Flat Connections in Open String Mirror Symmetry

TL;DR

This paper investigates a flat connection in open string mirror symmetry for Calabi-Yau threefolds with D-branes, establishing flat coordinates and superpotentials through integrability conditions, with explicit examples including orbifold invariants and non-Abelian brane stacks.

Contribution

It introduces a method to define flat coordinates and superpotentials in open-closed deformation spaces using flatness and integrability conditions, applicable to complex brane configurations.

Findings

Derived explicit flat coordinates for various deformation spaces.

Computed orbifold Gromov-Witten invariants in the context of open string mirror symmetry.

Analyzed brane configurations with non-Abelian symmetry, expanding the understanding of open string moduli.

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.