Landau-Ginzburg models, Monge-Ampère domains and (pre-)Frobenius manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910770077368320 |
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| author | Combe, Noémie C. |
| author_facet | Combe, Noémie C. |
| contents | Kontsevich suggested that the Landau-Ginzburg model presents a good formalism for homological mirror symmetry. In this paper we propose to investigate the LG theory from the viewpoint of Koopman-von Neumann's construction. New advances are thus provided, namely regarding a conjecture of Kontsevich-Soibelman (on a version of the Strominger-Yau-Zaslow mirror problem). We show that there exists a Monge-Ampère domain Y, generated by a space of probability densities parametrising mirror dual Calabi-Yau manifolds. This provides torus fibrations over Y. The mirror pairs are obtained via the Berglund-Hubsch-Krawitz construction. We also show that the Monge-Ampère manifolds are pre-Frobenius manifolds. Our method allows to recover certain results concerning Lagrangian torus fibrations. We illustrate our construction on a concrete toy model, which allows us, additionally to deduce a relation between von Neumann algebras, Monge-Ampère manifolds and pre-Frobenius manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00835 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Landau-Ginzburg models, Monge-Ampère domains and (pre-)Frobenius manifolds Combe, Noémie C. Algebraic Geometry Differential Geometry Kontsevich suggested that the Landau-Ginzburg model presents a good formalism for homological mirror symmetry. In this paper we propose to investigate the LG theory from the viewpoint of Koopman-von Neumann's construction. New advances are thus provided, namely regarding a conjecture of Kontsevich-Soibelman (on a version of the Strominger-Yau-Zaslow mirror problem). We show that there exists a Monge-Ampère domain Y, generated by a space of probability densities parametrising mirror dual Calabi-Yau manifolds. This provides torus fibrations over Y. The mirror pairs are obtained via the Berglund-Hubsch-Krawitz construction. We also show that the Monge-Ampère manifolds are pre-Frobenius manifolds. Our method allows to recover certain results concerning Lagrangian torus fibrations. We illustrate our construction on a concrete toy model, which allows us, additionally to deduce a relation between von Neumann algebras, Monge-Ampère manifolds and pre-Frobenius manifolds. |
| title | Landau-Ginzburg models, Monge-Ampère domains and (pre-)Frobenius manifolds |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2409.00835 |