arxiv: 2604.16096 · v1 · submitted 2026-04-17 · 🧮 math.DG

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

N. C. Combe

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Pith reviewed 2026-05-10 07:25 UTC · model grok-4.3

classification 🧮 math.DG

keywords Koopman-von NeumannLandau-Ginzburg theoryKontsevich-Soibelman conjectureStrominger-Yau-Zaslow fibrationCalabi-Yau orbifoldsMonge-Ampère domainmirror symmetryBerglund-Hübsch-Krawitz

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The pith

The open probability simplex is the base of dual Lagrangian torus fibrations for Berglund-Hübsch-Krawitz mirror pairs of Calabi-Yau orbifolds, proving the Kontsevich-Soibelman conjecture.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper connects the Koopman-von Neumann formulation of Landau-Ginzburg theory to the Strominger-Yau-Zaslow picture of mirror symmetry. It shows that the Hilbert space is parametrized by a real Monge-Ampère domain carrying a pre-Frobenius structure. For finite-dimensional exponential families, this domain reduces to the open probability simplex. The main result establishes that this simplex serves as the base for Lagrangian torus fibrations on both sides of every Berglund-Hübsch-Krawitz mirror pair arising from invertible polynomials, with the fibers being dual. This directly proves the 2001 Kontsevich-Soibelman conjecture in this setting by recovering the SYZ fibration from the Landau-Ginzburg framework.

Core claim

The Hilbert space of the Koopman-von Neumann formulation of Landau-Ginzburg theory is parametrized by a real Monge-Ampère domain, which carries a natural pre-Frobenius structure. Restricting to finite-dimensional dually flat exponential families, the parameter space becomes the open probability simplex. For every Berglund-Hübsch-Krawitz mirror pair of Calabi-Yau orbifolds from an invertible polynomial, this simplex is the base of a Lagrangian torus fibration on both the original and mirror hypersurface, with dual fibres in the Strominger-Yau-Zaslow sense.

What carries the argument

The real Monge-Ampère domain (open probability simplex) with its natural pre-Frobenius structure, serving as the SYZ base for dual torus fibrations on mirror pairs.

If this is right

The SYZ fibration base for these mirror pairs is the open probability simplex, a Monge-Ampère domain.
The Lagrangian torus fibrations on the hypersurface and its mirror are dual.
The SYZ picture is recovered from the Landau-Ginzburg-Koopman-von Neumann framework.
This holds for all Berglund-Hübsch-Krawitz mirror pairs of Calabi-Yau orbifolds from invertible polynomials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The pre-Frobenius structure on the simplex may allow new computations of periods or invariants in mirror symmetry.
Similar constructions could apply to other classes of mirror pairs beyond those from invertible polynomials.
The probability simplex interpretation suggests links to information geometry or statistical models in physics.
Low-dimensional examples like quintic Calabi-Yau could be checked numerically for the dual fibration property.

Load-bearing premise

That the parameter space of the Hilbert space in the Koopman-von Neumann Landau-Ginzburg theory restricts exactly to the open probability simplex as a Monge-Ampère domain with pre-Frobenius structure for the relevant mirror pairs.

What would settle it

A specific Berglund-Hübsch-Krawitz mirror pair of Calabi-Yau orbifolds where the open probability simplex fails to be the base of dual SYZ Lagrangian torus fibrations on the original and mirror sides.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Combe claims to prove the Kontsevich-Soibelman conjecture for all Berglund-Hübsch-Krawitz mirror pairs by showing that the open probability simplex from the Koopman-von Neumann Landau-Ginzburg setup is the base for dual SYZ fibrations.

read the letter

Combe's paper claims to prove the Kontsevich-Soibelman conjecture for all Berglund-Hübsch-Krawitz mirror pairs of Calabi-Yau orbifolds from invertible polynomials by reformulating Landau-Ginzburg theory in Koopman-von Neumann terms. The key move is showing that the parameter space of finite-dimensional dually flat exponential families in this setup is the open probability simplex, which then serves as the base for dual Lagrangian torus fibrations on both the original and mirror sides in the SYZ sense. This would organize a large class of mirror pairs under one geometric base and supply an explicit fibration construction. The approach is new in linking the statistical mechanics formulation directly to the pre-Frobenius structure and the Monge-Ampère domain for the SYZ fibration. It does a good job of recovering the SYZ picture from the LG-KvN framework and includes a toy model with positive definite matrix cones to illustrate the ideas. If the derivations check out, this could give a fresh route to the conjecture that avoids some of the usual algebraic geometry machinery. The potential soft spot is whether the identification of the simplex as the SYZ base is truly derived from the KvN Hilbert space parametrization or if it leans on existing mirror symmetry results in a way that makes the proof less independent. The abstract presents it as coming from restricting to exponential families, but one would need to verify the steps for circularity, especially since the full details aren't in the abstract. The central claim is ambitious, covering an infinite family, so the supporting arguments need to be solid and the pre-Frobenius structure must be shown to produce the duality without presupposing it. This paper is for specialists in mirror symmetry who are open to information-geometric or statistical physics inspired methods. A reader interested in new proofs of old conjectures or SYZ constructions would get value from it, assuming the math holds. It deserves a serious referee because a proof of KS for BHK pairs would be useful, even if the method requires careful checking. I would recommend sending it to peer review with requests for expanded derivations on the main theorem and explicit checks against circularity.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a Koopman-von Neumann formulation of Landau-Ginzburg theory in which the Hilbert space is parametrized by a real Monge-Ampère domain carrying a natural pre-Frobenius structure. Restricting to finite-dimensional dually flat exponential families yields the open probability simplex as parameter space. The central theorem asserts that, for every Berglund-Hübsch-Krawitz mirror pair of Calabi-Yau orbifolds arising from an invertible polynomial, this simplex is the base of dual Lagrangian torus fibrations (in the Strominger-Yau-Zaslow sense) on both the original and mirror hypersurfaces, thereby proving the Kontsevich-Soibelman conjecture for this class. A toy model based on cones of positive-definite matrices is used to illustrate the geometric structures.

Significance. If the main theorem and its supporting derivations hold, the work supplies a new information-geometric and statistical-mechanics route to the SYZ picture within Landau-Ginzburg models and resolves the Kontsevich-Soibelman conjecture for the entire family of Berglund-Hübsch-Krawitz mirror pairs from invertible polynomials. The introduction of pre-Frobenius structures on Monge-Ampère domains is a potentially reusable contribution that links mirror symmetry to dually flat geometry.

major comments (1)

The manuscript must supply the explicit derivation (presumably in the section containing the main theorem) showing that the open probability simplex arises independently as the SYZ base rather than being presupposed by the definition of the Monge-Ampère domain in the Koopman-von Neumann setup; without these steps the identification for both sides of the mirror pair remains unverified.

minor comments (2)

The abstract is compact; a single additional sentence clarifying the role of the pre-Frobenius structure would improve accessibility.
In the toy-model section, the precise manner in which the positive-definite-matrix-cone example prefigures the main construction should be stated explicitly rather than left implicit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We appreciate the acknowledgment of the manuscript's potential to connect information geometry with the SYZ picture in Landau-Ginzburg models. We address the single major comment below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

Referee: The manuscript must supply the explicit derivation (presumably in the section containing the main theorem) showing that the open probability simplex arises independently as the SYZ base rather than being presupposed by the definition of the Monge-Ampère domain in the Koopman-von Neumann setup; without these steps the identification for both sides of the mirror pair remains unverified.

Authors: We agree that the logical independence requires explicit derivation. In the manuscript, the Monge-Ampère domain is introduced abstractly in the Koopman-von Neumann-Landau-Ginzburg framework as the parameter space of the Hilbert space equipped with a pre-Frobenius structure from the dual flat connection. The restriction to finite-dimensional dually flat exponential families is then applied, which by the standard correspondence in information geometry produces the open probability simplex as the concrete realization. The main theorem then shows this simplex is the base of dual SYZ fibrations on both sides of each Berglund-Hübsch-Krawitz pair. To make this fully transparent, we will add a new subsection immediately preceding the statement of the main theorem that derives the simplex step by step from the exponential-family restriction alone, without any reference to SYZ geometry. We will then apply the theorem to verify the base property independently on the original hypersurface and on the mirror, confirming that the identification is derived rather than presupposed. This will resolve the verification for both sides of the mirror pair. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper first establishes that the Hilbert space in the Koopman-von Neumann formulation of Landau-Ginzburg theory is parametrized by a real Monge-Ampère domain carrying a pre-Frobenius structure. It then restricts to finite-dimensional dually flat exponential families, yielding the open probability simplex as the parameter space. The main theorem independently shows this simplex is the base of dual SYZ Lagrangian torus fibrations for every Berglund-Hübsch-Krawitz mirror pair of Calabi-Yau orbifolds from invertible polynomials, recovering the SYZ picture from the KvN-LG framework and thereby proving the Kontsevich-Soibelman conjecture in this class. No quoted step equates a claimed prediction or uniqueness result to its own inputs by construction, and no load-bearing premise reduces to a self-citation chain or ansatz smuggled via prior work by the same authors. The toy model of matrix cones is presented only illustratively.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assertion that the Koopman-von Neumann Hilbert space is parametrised by a real Monge-Ampère domain carrying a pre-Frobenius structure, and that finite-dimensional exponential families yield precisely the open probability simplex as the SYZ base. No free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)

domain assumption 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 structure.
Stated as the starting point of the construction in the abstract.
domain assumption Restricting to finite-dimensional (dually flat) exponential families turns the parameter space into a Monge-Ampère domain whose open probability simplex is the base of the required Lagrangian torus fibrations.
Invoked to obtain the concrete geometric object used in the main theorem.

pith-pipeline@v0.9.0 · 5551 in / 1833 out tokens · 58563 ms · 2026-05-10T07:25:10.932029+00:00 · methodology

discussion (0)

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