Gromov-Witten theory of K3 surfaces and a Kaneko-Zagier equation for Jacobi forms

TL;DR

This paper establishes the existence of special quasi-Jacobi form solutions to a differential equation related to Gromov-Witten theory of K3 surfaces, providing insights into double ramification cycle integrals.

Contribution

It introduces a novel differential equation for Jacobi forms and characterizes its solutions, linking them to Gromov-Witten invariants of K3 surfaces.

Findings

Existence of quasi-Jacobi form solutions with polynomial index dependence

Transformation properties under the Jacobi group derived

Explicit conjectural description for double ramification cycle integrals

Abstract

We prove the existence of quasi-Jacobi form solutions for an analogue of the Kaneko--Zagier differential equation for Jacobi forms. The transformation properties of the solutions under the Jacobi group are derived. A special feature of the solutions is the polynomial dependence of the index parameter. The results yield an explicit conjectural description for all double ramification cycle integrals in the Gromov--Witten theory of K3 surfaces.