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 ${\rm SL}(2,\mathbb{Z})$. Vanishing conditions and Nakajima's blow up equations allow then to fix these amplitudes explicitly.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/8493