Refined broccoli invariants
  We introduce a tropical enumerative invariant depending on a variable y which generalizes the tropical refined Severi degree. We show that this refined broccoli invariant is indeed independent of the point configuration, and that it specializes to a tropical descendant Gromov-Witten invariant for y=1 and to the corresponding broccoli invariant for y=-1. Furthermore, we define tropical refined descendant Gromov-Witten invariants which equal the corresponding refined broccoli invariants giving a new insight to the nature of broccoli invariants. We discuss various possible generalizations, e.g. to refinements of bridge curves and Welschinger curves. 
Lothar G\"ottsche, Franziska Schroeter
math.AG