Flat Connections in Open String Mirror Symmetry

TL;DR

This work develops a comprehensive framework for flat open-closed mirror symmetry by studying the flat Gauss-Manin connection on relative cohomology and extracting flat coordinates and the open-closed superpotential across multi-parameter deformation spaces. It shows that integrability imposes nontrivial, global relations among relative periods, which in simple cases reduce to a K3-structure and, in general, are captured by a gradient form $M_t=\partial_t R$, $M_{\hat t}=\partial_{\hat t} R$ of a generating matrix. The authors provide explicit implementations in local $\mathbb{P}^2$ and quintic geometries, analyzing phase structures, disk invariants at orbifold points, and open-closed period vectors, including non-Abelian brane configurations. The results illuminate how flat coordinates encode domain-wall tensions and brane superpotentials, enabling systematic extraction of open- string invariants and revealing how brane stacks realize gauge enhancements in the mirror. Overall, the paper advances the understanding of the open-closed variation of mixed Hodge structure and its physical implications for D-branes in Calabi–Yau compactifications.

Abstract

We study a flat connection defined on the open-closed deformation space of open string mirror symmetry for type II compactifications on Calabi-Yau threefolds with D-branes. We use flatness and integrability conditions to define distinguished flat coordinates and the superpotential function at an arbitrary point in the open-closed deformation space. Integrability conditions are given for concrete deformation spaces with several closed and open string deformations. We study explicit examples for expansions around different limit points, including orbifold Gromov-Witten invariants, and brane configurations with several brane moduli. In particular, the latter case covers stacks of parallel branes with non-Abelian symmetry.

Figures (1)

• Figure 1: Secondary fan for open-closed moduli space of local $\mathbbm{P}^2$.