Topological Space in Homological Mirror Symmetry

In the mirror symmetry including the T-duality, the observables coincide in the A- and B-model on different manifolds. Because the observables are determined by how the strings propagate on the manifolds, the observed geometry by the A- and B-model will coincide. In this paper, we prove that the moduli space of the pseudo holomorphic curves in the A-model on a symplectic torus is homeomorphic to a moduli space of Feynman diagrams in the configuration space of the morphisms in the B-model on the corresponding elliptic curve. These moduli spaces determine the $A_{\infty}$ structure of the both models. Therefore, this homeomorphic topological space will be the observed geometry by the strings.