Open string mirror maps from Picard- Fuchs equations on relative cohomology

TL;DR

The work develops a relative-cohomology–based framework to compute open-string mirror maps and superpotentials for Calabi–Yau geometries, using extended Picard–Fuchs (PF) equations and GKZ systems in toric settings. It shows that naively extending open-string mirror symmetry to compact Calabi–Yau manifolds is inconsistent and instead emphasizes a noncompact, local approach (e.g., the local $\mathbb{P}^2$) with a well-defined open-string PF system and superpotential. The paper provides concrete constructions of relative periods, GKZ operators on enlarged moduli, and explicit PF solutions, and validates the approach against established results in the noncompact case, illustrating correspondence with open Gromov–Witten invariants. The $K_{\mathbb{P}^2}$ example serves as a detailed testbed, yielding open and closed string data that agree with prior physical and mathematical predictions, thereby highlighting the utility and limitations of the relative-cohomology method for open string mirror symmetry.

Abstract

A method for computing the open string mirror map and superpotential for noncompact Calabi-Yaus, following the physical computations of Lerche and Mayr, is presented. It is also shown that the obvious extension of these techniques to the compact case is not consistent. As an example, the local CP^2 case is worked out in 2 ways.