symmetry on the Quintic
• Quantum periods - I. Semi-infinite variations of Hodge structures
• Disk enumeration on the quintic 3-fold
• Point-like bounding chains in open Gromov-Witten theory
• Relative quantum cohomology
• Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures
• Formulae in noncommutative Hodge theory
• Differential forms, Fukaya $A_{\infty }$ algebras, and Gromov-Witten axioms
• Categorical primitive forms of Calabi-Yau $A_{\infty }$-categories with semi-simple cohomology

Cited by in corpus (1)

• Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds