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
. We also
show the compatibility of the conjecture with the degeneration formula. As a corollary we
deduce that the Gromov–Witten potentials of the Schoen Calabi–Yau threefold (relative
to
)
are
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.