Refined floor diagrams from higher genera and lambda classes

We show that, after the change of variables q = e iu , refined floor diagrams for 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 P 1 . Combining this result with the similar looking refined tropical correspondence theorem for log Gromov–Witten invariants, we obtain a non-trivial relation between relative and log Gromov–Witten invariants for P 2 and Hirzebruch surfaces. We also prove that the Block–Göttsche invariants of F 0 and F 2 are related by the Abramovich–Bertram formula.