The GW/DT4 conjecture for genus-zero invariants of Calabi–Yau fourfolds

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Let XX be a sextic 44-fold, let β\beta be a curve class, and let ρ∈H4(X,Z)\rho\in H^{4}(X,\mathbb{Z}). Denote by GW0,β(γ)\mathrm{GW}_{0,\beta}(\gamma) the genus-zero Gromov–Witten invariant with insertion γ\gamma, and by DT4(β∣γ)\mathrm{DT}_{4}(\beta\mid\gamma) the corresponding Donaldson–Thomas 44 invariant of one-dimensional stable sheaves, with the orientation chosen as below. GW/DT4_4 conjecture. There is a choice of orientation for which

GW0,β(γ)=∑k∣β1k2⋅DT4(β/k∣γ).\mathrm{GW}_{0,\beta}(\gamma)=\sum_{k\mid\beta}\frac{1}{k^{2}}\cdot\mathrm{DT}_{4}(\beta/k\mid\gamma).

This conjecture proposes an interpretation of Klemm–Pandharipande's Gopakumar–Vafa type invariants on Calabi–Yau fourfolds in terms of Donaldson–Thomas 44 invariants. Its status is not resolved in the supplied source.

References

Primary source

Yalong Cao, “Counting conics on sextic 4-folds”, arXiv:1805.04696 (2019).

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