There has been important progress recently in our ability to compute the topological string partition function Z_top on elliptically fibered Calabi-Yau manifolds over non-compact bases. An important ingredient in the computation is identifying an appropriate basis of functions in which Z_top can be expressed, thus reducing its determination to a finite dimensional problem. In this talk, after reviewing some of the physics background and general setup, we will focus both on the basis of functions and on the boundary conditions required to solve the finite dimensional problem. We will describe how geometries leading to gauge groups SU(3) and SO(8) led us to introduce novel subrings of Weyl invariant Jacobi forms, and discuss the problem of completeness of boundary conditions in solving for Z_top.