The proper Landau-Ginzburg potential is the open mirror map
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914683912454144 |
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| author | Gräfnitz, Tim Ruddat, Helge Zaslow, Eric |
| author_facet | Gräfnitz, Tim Ruddat, Helge Zaslow, Eric |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_12249 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The proper Landau-Ginzburg potential is the open mirror map Gräfnitz, Tim Ruddat, Helge Zaslow, Eric Algebraic Geometry Mathematical Physics Symplectic Geometry 14J33, 53D37, 14J45, 14N10, 13F60, 14T20 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. |
| title | The proper Landau-Ginzburg potential is the open mirror map |
| topic | Algebraic Geometry Mathematical Physics Symplectic Geometry 14J33, 53D37, 14J45, 14N10, 13F60, 14T20 |
| url | https://arxiv.org/abs/2204.12249 |