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

Let XX be a sextic 44-fold, let β\beta be a curve class, and let ρH4(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(βγ)\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)=kβ1k2DT4(β/kγ).\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.

Sources & referencesView supporting material

Primary source

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

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