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Cosmic String, Harvey-Moore Conjecture and Family Seiberg-Witten Theory

Ai-Ko Liu
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Abstract

In this paper, we study the enumeration of virtual numbers of immersed nodal curves along certain Calabi-Yau K3 fibrations. By using the concept of cosmic strings, we verify the modularity conjeture of the generating function of immersed nodal curves from String theory (Harvey-Moore). Based on the theory defining virtual numbers of nodal curves on algebraic surface (applying to a pencil of lattice polarized K3), a mathematical definition of virtual numbers of immersed rational curves (so-called Gopakumar-Vafa numbers) for Calabi-Yau K3 fibrations is given and it is shown to match up with string theory prediction.

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Ai-Ko Liu. 2004-09-02. Cosmic String, Harvey-Moore Conjecture and Family Seiberg-Witten Theory. https://arxiv.org/abs/math/0409038

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