Mirror Symmetry And Enumerative Geometry

Mirror symmetry is an important topic in modern mathematical physics which is re-searched by mathematicians and physicists from different aspects.For example,some people used it as a tool to predict the number of rational curves on a generic quintic three-fold,while some other people researched the relationship between the derived category of coherent sheaves of a Calabi-Yau 3-fold and the derived Fukaya category of its mirror pair.In this article,we will introduce enumerative geometries about mirror symmetry,especially Givental’s proof of the mirror theorem of projective spaces.We will first in-troduce some definitions and properties about Gromov-Witten invariants,and use them to define quantum cohomology.Then following Givental’s approach,we will introduce quantum differential equations,which will be used to give the mathematical statement of mirror conjecture,using quantum cohomology.Next,we are to introduce some useful tools which are necessary for Givental’s proof,especially equivariant cohomology and the technique of localization,which can significantly simplify the computation of Gromov-Witten invariants of moduli spaces and hence make the proof possible.Lastly,we will use quantum differential equations to provide the mathematical statement of the mirror theorm and then prove it case by case:the Fano case,which is simplier and can be proved by a recursion formula derived from localization;and then the Calabi-Yau case,which requires much more work since the recursion formula is not enough and the mirror map is much more complex.