Higher genus Gromov-Witten invariants from projective bundles on smooth log Calabi-Yau pairs

Abstract

Let $(X,E)$ be a smooth log Calabi-Yau pair consisting of a smooth Fano surface $X$ and a smooth anticanonical divisor $E$. We obtain certain higher genus local Gromov-Witten invariants from the projectivization of the canonical bundle $Z:=\mathbb{P}(K_X\oplus {\mathcal{O}}_X)$, using the degeneration formula for stable log maps [KLR]. We evaluate an invariant in the degeneration using the relationship between $q$-refined tropical curve counting and logarithmic Gromov-Witten theory with ${\lambda }_g$-insertion [Bou]. As a corollary, we use flops to prove a blow up formula for higher genus invariants of $Z$. Additionally assuming $X$ is toric, we prove an all-genus correspondence between open invariants of an outer Aganagic-Vafa brane $L\subset K_X$ and closed invariants of $Z$ that generalizes a genus-0 open-closed equality of [Cha] to all-genus, by using an argument in [GRZZ].

36 pages, version 2 with extended introduction. Comments are welcome