Gromov-Witten theory of bicyclic pairs

A bicyclic pair is a smooth surface equipped with a pair of smooth divisors intersecting in two reduced points. Resolutions of self-nodal curves constitute an important special case. We investigate the logarithmic Gromov-Witten theory of bicyclic pairs. We establish correspondences with local Gromov-Witten theory and open Gromov-Witten theory in all genera, a correspondence with orbifold Gromov-Witten theory in genus zero, and correspondences between all-genus refined Gopakumar-Vafa invariants and refined quiver Donaldson-Thomas invariants. For self-nodal curves in $\mathbb{P}(1,1,r)$ we obtain closed formulae for the genus zero invariants and relate these to the invariants of local curves. We also establish a conceptual relationship between invariants relative a self-nodal plane cubic and invariants relative a smooth plane cubic. The technical heart of the paper is a qualitatively new analysis of the degeneration formula for stable logarithmic maps, involving a tight intertwining of tropical and intersection-theoretic vanishing arguments.