Open-String Gromov-Witten Invariants: Calculations and a Mirror "Theorem"

TL;DR

This paper develops localization methods for calculating open-string Gromov-Witten invariants on Calabi-Yau manifolds, providing explicit formulas and mirror symmetry tools that match physics predictions at genus zero and support conjectures at higher genera.

Contribution

It introduces a localization framework for open Gromov-Witten invariants and extends mirror symmetry techniques to compute these invariants to all orders.

Findings

Matching results with physics predictions at genus zero.

Verification of integrality conjectures at higher genera.

Explicit hypergeometric series for open-string invariants.

Abstract

We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant version of the mirror theorem allows us to write down a hypergeometric series, which together with a mirror map allows one to compute the invariants to all orders, similar to the closed string model or the physics approach via mirror symmetry. In the noncompact example where the Calabi-Yau is $K_{\PP^2},$ our results agree with physics predictions at genus zero obtained using mirror symmetry for open strings. At higher genera, our results satisfy strong integrality checks conjectured from physics.