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

Understand

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.