# Refined floor diagrams from higher genera and lambda classes

## TL;DR

This paper demonstrates that refined floor diagrams, after a specific variable change, encode generating series of higher genus relative Gromov-Witten invariants with lambda class insertions for certain surfaces, revealing deep relations between different invariants.

## Contribution

It establishes a new connection between refined floor diagrams and higher genus Gromov-Witten invariants with lambda classes, extending the understanding of their generating series and relations.

## Key findings

- Refined floor diagrams compute generating series of higher genus Gromov-Witten invariants.
- A relation between relative and log Gromov-Witten invariants is established.
- Block-Göttsche invariants of certain surfaces are related by the Abramovich-Bertram formula.

## Abstract

We show that, after the change of variables $q=e^{iu}$, refined floor diagrams for $\mathbb{P}^2$ and Hirzebruch surfaces compute generating series of higher genus relative Gromov-Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov-Witten theory and an explicit result in relative Gromov-Witten theory of $\mathbb{P}^1$. Combining this result with the similar looking refined tropical correspondence theorem for log Gromov-Witten invariants, we obtain some non-trivial relation between relative and log Gromov-Witten invariants for $\mathbb{P}^2$ and Hirzebruch surfaces. We also prove that the Block-G\"ottsche invariants of $\mathbb{F}_0$ and $\mathbb{F}_2$ are related by the Abramovich-Bertram formula.