  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.
