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Categorical Mirror Symmetry: The Elliptic Curve

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arxiv math/9801119 v3 pith:XOAIKTNT submitted 1998-01-26 math.AG hep-th

Alexander Polishchuk , Eric Zaslow This is my paper · ORCID
classification math.AGhep-th
keywords curvewidetildecategoryellipticequivalencemirrorcategoricalcategories
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abstract

We describe an isomorphism of categories conjectured by Kontsevich. If $M$ and $\widetilde{M}$ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on $M$ and a suitable version of Fukaya's category of Lagrangian submanifolds on $\widetilde{M}.$ We prove this equivalence when $M$ is an elliptic curve and $\widetilde{M}$ is its dual curve, exhibiting the dictionary in detail.

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