BPS states on elliptic Calabi-Yau, Jacobi-forms and 6d theories

Using the holomorphic anomaly equation we prove that the all genus topological
string theory partition function on elliptic Calabi-Yau  can be written in terms of
meromorphic Jacobi-Forms, where the elliptic argument is identified with the
genus counting parameter. We give strong evidence for an universal form of the
denominator with zero at the torsion points and argue that the numerator is a
weak Jacobi form. This gives strong all genus predictions in accordance with
algebraic geometry considerations. We show that if a 6d  theory can be decoupled
the formalism determines its  topological partition function completely.