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arXiv · 2409.00835

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

Noémie C. Combe
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Abstract

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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Noémie C. Combe. 2024-09-01. Landau-Ginzburg models, Monge-Amp\`ere domains and (pre-)Frobenius manifolds. https://arxiv.org/abs/2409.00835

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