Mirror symmetry and localizations

Abstract

We describe the applications of localization methods, in particular the functorial localization formula, in the proofs of several conjectures from string theory. The functorial localization formula pushes the computations on complicated moduli spaces to simple moduli spaces. It is a key technique in the proof of the general mirror formula, the proof of the Hori-Vafa formula for explicit expressions of basic hypergeometric series of homogeneous manifolds, and the proof of the Mariño-Vafa formula for Hodge integrals. The proposal of Strominger-Yau-Zaslow of mirror symmetry will also be discussed.