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Open Gromov-Witten invariants from the Fukaya category

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Published
Article number109542
<mark>Journal publication date</mark>30/04/2024
<mark>Journal</mark>Advances in Mathematics
Volume441
Number of pages44
Publication StatusPublished
Early online date21/02/24
<mark>Original language</mark>English

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 . We associate to an object in an A_\infty category an extension of the negative cyclic homology, called 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. This in particular includes the open Gromov-Witten invariants of the real locus of the quintic threefold considered in [23].