Theta functions and adiabatic curvature on an Abelian variety

For an ample line bundle $L$ on an Abelian variety $M$, we study the theta functions associated with the family of line bundles $L\otimes T$ on $M$ indexed by $T\in {\text{Pic}}^0(M)$. Combined with an appropriate differential geometric setting, this leads to an explicit curvature computation of the direct image bundle $E$ on ${\text{Pic}}^0(M)$, whose fiber $E_T$ is the vector space spanned by the theta functions for the line bundle $L\otimes T$ on $M$. Some algebro-geometric properties of $E$ are also remarked.