Gauss-Manin connection in disguise: Open Gromov-Witten invariants

TL;DR

This paper explores the relationship between open Gromov-Witten invariants and mirror symmetry by constructing a parameter space with a compatible Hodge structure and computing a related modular vector field.

Contribution

It introduces a new quasi-affine space parametrizing open Gromov-Witten invariants with a compatible mixed Hodge structure and computes an associated modular vector field.

Findings

Construction of a parameter space for open Gromov-Witten invariants.

Computation of a modular vector field related to the parameter space.

Establishment of a link between invariants and Hodge structures.

Abstract

In mirror symmetry, after the work by J. Walcher, the number of holomorphic disks with boundary on the real quintic lagrangian in a general quintic threefold is related to the periods of the mirror quintic family with boundary on two homologous rational curves. Following the ideias of H.Movasati, we construct a quasi-affine space parametrizing such objects enhanced with a frame for the relative de Rham cohomology with boundary at the curves compatible with the mixed Hodge structure. We also compute a modular vector field attached to such a parametrization.