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Categorical Mirror Symmetry: The Elliptic Curve

Alexander Polishchuk·Eric Zaslow
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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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Alexander Polishchuk, Eric Zaslow. 2000-04-07. Categorical Mirror Symmetry: The Elliptic Curve. https://arxiv.org/abs/math/9801119

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