Opening Mirror Symmetry on the Quintic

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 GromovWitten 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.

Real mirror symmetry for one-parameter hypersurfaces

“…As further comfort, we note that the corresponding predictions on the quintic [1] have been verified in [30] using the open Gromov-Witten theory of [31] and localization on the space of maps to the ambient P 4 . It would be interesting to verify our predictions in the weighted case by this or other methods.…”

“…for all M. The vacuum structure of a D-brane wrapped on L (6) [M ] or L (10) [M ] (think of a D6 or D4-brane in type IIA) is therefore very similar to the quintic [1]. In detail, since with boundary on the non-trivial one-cycle in RP 3 and with the remaining dimensions located in space-time [1].…”

“…In detail, since with boundary on the non-trivial one-cycle in RP 3 and with the remaining dimensions located in space-time [1]. The corresponding domainwall tension, which we will denote by T A , is the basic holomorphic observable associated with the D-brane configuration.…”

“…The reasoning that leads to the classical terms t/2 + 1/4 in (2.6) takes into account that the corresponding domainwall not only changes the vacuum on the brane (σ = ±1), but also the value of the Ramond-Ramond four and six form flux, N 4 and N 6 , through the corresponding cycles of the Calabi-Yau manifold. We have [1] …”

Towards open string mirror symmetry for one-parameter Calabi-Yau hypersurfaces

“…Recently, in a pioneering series of articles [10][11][12][13][14][15] Walcher and various collaborators took the first steps towards understanding open string mirror symmetry for compact Calabi-Yau three-folds. Further work includes [16][17][18][19][20] In the first paper, [10], disk instantons have been computed for the quintic using mostly A-model techniques.…”