Holomorphic anomaly equation for family Gromov–Witten classes

Establish the stated holomorphic anomaly equation governing the derivative with respect to the non-modular Eisenstein generator G_2 of every generating series of Gromov–Witten classes of the universal principally polarized abelian variety, including the genus-reduction, separating-gluing, and insertion terms involving the Lefschetz dual correspondence.

Background

The proposed anomaly equation specifies the dependence of the cycle-valued quasimodular series on the quasimodular generator G_2. Its right-hand side contains contributions from nonseparating and separating boundary gluings, together with corrections obtained by applying the Lefschetz dual correspondence associated with the principal polarization to insertions.

Together with the quasimodularity conjecture, the equation would strongly constrain the possible generating series and determine their non-modular dependence recursively. The paper verifies this equation in genus 2 after tautological projection.