Holomorphic anomaly equations and the Igusa cusp form conjecture
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
Gespeichert in:
| Hauptverfasser: | , |
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| Dokumenttyp: | Article (Journal) |
| Sprache: | Englisch |
| Veröffentlicht: |
August 2018
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| In: |
Inventiones mathematicae
Year: 2018, Jahrgang: 213, Heft: 2, Pages: 507-587 |
| ISSN: | 1432-1297 |
| DOI: | 10.1007/s00222-018-0794-0 |
| Online-Zugang: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1007/s00222-018-0794-0
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| Verfasserangaben: | Georg Oberdieck, Aaron Pixton |
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| 520 | |a 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 elliptic 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. | ||
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