Enumerative Geometry of Calabi-Yau Manifolds and String Theory

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We investigated analogs of Borcherds products for Calabi-Yau threefolds. From a geometrical viewpoint, such products should count sheaves or D-branes on the manifolds. It is also expected that they are related to the Gromov-Witten potentials through certain asymptotic expansions. In the context of string theory they should count BPS states in type IIA string theory compactified on the Calabi-Yau manifolds. In this research we tried to develop a systematic method to construct such products or partial products when the Calabi-Yaus are elliptically fibered or K3 fibered. To certain extent, we have succeeded in this mission and obtained concrete expressions. Some consistency checks have been made and functional properties are being investigated. I am currently preparing the manuscript reporting these results.As a related subject we have studied the properties of vortices on nodal and cuspidal curves.