Mirror Symmetry, D-Branes and Counting Holomorphic Discs

TL;DR

This paper explores the mirror symmetry correspondence for special Lagrangian submanifolds in Calabi-Yau manifolds, translating disc counting problems into classical geometric computations on the mirror, leading to new predictions.

Contribution

It identifies mirror partners of special Lagrangian submanifolds and relates disc counting to the Abel-Jacobi map, providing new insights and predictions in mirror symmetry.

Findings

Established a correspondence between holomorphic disc counting and Abel-Jacobi map

Recovered known results in mirror symmetry

Generated new non-trivial predictions for holomorphic discs

Abstract

We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical Abel-Jacobi map on the mirror. We recover some results already anticipated as well as obtain some highly non-trivial new predictions.