The all genus topological string amplitudes on Calabi-Yau 3-folds are generating functions of the symplectic invariants of holomorphic curves. We argue that the Fourier-Mukai transform on the A-model category as well as the holomorphic anomaly of the B-model restrict these amplitudes to be meromorphic Jacobi-forms of Gamma_0(N) subgroups of SL(2,ZZ). Vanishing conditions and Nakajima's blow up equations allow then to fix these amplitudes explicetly.

Albrecht Klemm (Physikalisches Institut der Universität Bonn)