Categorical Mirror Symmetry: The Elliptic Curve

We describe an isomorphism of categories conjectured by Kontsevich. If $M$ and $\tilde{M}$ 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 $\tilde{M}\mathrm{.}$ We prove this equivalence when $M$ is an elliptic curve and $\tilde{M}$ is its dual curve, exhibiting the dictionary in detail.