Floor diagrams and enumerative invariants of line bundles over an elliptic curve

TL;DR

This paper develops a tropical geometry method using floor diagrams to compute Gromov-Witten invariants of $ ext{CP}^1$-bundles over elliptic curves, revealing their piecewise polynomiality and quasi-modularity.

Contribution

It introduces a novel floor diagram algorithm for calculating invariants and refines tropical multiplicities with Block-Göttsche multiplicity, connecting to Fock space operators.

Findings

Floor diagram algorithm effectively computes invariants.

Refined tropical multiplicities lead to refined invariants.

Proves piecewise polynomiality and quasi-modularity of generating series.

Abstract

We use the tropical geometry approach to compute absolute and relative Gromov-Witten invariants of complex surfaces which are $\CC P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block-G\"ottsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.