Gromov–Witten/pairs descendent correspondence for toric $3$–folds

Abstract

We construct a fully equivariant correspondence between Gromov–Witten and stable pairs descendent theories for toric $3$–folds $X$. Our method uses geometric constraints on descendents, ${\mathcal{A}}_n$ surfaces and the topological vertex. The rationality of the stable pairs descendent theory plays a crucial role in the definition of the correspondence. We prove our correspondence has a non-equivariant limit.

As a result of the construction, we prove an explicit non-equivariant stationary descendent correspondence for $X$ (conjectured by MNOP). Using descendent methods, we establish the relative GW/Pairs correspondence for $X∕D$ in several basic new log Calabi–Yau geometries. Among the consequences is a rationality constraint for non-equivariant descendent Gromov–Witten series for $P^3$.