Tropical compactification and the Gromov--Witten theory of $\mathbb{P}^1$

TL;DR

This paper employs tropical and nonarchimedean geometry to analyze the moduli space of genus 0 stable maps to 1, establishing a tropical compactification and confirming the tropicalization of Hurwitz cycles, leading to a descendant correspondence.

Contribution

It introduces a tropical compactification of the moduli space of genus 0 stable maps to 1 and confirms the tropicalization of Hurwitz cycles, providing new insights into relative Gromov-Witten invariants.

Findings

The moduli space is a tropical compactification in a toric variety.

Tropical Hurwitz cycles are shown to be tropicalizations of classical Hurwitz cycles.

A full descendant correspondence for genus 0 relative invariants of 1 is established.

Abstract

We use tropical and nonarchimedean geometry to study the moduli space of genus $0$ stable maps to $\mathbb{P}^1$ relative to two points. This space is exhibited as a tropical compactification in a toric variety. Moreover, the fan of this toric variety may be interpreted as a moduli space for tropical relative stable maps with the same discrete data. As a consequence, we confirm an expectation of Bertram and the first two authors, that the tropical Hurwitz cycles are tropicalizations of classical Hurwitz cycles. As a second application, we obtain a full descendant correspondence for genus $0$ relative invariants of $\mathbb{P}^1$.