TY - JOUR

T1 - Open Gromov-Witten invariants from the Fukaya category

AU - Hugtenburg, Kai

PY - 2024/4/30

Y1 - 2024/4/30

N2 - 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].

AB - 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].

KW - Open Gromov-Witten invariants

KW - Fukaya category

KW - Open-closed map

U2 - 10.1016/j.aim.2024.109542

DO - 10.1016/j.aim.2024.109542

M3 - Journal article

VL - 441

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 109542

ER -