Correlated Gromov-Witten Invariants

We introduce a geometric refinement of Gromov-Witten invariants for ${\mathbb{P}}^1$-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore, we prove a refinement of the degeneration formula keeping track of the correlation. Finally, combining certain invariance properties of the correlated invariant, a local computation and the refined degeneration formula we follow floor diagram techniques to prove regularity results for the generating series of the invariants in the case of ${\mathbb{P}}^1$-bundles over elliptic curves. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces.