Opening Mirror Symmetry on the Quintic

TL;DR

The paper addresses open string mirror symmetry for a compact Calabi-Yau by studying holomorphic disks ending on the real Lagrangian in the quintic. It develops an ${\cal N}=1$ open-geometry extension of the Picard-Fuchs system, introducing an open period $\tau(z)$ that together with the closed periods $w^1,w^2$ determines the brane domain-wall tension via ${\cal T}_{\pm}=\tfrac{w^1}{2}\pm\tfrac{w^2}{4}\pm\tfrac{15}{\pi^2}\tau(z)$, and uses mirror symmetry to extract the open-string instanton expansion. The work validates the results by analytic continuation across the moduli space, consistent monodromies, and direct A-model localization, yielding integral open invariants $n_d$ and confirming Ooguri-Vafa integrality. This constitutes the first exact open-string mirror-symmetry result for a compact Calabi-Yau, with implications for Floer theory, the Fukaya category, and real enumerative geometry.

Abstract

Aided by mirror symmetry, we determine the number of holomorphic disks ending on the real Lagrangian in the quintic threefold. The tension of the domainwall between the two vacua on the brane, which is the generating function for the open Gromov-Witten invariants, satisfies a certain extension of the Picard-Fuchs differential equation governing periods of the mirror quintic. We verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space. We reproduce the first few instanton numbers by a localization computation directly in the A-model, and check Ooguri-Vafa integrality. This is the first exact result on open string mirror symmetry for a compact Calabi-Yau manifold.