Mirror symmetry of abelian varieties and multi-theta functions

Abstract

We study homological mirror symmetry conjecture of symplectic and complex torus. We will associate a mirror torus$(T^{2n},\omega +\sqrt {-1}B)^{\wedge }$to each symplectic torus$(T^{2n},\omega )$together with a closed 2 form$B$which we call a$B$-field. We will associate a coherent sheaf${\mathcal E}(L,{\mathcal L})$on$(T^{2n},\omega +\sqrt {-1}B)^{\wedge }$to each pair$(L,{\mathcal L})$of affine Lagrangian submanifolds$L$and a flat complex line bundle${\mathcal L}$on$L$. In the case of affine Lagrangian submanifolds, we show that the Floer homology of Langrangian submanifolds is isomorphic to the extension of the mirror sheaf${\mathcal E}(L,{\mathcal L})$. We construct a canonical isomorphism in the case when a certain transversality condition is satisfied. Our isomorphism then is functorial.