Gromov-Witten theory of bicyclic pairs

TL;DR

This paper advances the understanding of Gromov-Witten invariants for bicyclic pairs, establishing multiple correspondences with other theories and providing explicit formulas for special cases, through novel degeneration analysis.

Contribution

It introduces new methods for analyzing logarithmic Gromov-Witten invariants of bicyclic pairs, connecting them with local, open, orbifold, and refined invariants, and offers explicit formulas for self-nodal curves.

Findings

Established correspondences with local, open, and orbifold Gromov-Witten theories.

Derived closed formulas for genus zero invariants of self-nodal curves.

Developed new degeneration formula analysis involving tropical and intersection theory.

Abstract

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.