Tropical descendant invariants with line constraints

Abstract

AbstractVia 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 (generalised) 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, that is, 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, because it shows up in mirror symmetry as contributions to coefficients of the ‐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 1.