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

TL;DR

This paper explores Landau-Ginzburg models and Monge-Ampère domains in relation to mirror symmetry, proposing a novel approach using Koopman-von Neumann's construction to connect these concepts with pre-Frobenius manifolds.

Contribution

It introduces a new framework for studying LG theory via Koopman-von Neumann's construction, linking Monge-Ampère domains, mirror symmetry, and pre-Frobenius manifolds.

Findings

Existence of a Monge-Ampère domain generating torus fibrations.

Construction of mirror pairs via Berglund-Hubsch-Krawitz method.

Relation between von Neumann algebras, Monge-Ampère manifolds, and pre-Frobenius manifolds.

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.