Gopakumar–Vafa invariant

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. In the same papers, the authors also derived the following formula which relates the Gromov–Witten invariants and the Gopakumar-Vafa invariants.

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

where

• ${\displaystyle \beta }$ is the class of holomorphic curves with genus g,
• ${\displaystyle \lambda }$ is the topological string coupling, mathematically a formal variable,
• ${\displaystyle q^{\beta }=\exp(2\pi it_{\beta })}$ with ${\displaystyle t_{\beta }}$ the Kähler parameter of the curve class ${\displaystyle \beta }$,
• ${\displaystyle {\text{GW}}(g,\beta )}$ are the Gromov–Witten invariants of curve class ${\displaystyle \beta }$ at genus ${\displaystyle g}$,
• ${\displaystyle {\text{GV}}(g,\beta )}$ are the Gopakumar–Vafa invariants of curve class ${\displaystyle \beta }$ at genus ${\displaystyle g}$.

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:

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

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