Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 046, 49 pages      arXiv:2409.09472      https://doi.org/10.3842/SIGMA.2025.046

Correlated Gromov-Witten Invariants

Thomas Blomme a and Francesca Carocci b
a) Université de Neuchâtel, rue Émile Argan 11, Neuchâtel 2000, Switzerland
b) Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma 00133, Italy

Received September 24, 2024, in final form June 09, 2025; Published online June 18, 2025

Abstract
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.

Key words: Gromov-Witten invariants; enumerative geometry; elliptic curves; decomposition formula.

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