An Introduction To Mirror Symmetry

This is a review articles of mirror symmetry in mathematical physics.Mirror sym-metry came out in string theory around 1990 at first,and then attracted both mathemati-cians' and physicists' attentions.Besides the applications by physicists,mathematicians are interested in the specific features and the proofs of the conjecture,and have devel-oped two main methods in algebra/geometry,and have achieved lots of developments.Today,mirror symmetry has become the main stream intersecting theoretical physics,symplectic geometry,algebraic geometry,also one of the most promising aspects of mathematical physics.This article is going to introduce mirror symmetry from several aspects,to give readers the first look of it:Chapter 1,we introduce the history and the basic aspects of mirror symmetry,and related study materials.Chapter 2,we briefly introduce sigma model and an example of mirror symmetry in it.Chapter 3,we introduce homological mirror symmetry established by Maxim Kont-sevich in 1994[1]using algebraic methods,it has great influence to the early study of the mathematical aspects of mirror symmetry.Chapter 4,we briefly introduce SYZ mirror symmetry established by A.Stro-minger,S.-T.Yau,and E.Zaslowin in 1996[2]using geometry methods,and give an example of T-duality.Chapter 5,we introduce A?-categories which Fukaya categories in homological mirror symmetry belong to.Chapter 6,we introduce symplectic manifolds with corners developed by homo-logical mirror symmetry.