Details
| Original language | English |
|---|---|
| Publication status | E-pub ahead of print - 18 Feb 2024 |
Abstract
Keywords
- math.AG, math-ph, math.MP, math.SG, 14J33, 53D37, 14J45, 14N10, 13F60, 14T20
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2024.
Research output: Working paper/Preprint › Preprint
}
TY - UNPB
T1 - The proper Landau-Ginzburg potential is the open mirror map
AU - Gräfnitz, Tim
AU - Ruddat, Helge
AU - Zaslow, Eric
N1 - 69 pages, sections 3.1 and 4.4 added, further improvements based on report at AiM
PY - 2024/2/18
Y1 - 2024/2/18
N2 - 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 the superpotential in terms of tropical disks 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.
AB - 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 the superpotential in terms of tropical disks 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.
KW - math.AG
KW - math-ph
KW - math.MP
KW - math.SG
KW - 14J33, 53D37, 14J45, 14N10, 13F60, 14T20
M3 - Preprint
BT - The proper Landau-Ginzburg potential is the open mirror map
ER -