Psi-floor diagrams and a Caporaso-Harris type recursion

TL;DR

This paper introduces Psi-floor diagrams to count tropical curves with point and Psi-class conditions, generalizes to relative cases, and proves a Caporaso-Harris type recursion linking tropical and classical Gromov-Witten invariants.

Contribution

It defines Psi-floor diagrams for counting tropical curves with Psi-conditions and establishes a recursion formula matching classical invariants.

Findings

Psi-floor diagrams effectively count tropical curves with Psi-conditions.

The Caporaso-Harris type recursion is proved for relative Psi-floor diagrams.

Tropical and classical relative Gromov-Witten invariants are shown to coincide.

Abstract

Floor diagrams are combinatorial objects which organize the count of tropical plane curves satisfying point conditions. In this paper we introduce Psi-floor diagrams which count tropical curves satisfying not only point conditions but also conditions given by Psi-classes (together with points). We then generalize our definition to relative Psi-floor diagrams and prove a Caporaso-Harris type formula for the corresponding numbers. This formula is shown to coincide with the classical Caporaso-Harris formula for relative plane descendant Gromov-Witten invariants. As a consequence, we can conclude that in our case relative descendant Gromov-Witten invariants equal their tropical counterparts.