Gopakumar–Vafa invariant

Template:Short description 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.

$\sum _{g=0}^{\mathrm{\infty }}\sum _{\beta \in H_2(M,\mathbb{Z})}\text{GW}(g,\beta )q^{\beta }{\lambda }^{2g-2}=\sum _{g=0}^{\mathrm{\infty }}\sum _{k=1}^{\mathrm{\infty }}\sum _{\beta \in H_2(M,\mathbb{Z})}\text{GV}(g,\beta )\frac{1}{k}{(2\sin \left(\frac{k\lambda }{2}\right))}^{2g-2}{q}^{k\beta }$ ,

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:

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

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.

References

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