Gopakumar–Vafa invariant

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:


 * $$\sum_{g=0}^\infty~\sum_{\beta\in H_2(M,\mathbb{Z})} \text{GW}(g,\beta)q^{\beta}\lambda^{2g-2}=\sum_{g=0}^\infty~\sum_{k=1}^\infty~\sum_{\beta\in H_2(M,\mathbb{Z})}\text{BPS}(g,\beta)\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}q^{k\beta}$$ ,

where
 * $$\beta$$ is the class of pseudoholomorphic curves with genus g,
 * $$\lambda$$ is the topological string coupling,
 * $$q^\beta=\exp(2\pi i t_\beta)$$ 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 $$g$$,
 * $$\text{BPS}(g,\beta)$$ are the number of BPS states (the Gopakumar–Vafa invariants) of curve class $$\beta$$ at genus $$g$$.

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:


 * $$Z_{top}=\exp\left[\sum_{g=0}^\infty~\sum_{k=1}^\infty~\sum_{\beta\in H_2(M,\mathbb{Z})}\text{BPS}(g,\beta)\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}q^{k\beta}\right]\ .$$