The Gopakumar-Vafa formula for symplectic manifolds


The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau $3$-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.


Eleny-Nicoleta Ionel

Department of Mathematics, Stanford University, Stanford, CA 94305

Thomas H. Parker

Department of Mathematics, Michigan State University, East Lansing, MI 48824