Refined descendant invariants of toric surfaces

TL;DR

This paper introduces refined tropical enumerative invariants for toric surfaces that unify and extend previous invariants, providing new computational tools and demonstrating their unique properties.

Contribution

It constructs a new class of refined tropical invariants for toric surfaces that generalize existing invariants and are computable via a lattice path algorithm.

Findings

Refined invariants specialize to known tropical descendant invariants at quantum parameter 1.

For trivalent tropical curves, invariants match Goettsche-Schroeter refined broccoli invariants.

The invariants are shown to be the unique refinement extending previous work.

Abstract

We construct refined tropical enumerative genus zero invariants of toric surfaces that specialize to the tropical descendant genus zero invariants introduced by Markwig and Rau when the quantum parameter tends to $1$. In the case of trivalent tropical curves our invariants turn to be the Goettsche-Schroeter refined broccoli invariants. We show that this is the only possible refinement of the Markwig-Rau descendant invariants that generalizes the Goettsche-Schroeter refined broccoli invariants. We discuss also the computational aspect (a lattice path algorithm) and exhibit some examples.