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.
Highlights
1.1 K3 surfacesThe Yau–Zaslow formula evaluates the generating series of counts of rational curves on K3 surfaces in primitive classes as the inverse of the discriminant (τ ) = q (1 − qn)24 n≥1 64 Page 2 of 30J.-W. van Ittersum et al.where q = e2πiτ and τ ∈ H is the standard variable of the upper half-plane
Where q = e2πiτ and τ ∈ H is the standard variable of the upper half-plane
More general curve counts on K3 surfaces are defined by the Gromov–Witten invariants α; γ1, . . . , γn g,β :=
Summary
The Yau–Zaslow formula (proven by Beauville [2] and Bryan–Leung [3]) evaluates the generating series of counts of rational curves on K3 surfaces in primitive classes as the inverse of the discriminant (τ ) = q (1 − qn) n≥1
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