In theoretical physics, Rajesh Gopakumar and Cumrun Vafa introduced in a series of papers new topological invariants, called Gopakumar–Vafa invariants, that represent the number of BPS states on a Calabi–Yau 3-fold. They lead to the following generating function for the Gromov–Witten invariants on a Calabi–Yau 3-fold M :