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Landau-Ginzburg models, Monge-Amp\`ere domains and (pre-)Frobenius manifolds

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arxiv 2409.00835 v2 pith:AGK7F5HC submitted 2024-09-01 math.AG math.DG

Noémie C. Combe This is my paper · ORCID
classification math.AGmath.DG
keywords manifoldsmirrormonge-ampconstructionallowsfibrationslandau-ginzburgmodel
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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-Amp\`ere 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-Amp\`ere 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-Amp\`ere manifolds and pre-Frobenius manifolds.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Maximum Likelihood, permutohedra and Associativity Equations

    math.AG 2025-01 reject novelty 5.0 of 10

    The paper asserts that concentration cones and diagonal spectrahedra carry Frobenius and Monge-Ampere structures, with maximum likelihood degree indexed by Frobenius residuals.

  2. On the geometry of K\"ahler--Frobenius manifolds and their classification

    math.DG 2024-11 reject novelty 4.0 of 10

    The paper asserts that flat compact Kähler manifolds are Frobenius manifolds and classifies them, but the core construction is invalid.

  3. Wishart cones and quantum geometry

    math.OA 2024-12 reject novelty 3.0 of 10

    The paper asserts, without a full derivation, that finite-dimensional Connes Araki Haagerup cones carry Wishart laws, linking modular theory to information geometry.

  4. Learning on hexagonal structures and Monge-Amp\`ere operators

    math.DG 2024-12 reject novelty 2.0 of 10

    Dually flat manifolds are shown, essentially by definition, to be Monge-Ampere manifolds, and a web-theoretic argument claims hexagonal structures for learning, but the key learning claims rest on citations and circul...

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