Gromov-Witten Invariants and Mirror Symmetry for Non-Fano Varieties Using Scattering Diagrams

Research output: Working paper/Preprint › Preprint

Authors

  • Per Berglund
  • Tim Gräfnitz
  • Michael Lathwood

Research Organisations

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Original languageEnglish
Publication statusE-pub ahead of print - 25 Apr 2024

Abstract

Gromov-Witten invariants arise in the topological A-model as counts of worldsheet instantons. On the A-side, these invariants can be computed for a Fano or semi-Fano toric variety using generating functions associated to the toric divisors. On the B-side, the same invariants can be computed from the periods of the mirror. We utilize scattering diagrams (aka wall structures) in the Gross-Siebert mirror symmetry program to extend the calculation of Gromov-Witten invariants to non-Fano toric varieties. Following the work of Carl-Pumperla-Siebert, we compute corrected mirror superpotentials $\vartheta_1(\mathbb{F}_m)$ and their periods for the Hirzebruch surfaces $\mathbb{F}_m$ with $m \ge 2$.

Keywords

    math.AG, hep-th, math-ph, math.MP

Cite this

Gromov-Witten Invariants and Mirror Symmetry for Non-Fano Varieties Using Scattering Diagrams. / Berglund, Per; Gräfnitz, Tim; Lathwood, Michael.
2024.

Research output: Working paper/Preprint › Preprint

Berglund, P., Gräfnitz, T., & Lathwood, M. (2024). Gromov-Witten Invariants and Mirror Symmetry for Non-Fano Varieties Using Scattering Diagrams. Advance online publication.
Berglund P, Gräfnitz T, Lathwood M. Gromov-Witten Invariants and Mirror Symmetry for Non-Fano Varieties Using Scattering Diagrams. 2024 Apr 25. Epub 2024 Apr 25.
Berglund, Per ; Gräfnitz, Tim ; Lathwood, Michael. / Gromov-Witten Invariants and Mirror Symmetry for Non-Fano Varieties Using Scattering Diagrams. 2024.
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abstract = " Gromov-Witten invariants arise in the topological A-model as counts of worldsheet instantons. On the A-side, these invariants can be computed for a Fano or semi-Fano toric variety using generating functions associated to the toric divisors. On the B-side, the same invariants can be computed from the periods of the mirror. We utilize scattering diagrams (aka wall structures) in the Gross-Siebert mirror symmetry program to extend the calculation of Gromov-Witten invariants to non-Fano toric varieties. Following the work of Carl-Pumperla-Siebert, we compute corrected mirror superpotentials $\vartheta_1(\mathbb{F}_m)$ and their periods for the Hirzebruch surfaces $\mathbb{F}_m$ with $m \ge 2$. ",
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