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

Abstract

We show that the Hilbert space of the Koopman--von Neumann formulation of Landau--Ginzburg theory is parametrised by a real Monge--Amp\`ere domain, which carries a natural pre-Frobenius. Restricting to finite-dimensional (dually flat) exponential families, the parameter space becomes a Monge--Amp\`ere domain and a pre-Frobenius manifold. Our main theorem proves that for every Berglund--H\"ubsch--Krawitz mirror pair of Calabi--Yau orbifolds arising from an invertible polynomial, this Monge--Amp\`ere 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--H\"ubsch--Krawitz mirror pairs: the base of the SYZ fibration is a Monge--Amp\`ere 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.