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Gromov–Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations

Georg Oberdieck and Aaron Pixton

Geometry & Topology 23 (2019) 1415–1489
Abstract

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 E8. 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 E8 × E8 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.

Keywords
Gromov–Witten theory, elliptic fibrations, holomorphic anomaly equation, rational elliptic surface, Schoen, Jacobi forms, quasimodular, Siegel modular forms, abelian surfaces
Mathematical Subject Classification 2010
Primary: 14N35
References
Publication
Received: 25 September 2017
Revised: 15 August 2018
Accepted: 30 September 2018
Published: 28 May 2019
Proposed: Jim Bryan
Seconded: Richard Thomas, Lothar Göttsche
Authors
Georg Oberdieck
Mathematisches Institut
Universität Bonn
Bonn
Germany
Aaron Pixton
Department of Mathematics
MIT
Cambridge, MA
United States