Landau-Ginzburg models, Monge-Ampère domains and (pre-)Frobenius manifolds
Noémie C. Combe
TL;DR
The paper develops a KvN reformulation of Landau–Ginzburg theory to address a Kontsevich–Soibelman–SYZ mirror problem, formulating a real Monge–Ampère domain $\mathscr{Y}$ parameterizing a KvN Hilbert space and yielding torus fibrations that capture mirror Calabi–Yau pairs via the Berglund–Hubsch–Krawitz construction. It proves that the Monge–Ampère domain is a potential pre-Frobenius manifold and, in a concrete toy model based on cones of positive definite matrices over division algebras, exhibits a Frobenius locus and an isometrically immersed Frobenius manifold (an algebraic torus). The framework bridges LG theory, Monge–Ampère geometry, and Frobenius manifold theory, providing geometric realizations of Lagrangian torus fibrations and a pathway to connect noncommutative/von Neumann–algebra structures with mirror symmetry. The work thus offers a novel geometric toolkit for generating Frobenius manifolds from EMA/Monge–Ampère data and for reinterpreting LG models within a KvN/MALS/Machine-Learning-flavored context, with potential applications to Calabi–Yau moduli and quantum-state geometry.
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-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.