A monoidal structure for the Fukaya category

Homological mirror symmetry relates Fukaya categories of certain symplectic spaces to derived categories of coherent sheaves of certain "dual" algebraic spaces. In this work we study the analogue in symplectic geometry of the structure given by the tensor product on the derived category of coherent sheaves, and the symplectic techniques necessary for its definition. We use the theory of triangulated categories with tensor structure to propose that one may think of homological mirror symmetry as the study of how the geometries of certain symplectic manifolds induce natural tensor product structures on their Fukaya categories. This is also the main motivation for this work.;We show that for a Lagrangian torus fibration with a distinguished Lagrangian section, which is the classical setting for mirror symmetry, there exists a natural monoidal structure on a category closely related to the Fukaya category, induced by the group structure on the toric fibers. Having defined the monoidal structure we show that for a mirror pair of elliptic curves, the tensor product can be defined for a version of the Fukaya category used in the proof of homological mirror symmetry for elliptic curves, and that with this tensor structure it will be exactly mirror to the derived category of coherent sheaves of the dual elliptic curve with its standard tensor structure. This gives evidence that our construction agrees with expectations for the mirror tensor product. Also, as a consequence of this theorem one gets a geometric way of computing the ring structure on vector bundles on an elliptic curve.