Holomorphic anomaly equations and the Igusa cusp form conjecture
| Autor: | Aaron Pixton, Georg Oberdieck |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Surface (mathematics)
High Energy Physics - Theory Pure mathematics Conjecture 010308 nuclear & particles physics General Mathematics 010102 general mathematics Fibration Holomorphic function FOS: Physical sciences 01 natural sciences Cusp form K3 surface Elliptic curve Mathematics - Algebraic Geometry High Energy Physics::Theory Mathematics::Algebraic Geometry High Energy Physics - Theory (hep-th) 0103 physical sciences FOS: Mathematics 0101 mathematics Anomaly (physics) Mathematics::Symplectic Geometry Algebraic Geometry (math.AG) Mathematics |
| Zdroj: | arXiv |
| DOI: | 10.48550/arxiv.1706.10100 |
| Popis: | 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. Comment: 68 pages |
| Databáze: | OpenAIRE |
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