Gopakumar–Vafa invariant

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.

:$\sum_^\infty~\sum_ \text(g,\beta)q^\lambda^=\sum_^\infty~\sum_^\infty~\sum_\text(g,\beta)\frac\left(2\sin\left(\frac\right)\right)^q^$ ,

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 $\beta$,
* $\text(g,\beta)$ are the Gromov–Witten invariants of curve class $\beta$ at genus $g$,
* $\text(g,\beta)$ 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_=\exp\left sum_^\infty~\sum_^\infty~\sum_\text(g,\beta)\frac\left(2\sin\left(\frac\right)\right)^q^\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

Notes

References

*
*
*
*
*

Quantum field theory
Algebraic geometry
String theory

{{quantum-stub