Topological String on elliptic CY 3-folds with N-sections and Jacobi forms

Albrecht Klemm

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_1(N) subgroups of SL(2,\mathbb{Z}). Vanishing conditions and in the non-compact case Nakajima’s blow up equations allow to fix the amplitudes explicitly.

http://scgp.stonybrook.edu/video_portal/video.php?id=4260

Simons- Special Holonomy in Geometry, Analysis, and Physics: Progress and Open Problems