The proper Landau-Ginzburg potential is the open mirror map

Tim Gräfnitz, Helge Ruddat*, Eric Zaslow

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The mirror dual of a smooth toric Fano surface X equipped with an anticanonical divisor E is a Landau–Ginzburg model with superpotential, W. Carl–Pumperla–Siebert give a definition of the superpotential in terms of tropical disks [13] using a toric degeneration of the pair (X,E). When E is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic–Vafa branes in the canonical bundle KX, in framing zero. As a consequence, the proper Landau–Ginzburg potential is a solution to the Lerche–Mayr Picard–Fuchs equation. Along the way, we prove a generalization of a result about relative Gromov–Witten invariants by Cadman–Chen to arbitrary genus using the multiplication rule of quantum theta functions. In addition, we generalize a theorem of Hu that relates Gromov–Witten invariants of a surface under a blow-up from the absolute to the relative case. One of the two proofs that we give introduces birational modifications of a scattering diagram. We also demonstrate how the Hori–Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram.

Original languageEnglish (US)
Article number109639
JournalAdvances in Mathematics
Volume447
DOIs
StatePublished - Jun 2024

Keywords

  • Gromov-Witten
  • Lerche-mayr
  • Scattering diagram
  • Theta function
  • Wall-crossing

ASJC Scopus subject areas

  • General Mathematics

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