Let X be the smooth projective variety under consideration, let N^operatornameGW_g,beta(X) denote its Gromov–Witten invariants, let u be a formal variable, and let t^beta record curve classes beta∈ H_2(X, Z). The symbols beta≥ 0 and beta>0 denote, respectively, the effective classes including the zero class and the nonzero effective classes. Gopakumar–Vafa integrality conjecture. There exist integers n_g^beta∈ Z, for g≥ 0 and beta∈ H_2(X, Z), such that ∑_g≥ 0,beta≥ 0N^operatornameGW_g,beta(X)u^2g-2t^beta=∑_substackg≥ 0,beta> 0 k∈ Z_≥ 1fracn_g^betak(2 sin(fracku2)^2g-2)t^k beta. The conjecture asserts the integrality of the Gopakumar–Vafa invariants and is motivated by string duality between type IIA string theory and M-theory; the displayed formula relates them to the generally rational Gromov–Witten invariants.

Gopakumar–Vafa integrality conjecture for Gromov–Witten invariants

At least 25 years old · documented by 2000

Let $X$ be the smooth projective variety under consideration, let $N^{\operatorname{GW}}_{g,\beta}(X)$ denote its Gromov–Witten invariants, let $u$ be a formal variable, and let $t^\beta$ record curve classes $\beta\in H_2(X,\mathbb Z)$ . The symbols $\beta\geq 0$ and $\beta>0$ denote, respectively, the effective classes including the zero class and the nonzero effective classes. Gopakumar–Vafa integrality conjecture. There exist integers $n_g^\beta\in\mathbb Z$ , for $g\geq 0$ and $\beta\in H_2(X,\mathbb Z)$ , such that

\sum_{g\geq 0,\beta\geq 0}N^{\operatorname{GW}}_{g,\beta}(X)u^{2g-2}t^\beta=\sum_{\substack{g\geq 0,\beta> 0\\ k\in\mathbb Z_{\geq 1}}}\frac{n_g^\beta}{k}\left(2\sin\left(\frac{ku}{2}\right)^{2g-2}\right)t^{k\beta}.

The conjecture asserts the integrality of the Gopakumar–Vafa invariants and is motivated by string duality between type IIA string theory and M-theory; the displayed formula relates them to the generally rational Gromov–Witten invariants.

References

Primary source

Yunfeng Jiang and Hsian-Hua Tseng, “On multiple cover formula for local K3 gerbes”, arXiv:2201.09315 (2022).

Additional references

10 papers in this index state this conjecture (2000–2022). The statement above is taken from the most recent of them; the others are arXiv:1910.12338, arXiv:1905.07085, arXiv:1404.4684, arXiv:1103.4229, arXiv:0707.1643, arXiv:math/0410540, arXiv:math/0302077, arXiv:math/0105148, arXiv:math/0009025.

gopakumar-vafa-invariants gromov-witten-theory

Posted 1 month ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

The claim that the relevant curve-counting numbers are integers was proved by Ionel and Parker, but this report does not independently verify their proof.

Gopakumar and Vafa conjectured that the rational Gromov–Witten series can be rewritten using integer BPS invariants through the multiple-cover formula. The statement is recorded for symplectic six-manifolds and includes the smooth projective setting considered here.

Known results

Ionel and Parker proved the integrality assertion (2018).
Doan, Ionel, and Walpuski proved the separate finiteness assertion (2021).
A direct definition of the Gopakumar–Vafa invariants is reportedly still unavailable.

September 2021 status report

A seminar document dated September 9, 2021 states explicitly that Ionel and Parker proved the Gopakumar–Vafa integrality conjecture. The Annals source independently describes integrality as proved earlier by Ionel and Parker, while treating finiteness as the remaining result; the proof claim is reported here as unverified.

Current status (as of September 2026): the integrality assertion is claimed proved by Ionel and Parker, but this automated report does not independently verify the proof; finiteness is a separate result.

Sources

No solutions have been posted yet.