Gromov–Witten Theory of $\text{K3} \times {\mathbb{P}}^1$ and Quasi-Jacobi Forms

Abstract

Let S be a K3 surface with primitive curve class β. We solve the relative Gromov-Witten theory of S × P 1 in classes (β, 1) and (β, 2). The generating series are quasi-Jacobi forms and equal to a corresponding series of genus 0 Gromov-Witten invariants on the Hilbert scheme of points of S. This proves a special case of a conjecture of Pandharipande and the author. The new geometric input of the paper is a genus bound for hyperelliptic curves on K3 surfaces proven by Ciliberto and Knutsen. By exploiting various formal properties we find that a key generating series is determined by the very first few coefficients.Let E be an elliptic curve. As collorary of our computations we prove that Gromov-Witten invariants of S × E in classes (β, 1) and (β, 2) are coefficients of the reciprocal of the Igusa cusp form. We also calculate several linear Hodge integrals on the moduli space of stable maps to a K3 surface and the Gromov-Witten invariants of an abelian threefold in classes of type (1, 1, d).Contents 1 Jacobi forms are two-parameter generalizations of classical modular forms. A quasi-Jacobi forms is the holomorphic part of a almost-holomorphic Jacobi form, see [24] for the definition and [30, Sec.1] for an introduction. In this paper we will use the explicit presentation of the quasi-Jacobi form algebra presented in [28, Appendix B].2 The shift r = 1 − g − d is related to a similar shift in the GW/Pairs correspondence [34,35].where α 1 , . . . , α n ∈ H * (S × P 1 , Q) are cohomology classes and ψ i is the cotangent line class at the ith non-relative marked point. Since all cohomology of S is even, the integral is independent of the choice of ordering of µ i . The automorphism factors correct for the choice of an ordering.3 The moduli space in the disconnected case is always denoted by a • here. 4 The same as a set but with possible repetitions. 5 The pullback of the symplectic form from the K3 surface yields a trivial quotient of the standard perfect-obstruction theory on the moduli space. The reduction by this quotient defines the reduced virtual class, see [22] for a modern treatment of this process.

Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

“…Note added: after this article appeared on the arxiv, Georg Oberdieck pointed out that our results in section 4.5 match with his and Aaron Pixton's conjectured holomorphic anomaly equation on Calabi-Yau n-folds appearing in [23,24]. Moreover, he explained to us the explicit form of the generalized holomorphic anomaly equation for the Gromov-Witten potentials on Calabi-Yau fourfolds, which we include now in appendix B.…”

“…For the non π-vertical cycles the conjectured anomaly equation agrees with our results for M E 8 P 3 and we also checked it for the Gromov-Witten potentials of M E 8 P 1 ×P 2 . Our results on the modular structure can therefore be seen as a partial derivation and non-trivial check of the holomorphic anomaly equation conjectured in [23,24] for Calabi-Yau fourfolds. We will now briefly describe the general form of the holomorphic anomaly equations for genus zero Gromov-Witten potentials.…”

“…After releasing a pre-print of this paper it was brought to our attention by Georg Oberdieck that there is a conjectured modular anomaly equation for elliptic Calabi-Yau n-folds in [23,24]. For n = 4 the conjecture implies the modular anomaly equations for the GromovWitten potentials associated to π-vertical cycles and for genus one free energies that we derived in this paper.…”

“…On the other hand, conjecture B of [23,24] implies a general modular anomaly equation for F (g) γ 1 ,...,γm . Following the discussion of sections 3.1 and 4.4, we can make a generalization of the pure modular basis by taking the 4-cycles,…”

“…In particular, we can rewrite the instanton part of the string amplitudes in the coordinates (3.10) as 24) where each factor in the parenthesis transforms as a modular form of weight -6c 1 (B) · β.…”