## 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 K_{X}, 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 language | English (US) |
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Article number | 109639 |

Journal | Advances in Mathematics |

Volume | 447 |

DOIs | |

State | Published - Jun 2024 |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics