The Gopakumar-vafa Formula for Symplectic Manifolds

The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a CalabiYau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa conjecture holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds. The Gopakumar-Vafa conjecture [GV] predicts that the Gromov-Witten invariants GWA,g of a Calabi-Yau 3-fold can be expressed in terms of some other invariants nA,h, called BPS numbers, by a transform between their generating functions: ∑ A 6=0 g GWA,g t 2g−2qA = ∑