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Gromov-Witten invariants of Calabi-Yau fibrations
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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.
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Finite generation and holomorphic anomaly equation for equivariant Gromov-Witten invariants of $K_{\mathbb{P}^1\times\mathbb{P}^1}$
The equivariant Gromov-Witten potentials of local P^1×P^1 are proven to lie in an explicit finitely generated ring and to satisfy a holomorphic anomaly equation, via Givental-Teleman graph sums.
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