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.

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

where

• $\beta$ is the class of holomorphic curves with genus g,
• $\lambda$ is the topological string coupling, mathematically a formal variable,
• $q^{\beta }=\exp(2\pi it_{\beta })$ 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.

References

• Gopakumar, Rajesh; Vafa, Cumrun (1998a), M-Theory and Topological strings-I, arXiv:hep-th/9809187, Bibcode:1998hep.th....9187G
• Gopakumar, Rajesh; Vafa, Cumrun (1998b), M-Theory and Topological strings-II, arXiv:hep-th/9812127, Bibcode:1998hep.th...12127G
• Gopakumar, Rajesh; Vafa, Cumrun (1999), "On the Gauge Theory/Geometry Correspondence", Adv. Theor. Math. Phys., 3 (5): 1415–1443, arXiv:hep-th/9811131, Bibcode:1998hep.th...11131G, doi:10.4310/ATMP.1999.v3.n5.a5, S2CID 13824856
• Gopakumar, Rajesh; Vafa, Cumrun (1998d), "Topological Gravity as Large N Topological Gauge Theory", Adv. Theor. Math. Phys., 2 (2): 413–442, arXiv:hep-th/9802016, Bibcode:1998hep.th....2016G, doi:10.4310/ATMP.1998.v2.n2.a8, S2CID 16676561
• Ionel, Eleny-Nicoleta; Parker, Thomas H. (2018), "The Gopakumar–Vafa formula for symplectic manifolds", Annals of Mathematics, Second Series, 187 (1): 1–64, arXiv:1306.1516, doi:10.4007/annals.2018.187.1.1, MR 3739228, S2CID 7070264