All-degree verification of the genus-one GW/DT₄ correspondence for the local projective plane

Prove the genus-one descendant GW/DT₄ correspondence for the local projective plane X = Tot_{P²}(O_{P²}(-1) ⊕ O_{P²}(-2)) with the specified orientation for every degree d, rather than only for degrees d ≤ 100.

Background

The paper computes the descendant DT₄ invariant ⟨τ₁(π*H)⟩ᵈ_DT₄ for every positive degree d. To compare this formula with the genus-one Gopakumar–Vafa invariants and the associated meeting invariants, however, the authors use a recursive computation that they carry out only through degree 100.

Consequently, the genus-one part of the conjectural correspondence is verified for the local projective plane only in degrees d ≤ 100. Extending this verification to all degrees would establish the unresolved portion of the correspondence in this geometry.

References

Although we have not found a proof of Conjecture \ref{main conj}\,(2) for all degree curve classes, the following gives a computational check for $d\leqslant 100$.

— A proof of GW/DT4 conjecture on local projective plane  (2610.06024 - Cao, 5 Oct 2026) in Section 2, immediately before Corollary 2.5; Corollary 2.5