MNOP correspondence for projective threefolds

Let $X$ be a projective threefold, let $\beta\in H_2(X)$ be a homology class, and let $\gamma_1,\ldots,\gamma_r\in H^*(X)$ be cohomology classes. Let $\mathrm{GW}(X,\beta;\gamma_1,\ldots,\gamma_r)\in\mathbb Q((u))$ and $\mathrm{PT}(X,\beta;\gamma_1,\ldots,\gamma_r)\in\mathbb Z((q))$ be the Gromov–Witten and Pandharipande–Thomas invariants, respectively. A pair of formal Laurent series satisfies the MNOP correspondence when the PT series is a rational function of $q$ whose evaluation at $q=-e^{iu}$ equals the GW series.

MNOP correspondence. The invariants

$$(-iu)^{\langle c_1(TX),\beta\rangle}\mathrm{GW}(X,\beta;\gamma_1,\ldots,\gamma_r)$$

and

$$(-q)^{-\langle c_1(TX),\beta\rangle/2}\mathrm{PT}(X,\beta;\gamma_1,\ldots,\gamma_r)$$

satisfy the MNOP correspondence. This is the Gromov–Witten/Pandharipande–Thomas form of the original MNOP conjecture; the source attributes the related conjectures to Maulik–Nekrasov–Okounkov–Pandharipande and to Pandharipande–Thomas, with Donaldson–Thomas/Pandharipande–Thomas equivalence supplied by Bridgeland.