In theoretical physics, Rajesh Gopakumar and Cumrun Vafa introduced in a series of papers[1][2][3][4] numerical invariants of Calabi-Yau threefolds, later referred to as the Gopakumar–Vafa invariants.
These physically defined invariants represent the number of BPS states on a Calabi–Yau threefold.
Gopakumar–Vafa invariants can be viewed as a partition function in topological quantum field theory.
They are proposed to be the partition function in Gopakumar–Vafa form: While Gromov-Witten invariants have rigorous mathematical definitions (both in symplectic and algebraic geometry), there is no mathematically rigorous definition of the Gopakumar-Vafa invariants, except for very special cases.
Ionel-Parker proved that these expressions are indeed integers.