Holomorphic anomaly equations and the Igusa cusp form conjecture

Abstract

Let $S$ be a K3 surface and let $E$ be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold $S\times E$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov-Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for ellliptic Calabi-Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov-Witten theory of K3 surfaces in primitive classes.

arXiv comment68 pages

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Cited by in corpus (27)

• Double ramification cycles with target varieties
• Gromov-Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations
• Holomorphic anomaly equation for $({\mathbb{P}}^2,E)$ and the Nekrasov-Shatashvili limit of local ${\mathbb{P}}^2$
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• Quasimaps and stable pairs
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• The Donaldson-Thomas partition function of the banana manifold
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• Higher pullbacks of modular forms on orthogonal groups
• Quasimaps to moduli spaces of sheaves on a $K3$ surface
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• Gromov-Witten classes of K3 surfaces
• Regularized Integrals on Riemann Surfaces and Modular Forms
• Resurgence and Riemann-Hilbert problems for elliptic Calabi-Yau threefolds
• Regularized Integrals on Elliptic Curves and Holomorphic Anomaly Equations
• Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface
• Quasimaps to moduli spaces of sheaves
• Higher Genus FJRW Invariants of a Fermat Cubic
• Lost Chapters in CHL Black Holes: Untwisted Quarter-BPS Dyons in the ${\mathbb{Z}}_2$ Model
• Unweighted Donaldson-Thomas theory of the banana 3-fold with section classes
• Faber's socle intersection numbers via Gromov--Witten theory of elliptic curve
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• d-elliptic loci in genus 2 and 3
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• Quasi-modularity and holomorphic anomaly equation for the twisted Gromov-Witten theory: $\mathcal{O}(3)$ over ${\mathbb{P}}^2$