The GW/PT conjectures for toric pairs

TL;DR

This paper proves the conjectural correspondence between logarithmic Gromov-Witten and Donaldson-Thomas theories for toric pairs, extending known results to singular divisors and establishing new polynomiality properties.

Contribution

It provides the first verification of the conjecture in the fully logarithmic setting and introduces methods that strengthen existing toric correspondence results.

Findings

PT series is a Laurent polynomial under positivity conditions

Proved the capped vertex is a Laurent polynomial (2008 conjecture)

Verified the logarithmic DT/PT conjecture 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 $\mathbb{P}^3$.