Overview
In theoretical physics, Rajesh Gopakumar and Cumrun Vafa introduced in a series of papers 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.
In the same papers, the authors also derived the following formula which relates the Gromov–Witten invariants and the Gopakumar-Vafa invariants. : \sum_ g=0 ^\infty~\sum_ \beta\in H_2(M,\mathbb Z ) \text GW (g,\beta)q^ \beta \lambda^ 2g-2 =\sum_ g=0 ^\infty~\sum_ k=1 ^\infty~\sum_ \beta\in H_2(M,\mathbb Z ) \text GV (g,\beta)\frac 1 k \left(2\sin\left(\frac k\lambda 2 \right)\right)^ 2g-2 q^ k\beta , where * \beta is the class of holomorphic curves with genus g, * \lambda is the topological string coupling, mathematically a formal variable, * q^\beta=\exp(2\pi i t_\beta) with t_\beta the Kähler parameter of the curve class \beta , * \text GW (g,\beta) are the Gromov–Witten invariants of curve class \beta at genus g , * \text GV (g,\beta) are the Gopakumar–Vafa invariants of curve class \beta at genus g .
Notably, Gromov-Witten invariants are generally rational numbers while Gopakumar-Vafa invariants are always integers.
As a partition function in topological quantum field theory
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: : Z_ top =\exp\left \sum_ g=0 ^\infty~\sum_ k=1 ^\infty~\sum_ \beta\in H_2(M,\mathbb Z ) \text GV (g,\beta)\frac 1 k \left(2\sin\left(\frac k\lambda 2 \right)\right)^ 2g-2 q^ k\beta \right \ .
Mathematical approaches
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. On the other hand, Gopakumar-Vafa's formula implies that Gromov-Witten invariants and Gopakumar-Vafa invariants determine each other. One can solve Gopakumar-Vafa invariants from Gromov-Witten invariants, while the solutions are a priori rational numbers. Ionel-Parker proved that these expressions are indeed integers.
See also
* Gopakumar–Vafa duality