Problems in Enumerative Geometry Related to String Theory and Their Relations to Automorphic Forms

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The first result of this research project is that, based on string duality conjecture, I uniformly constructed conjectural expressions of the Gromov-Witten potentials of elliptic Calabi-Yau threefolds with sections fibered over Hirzebruch surfaces in a certain limit by investigating appropriate conformal field theories on elliptic curves. I made various non-trivial consistency checks on this result. This research revealed various rich relations to elliptic cohomology, Borcherds products, Jacobi forms and hyperbolic reflection groups. I am currently preparing a paper for the results I obtained.The second result of this project has already been reported as an article ("String and Vortex"). It is about the properties of Gromov-Witten potentials of generic Calabi-Yau threefolds. In my previous work with Kota Yoshioka we proposed some connections between relative Hilbert schemes of points on curves and Gromov-Witten invariants. This was based on the experience on specific Calabi-Yau threefolds. In the present work I extended this proposal for more general cases. On the other hand there was another proposal by Gopakumar and Vafa in which the relative compactified Jacobians are relevant. I discussed the consistency of the two proposals in favorable situations. Along the way I found a formula for the Euler characteristic of Hilbert schemes of points on nodal curves which may be viewed as a non-trivial generalization of Macdonald's classical formula on the Euler characteristic of symmetric products of non-singular curves.