Intersections on tropical moduli spaces

TL;DR

This paper demonstrates that key algebro-geometric properties of rational descendant Gromov-Witten invariants can be extended to tropical geometry, establishing equivalences with classical invariants for certain toric varieties.

Contribution

It develops tropical analogues of fundamental Gromov-Witten equations and proves the coincidence of tropical and classical invariants for specific toric varieties.

Findings

Tropical versions of string, divisor, and dilaton equations are established.

A splitting lemma for boundary divisor intersections is proved.

Tropical and classical descendant Gromov-Witten invariants coincide for certain toric varieties.

Abstract

This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we prove general tropical versions of the WDVV resp. topological recursion equations (under some assumptions). As a direct application, we prove that the toric varieties $\mathbb{P}^1$, $\mathbb{P}^2$, $\mathbb{P}^1 \times \mathbb{P}^1$ and with Psi-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for $\mathbb{P}^2$ in Markwig-Rau-2008). Our approach uses tropical intersection theory and can unify and simplify some parts of the existing tropical enumerative geometry (for rational curves).