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2019 Gromov–Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations
Georg Oberdieck, Aaron Pixton
Geom. Topol. 23(3): 1415-1489 (2019). DOI: 10.2140/gt.2019.23.1415


We conjecture that the relative Gromov–Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice E 8 . We also show the compatibility of the conjecture with the degeneration formula. As a corollary we deduce that the Gromov–Witten potentials of the Schoen Calabi–Yau threefold (relative to 1 ) are E 8 × E 8 quasi-bi-Jacobi forms and satisfy a holomorphic anomaly equation. This yields a partial verification of the BCOV holomorphic anomaly equation for Calabi–Yau threefolds. For abelian surfaces the holomorphic anomaly equation is proven numerically in primitive classes. The theory of lattice quasi-Jacobi forms is reviewed.

In the appendix the conjectural holomorphic anomaly equation is expressed as a matrix action on the space of (generalized) cohomological field theories. The compatibility of the matrix action with the Jacobi Lie algebra is proven. Holomorphic anomaly equations for K3 fibrations are discussed in an example.


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Georg Oberdieck. Aaron Pixton. "Gromov–Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations." Geom. Topol. 23 (3) 1415 - 1489, 2019.


Received: 25 September 2017; Revised: 15 August 2018; Accepted: 30 September 2018; Published: 2019
First available in Project Euclid: 5 June 2019

zbMATH: 07079061
MathSciNet: MR3956895
Digital Object Identifier: 10.2140/gt.2019.23.1415

Primary: 14N35

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.23 • No. 3 • 2019
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