The GW/PT conjectures for toric pairs

Abstract

We prove the conjectural correspondence between logarithmic Gromov-Witten theory and logarithmic Donaldson/Pandharipande-Thomas theory for pairs $(Y|\partial Y)$ consisting of a toric threefold $Y$ and any torus invariant divisor $\partial Y$, with primary insertions. The results are the first verifications of this conjecture when $\partial Y$ is singular, i.e., the ``fully logarithmic'' setting, and the first proof of the equivariant toric correspondence for pairs when $\partial Y$ is nonempty. When $\partial Y$ is empty, we get a new proof of the known toric correspondence, but our methods also lead to stronger conclusions. In particular, we show the PT series is a Laurent polynomial in the presence of sufficient positivity and prove a 2008 conjecture of Oblomkov, Okounkov, Pandharipande, and the first author stating the capped vertex is a Laurent polynomial. The methods also verify the logarithmic DT/PT conjecture for toric threefold pairs. Using the constraints of the logarithmic theory, the complete evaluation of toric pairs is determined by a single calculation -- the degree $1$ series of $\mathbb{P}^3$.