Tropical descendant invariants with line constraints

TL;DR

This paper develops recursive tropical formulas to compute non-stationary rational descendant log Gromov--Witten invariants with line constraints, extending existing algorithms to new cases relevant in mirror symmetry.

Contribution

It introduces recursive methods for calculating non-stationary rational descendant invariants with line constraints using tropical geometry, filling a gap in existing computational techniques.

Findings

Derived recursive formulas for specific Gromov--Witten invariants

Extended tropical methods to cases involving two lines with Psi-conditions

Connected tropical computations to mirror symmetry coefficients

Abstract

Via correspondence theorems, rational log Gromov--Witten invariants of the plane can be computed in terms of tropical geometry. For many cases, there exists a range of algorithms to compute tropically: for instance, there are (generalized) lattice path counts and floor diagram techniques. So far, the cases for which there exist algorithms do not extend to non-stationary rational descendant log Gromov--Witten invariants, i.e.\ those where Psi-conditions do not have to be matched up with the evaluation of a point. The case of rational descendant log Gromov--Witten invariants satisfying point conditions (without Psi-conditions) and one Psi-condition of any power combined with a line plays a particularly important role, since it shows up in mirror symmetry as coefficients of the $J$-function. We provide recursive formulas to compute those numbers via tropical methods. Our method is inspired by the tropical proof of the WDVV equations. We also extend our study to counts involving two lines, both paired up with a Psi-condition, appearing with power one.