Abstract

Co-authors: Thomas H Parker (MSU); Penka Georgieva (IMJ-PRG). 

In the late nineties string theorists Gopakumar and Vafa conjectured that the Gromov-Witten invariants of Calabi-Yau 3-folds have a hidden structure: they are obtained, by a specific transform, from a set of more fundamental “BPS numbers”, which are integers. In joint work with Tom Parker, we proved this conjecture by decomposing the GW invariants into contributions of ``clusters” of curves, deforming the almost complex structure and reducing it to a local calculation. This talk presents some of the background and geometric ingredients of our proof, as well as recent progress, joint with Penka Georgieva, towards proving that a similar structure theorem holds for the real GW invariants of Calabi-Yau 3-folds with an anti-symplectic involution.