Gromov-Witten invariants of Calabi-Yau fibrations

TL;DR

This paper investigates the quasimap invariants of elliptic and K3 fibrations, proposing conjectures on their modular properties and finite generation, supported by examples and potential generalizations to higher-dimensional Calabi-Yau fibrations.

Contribution

It introduces new conjectures relating quasimap invariants to modular forms and finite generation, extending previous Gromov-Witten theory insights to elliptic and K3 fibrations.

Findings

Evidence from examples supports the conjectures.

Proposed quasi-modularity of Gromov-Witten potentials.

Potential for generalization to higher-dimensional Calabi-Yau fibrations.

Abstract

We study the quasimap invariants of elliptic and K3 fibrations. Oberdieck and Pixton conjectured that the Gromov-Witten potentials of elliptic fibrations are quasi-modular forms. Analogously, we propose similar conjecture for the quasimap potentials of elliptic fibrations. We also conjecture some finite generation properties of quasimap potentials of K3 fibrations. Via wall-crossing conjecture, this will imply some quasi-modularity of the Gromov-Witten potentials of K3 fibrations. We provide some evidences for our conjectures through several examples. The method here can be further generalized to arbitrary n-dimensional Calabi-Yau fibrations.