Over twenty years ago, Candelas and his collaborators proposed the celebrated mirror conjecture, which related the genus zero Gromov-Witten invariants of the quintic 3-folds with the periods on their mirror family. This classical mirror symmetry is considered as a local duality near the large Kahler/complex structure limit points on the two moduli spaces. About ten years ago, Fan-Javis-Ruan introduced a new mathematical quantum singularity theory. Later Chiodo-Ruan proved that for the quintic case, this theory is the A-theory which is mirror to the B-theory near the Gepner point. However, a mathematically A-theory corresponding to the conifold point is still missing. In this talk, I will present an uniform computation method for A-model theories on all these three points. For simlicity we will consider the case of local CP^2. We will show how to prove certain important properties such as polynomiality and gap condition for higher genus free energy, there properties are orginally conjectured by phisists from the B-model theories. This is still a work in progress.
Speaker(s): Shui Guo (Peking University)