Gromov–Witten/Donaldson–Thomas correspondence for logarithmic Calabi–Yau 3-folds

Let $\bold X$ be a logarithmic Calabi–Yau 3-fold, let $\mathcal L$ be the associated holomorphic lagrangian correspondence, and let $Z_{GW}(\bold X)^\circ$ and $Z_{PT}(\bold X)$ be the Gromov–Witten and stable-pair partition functions. For contact data $\mathbf p$ and curve class $\beta$, let $\Delta_{\mathbf p}$ be the corresponding lagrangian cycle and let $\star$ denote composition of lagrangian correspondences. Let $q$ and $\hbar$ satisfy

$$q^{\frac 12}=ie^{i\hbar/2}.$$

Gromov–Witten/Donaldson–Thomas correspondence conjecture. With this change of variables,

$$Z_{GW}(\bold X)^\circ =\mathcal L\star Z_{PT}(\bold X),\qquad \mathcal L^\dagger\star Z_{GW}(\bold X)^\circ =Z_{PT}(\bold X).$$

In particular,

$$Z_{GW}(\bold X)^\circ _{\beta,\mathbf p}=\left(\prod_v i^{\mathbf p(v)(|v|-1)}\right)\Delta_{\mathbf p}\star_{D^{[\beta]}}Z_{PT}(\bold X)_\beta.$$

The claim is the logarithmic version of the Gromov–Witten/Donaldson–Thomas correspondence and is intended to encode Gromov–Witten invariants through the canonical holomorphic lagrangian correspondence. The supplied status evidence says that the surrounding integrality conjecture is false without an additional contact-data hypothesis; it does not directly resolve this correspondence claim.