Refined floor diagrams from higher genera and lambda classes

Abstract

We show that, after the change of variables $q=e^{iu}$, refined floor diagrams for ${\mathbb{P}}^2$ 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 ${\mathbb{P}}^1$. Combining this result with the similar looking refined tropical correspondence theorem for log Gromov-Witten invariants, we obtain some non-trivial relation between relative and log Gromov-Witten invariants for ${\mathbb{P}}^2$ and Hirzebruch surfaces. We also prove that the Block-Göttsche invariants of ${\mathbb{F}}_0$ and ${\mathbb{F}}_2$ are related by the Abramovich-Bertram formula.

44 pages, 8 figures, revised version, exposition greatly improved, main results unchanged, published in Selecta Mathematica

References in corpus (6)

• Enumerative tropical algebraic geometry in R2
• Tropical refined curve counting from higher genera and lambda classes
• The quantum tropical vertex
• Three dimensional tropical correspondence formula
• A Fock Space approach to Severi Degrees of Hirzebruch Surfaces
• On an example of quiver DT/relative GW correspondence

Cited by in corpus (2)

• The log-open correspondence for two-component Looijenga pairs
• Polynomiality properties of tropical refined invariants