A mathematical bridge links two ways of describing the universe and proves a 24-year-old theory.
April 20, 2026
Original Paper
The Koopman--von Neumann--Landau--Ginzburg theory and a Proof of the Kontsevich--Soibelman Conjecture
arXiv · 2604.16096
The Takeaway
Mirror symmetry is a concept in physics where two distinct mathematical worlds can represent the exact same physical reality. The Kontsevich-Soibelman conjecture was a major missing link in understanding how the geometry of these two worlds connects at a fundamental level. This proof provides a rigorous geometric solution that works for an entire class of complex mirror pairs. It effectively solves a massive puzzle in high-energy physics regarding how extra dimensions are structured. This work gives physicists a reliable map to navigate the complex interactions between geometry and particle physics in string theory. It provides the mathematical proof needed to confirm that these two different views of the universe are perfectly identical.
From the abstract
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 doma