Gromov-Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations

TL;DR

This paper explores the Gromov-Witten theory of elliptic fibrations, proposing that their potentials are lattice quasi-Jacobi forms satisfying a holomorphic anomaly equation, with proofs and verifications in specific cases.

Contribution

It conjectures a new structure for Gromov-Witten potentials as lattice quasi-Jacobi forms and proves this for rational elliptic surfaces and abelian surfaces in certain cases.

Findings

Proved the conjecture for rational elliptic surfaces in all genera and classes.

Generated series are quasi-Jacobi forms for the lattice E8.

Verified the holomorphic anomaly equation numerically for abelian surfaces.

Abstract

We conjecture that the relative Gromov-Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice $E_8$. We also show the compatibility of the conjecture with the degeneration formula. As Corollary we deduce that the Gromov-Witten potentials of the Schoen Calabi-Yau threefold (relative to $\mathbb{P}^1$) are $E_8 \times E_8$ quasi-bi-Jacobi forms and satisfy a holomorphic anomaly equation. This yields a partial verification of the BCOV holomorphic anomaly equation for Calabi-Yau threefolds. For abelian surfaces the holomorphic anomaly equation is proven numerically in primitive classes. The theory of lattice quasi-Jacobi forms is reviewed. In the Appendix the conjectural holomorphic anomaly equation is expressed as a matrix action on the space of (generalized) cohomological field theories. The compatibility of the matrix action with the Jacobi Lie algebra is proven. Holomorphic anomaly equations for K3 fibrations are discussed in an example.