Gauss-Manin connection in disguise: Open Gromov-Witten invariants
Felipe Espreafico
TL;DR
This work develops a Gauss-Manin connection in disguise (GMCD) framework for open Gromov-Witten invariants of the quintic by working with the relative algebraic de Rham cohomology $H^3_{dR}(X, C_+\cup C_-)$ and a nine-parameter moduli space ${\sf T_{op}}$ of enhanced triples $(X, C_\pm, [\alpha])$. It constructs a unique modular vector field ${\sf R}$ on ${\sf T_{op}}$ and derives Ramanujan-type differential relations among the coordinates $s_i$, linking disk counts to period data and the Yukawa-type couplings through a non-homogeneous Picard-Fuchs equation. The paper also elucidates how relative periods organize into a generalized period domain with a $\tau$-matrix, and demonstrates the interplay between the Gauss-Manin connection, mixed Hodge structures, and period maps, establishing a modular-forms–type structure for open invariants. Further, it sketches extensions to higher-genus real/open Gromov-Witten theory via BCOV-type holomorphic anomaly equations and discusses moving-family generalizations, highlighting the potential for Jacobi-like forms and broader GMCD applications in open string enumerative geometry.
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.