Summary: 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 \(K_{\mathbb{P}^2}\), our results agree with physics predictions at genus zero obtained using mirror symmetry for open strings. At higher genera and with multiple boundary components, our results satisfy strong integrality checks conjectured from physics.
For the entire collection see [Zbl 1003.00015].