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

TL;DR

This work develops a comprehensive localization framework for open-string Gromov-Witten invariants on noncompact Calabi–Yau 3-folds, showing that genus-zero open invariants can be computed from one-pointed maps via a winding-dependent leg factor. It then proves an equivariant open-string genus-zero mirror theorem, encoding the invariants in a hypergeometric series with a mirror map that yields all-order results. The authors verify the predictions against physics in the $K_{\mathbb{P}^2}$ example and demonstrate strong integrality constraints at higher genus, using explicit calculations and a Maple implementation. By connecting A-model localization to B-model mirror symmetry and providing concrete computations for disks and higher-genus open invariants, the paper advances rigorous open-string enumerative geometry and cross-checks against established physics frameworks.

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.