The notion of Mirror Symmetry arose in physics while examining superconformal field theories associated to Calabi-Yau manifolds. The Homological Mirror Symmetry program is an attempt to explain Mirror Symmetry in terms of equivalence between certain triangulated categories attached to Calabi-Yaus (the derived category of coherent sheaves and the derived Fukaya category). We explain the physical intuition behind the Homological Mirror Symmetry, which is based on the notion of a D-brane, and discuss mirror symmetry for flat tori as an instructive example. We also discuss the evidence that the Fukaya category must be enlarged with objects other than Lagrangian submanifolds in order for the Homological Mirror Symmetry to work.
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