Refined descendant invariants of toric surfaces

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.