Open-string Gromov-Witten invariants: calculations and a mirror “theorem”

Abstract

AbstractWe propose localization techniques for computing Gromov-WitteninvariantsofmapsfromRiemannsurfaceswith boundariesintoaCalabi-Yau, with the boundaries mapped to a Lagrangian submanifold. Thecomputations can be expressed in terms of Gromov-Witten invariantsof one-pointed maps. In genus zero, an equivariant version of the mir-ror theorem allows us to write down a hypergeometric series, whichtogether with a mirror map allows one to compute the invariants to allorders, similar to the closed string model or the physics approach viamirror symmetry. In the noncompact example where the Calabi-Yau isK P 2 ,our results agree with physics predictions at genus zero obtainedusing mirror symmetry for open strings. At higher genera, our resultssatisfy strong integrality checks conjectured from physics. 1 Introduction 1.1 The Physics Mirror symmetry is famous for being able to predict Gromov-Witten invari-ants of Calabi-Yau manifolds. The basic conjecture is that there is a dualitybetween string theories on mirror Calabi-Yau manifolds. As a consequence,the topological field theory defined from the A-twist of one Calabi-Yau man-ifold is equal to the topological B-twist of the mirror. Both twists can beperformed on Calabi-Yau target manifolds. From a practical point of view,in order to obtain enumerative predictions, one needs to know the theoryon the B-model (in this case, defined through classical period integrals) aswell as an identification of the parameter spaces for both theories – the“mirror map.” To extract integer-valued invariants, one needs an all-genus“multiple-cover” formula. The technology for finding mirror manifolds [3]1