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Title:
How Homological Mirror Symmetry is Expected to Imply (classical) Mirror Symmetry
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Abstract:
When Mirror symmetry was discovered, what surprised mathematicians was its application to counting of the number of holomorphic spheres (for example in the quintic 3 fold). Later on, mirror symmetry was also formulated as an equivalence of categories (by Kontsevich). That version is called a homological mirror symmetry. Equivalence of categories usually contains more information than the coincidence of numbers, (for example, in this case the equivalence of categories implies relations between quantum cohomology and deformation theory of complex structures, of period integrals associated with singularities, and even possibly information about higher genus curves in the varieties.) Coincidence of the numbers have been checked in various situations. The deeper statements of Homological mirror symmetry have also been studied a lot recently. I will sketch some of the expected relations between homological mirror symmetry and other versions of mirror symmetry.
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