Caporaso–Griguolo–Mariño–Pasquetti–Seminara higher-genus ansatz and double-scaling conjecture

For every integer $p\ge 2$ , determine whether the higher-genus closed Gromov–Witten potentials of the toric Calabi–Yau threefold $X_p=\operatorname{Tot}\!\left(\mathcal{O}_{\mathbb{P}^{1}}(p-1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-p-1)\right)$ satisfy the higher-genus ansatz of Caporaso–Griguolo–Mariño–Pasquetti–Seminara and whether their degeneration to the relevant critical regime satisfies the corresponding double-scaling conjecture. The reported verification assumes the nonvanishing condition on the torus weights required for the open–closed mirror-symmetry construction.

References

Primary source

arXiv

Additional references

Structure of higher-genus open-closed Gromov--Witten theory of O_P1(p-1)⊕O_P1(-p-1) — arXiv — Shuai Guo, Jingyi Xu, Qingsheng Zhang

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 preprint claims to verify the conjectured higher-genus picture for one explicit geometric family, but the claim has not been independently checked.

The problem concerns the Caporaso–Griguolo–Mariño–Pasquetti–Seminara conjectural description of higher-genus structures and their double-scaling limit. The reported result addresses the specific toric Calabi–Yau family $\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^1}(p-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-p-1))$ .

September 30, 2026 claimed verification

Shuai Guo, Jingyi Xu, and Qingsheng Zhang report mirror-symmetry, polynomial-structure, and double-scaling results for the stated family, describing them as an all-genus verification of the conjectural structures. The result is conditional on the stated nonvanishing torus-weight assumptions and has not been independently verified in the retrieved sources.

Current status (as of October 2026): The conjecture is claimed solved for the stated family under the stated weight conditions, but the claim remains unverified and no broader generality is established.

Sources

No solutions have been posted yet.