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(T2n,ω+−1B)∧(T^{2n},\omega +\sqrt {-1}B)^{\wedge }to each symplectic torus(T2n,ω)(T^{2n},\omega )together with a closed 2 formBBwhich we call aBB-field. We will associate a coherent sheafE(L,L){\mathcal E}(L,{\mathcal L})on(T2n,ω+−1B)∧(T^{2n},\omega +\sqrt {-1}B)^{\wedge }to each pair(L,L)(L,{\mathcal L})of affine Lagrangian submanifoldsLLand a flat complex line bundleL{\mathcal L}onLL. In the case of affine Lagrangian submanifolds, we show that the Floer homology of Langrangian submanifolds is isomorphic to the extension of the mirror sheafE(L,L){\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.