The Koopman--von Neumann--Landau--Ginzburg theory and a Proof of the Kontsevich--Soibelman Conjecture

We show that the Hilbert space of the Koopman–von Neumann formulation of Landau–Ginzburg theory is parametrised by a real Monge–Ampère domain, which carries a natural pre-Frobenius. Restricting to finite-dimensional (dually flat) exponential families, the parameter space becomes a Monge–Ampère domain and a pre-Frobenius manifold. Our main theorem proves that for every Berglund–Hübsch–Krawitz mirror pair of Calabi–Yau orbifolds arising from an invertible polynomial, this Monge–Ampère domain (the open probability simplex) is the base of a Lagrangian torus fibration on both the original and the mirror hypersurface, with dual fibres in the sense of Strominger–Yau–Zaslow. The construction recovers the SYZ picture from the Landau–Ginzburg–Koopman–von Neumann framework. In particular, this proves the Kontsevich–Soibelman conjecture (2001) for all Berglund–Hübsch–Krawitz mirror pairs: the base of the SYZ fibration is a Monge–Ampère domain (the open simplex), and the torus fibrations on the mirror pair are dual. A toy model of cones of positive definite matrices illustrates the geometric structures.