Categorical Mirror Symmetry: The Elliptic Curve

Abstract

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 $\tilde{M}.$ We prove this equivalence when $M$ is an elliptic curve and $\tilde{M}$ is its dual curve, exhibiting the dictionary in detail.

arXiv commentharvmac, 29 pages (big); updated with corrections