Mirror Landau–Ginzburg Model

• Mirror Landau–Ginzburg models are defined as dual pairs combining non-compact Kähler manifolds with superpotentials to encapsulate mirror symmetry.
• They employ the BHK construction and real Monge–Ampère domains to connect deformation theory with Hessian metrics and categorical dualities.
• Their structure integrates torus fibrations and pre-Frobenius manifold properties, facilitating derived equivalences between Fukaya categories and matrix factorizations.

A mirror Landau–Ginzburg (LG) model is a dual pair of holomorphic data—typically a non-compact Kähler (or Fano/log Calabi–Yau) manifold equipped with a superpotential—arising from mirror symmetry constructions. These models exhibit deep equivalences between the geometry, deformation theory, homological invariants, and enumerative structures of dual spaces, with the BHK (Berglund–Hübsch–Krawitz) construction and the associated category-level correspondences serving as foundational frameworks. Recent geometric advancements recast the mirror correspondence in terms of real Monge–Ampère domains, Hessian metrics, and (pre-)Frobenius manifold structures, with state-space and categorical dualities realized via torus fibrations over spaces of probability densities, classical singularity theory, and optimal transport (Combe, 2024).

1. Construction of the Mirror Landau–Ginzburg Pair

On the A-side, an invertible, non-degenerate quasi-homogeneous polynomial

$$W(x_0, \ldots, x_n) = \sum_{i=0}^n c_i\, x_0^{m_{i0}} x_1^{m_{i1}} \cdots x_n^{m_{in}}$$

in $\mathbb{C}^{n+1}$, with integer weights $(w_0, \ldots, w_n; d)$ satisfying

$$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$

defines the hypersurface

$$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$

The LG A-model is the pair $(X_W, W)$, where $W$ acts as a superpotential.

The B-side mirror LG model is constructed via the BHK transpose: form the polynomial

$$W^\vee(y_0, \ldots, y_n) = \sum_{j=0}^n c_j^\vee\, y_0^{m_{0j}} y_1^{m_{1j}} \cdots y_n^{m_{nj}}$$

and corresponding dual group $G^\vee$. The mirror variety is then

$$X_{W^\vee} = \{ W^\vee(y) = 0 \} / \mathbb{C}^\times \subset \mathbb{P}_{(w_0^\vee, \ldots, w_n^\vee)},$$

paired with the superpotential $\mathbb{C}^{n+1}$0.

Mirror duality is encoded in the orbifold cohomology and Milnor (Jacobian) rings, with the state-space isomorphism

$\mathbb{C}^{n+1}$1

where the sum runs over sectors dictated by the group action (Combe, 2024).

2. The Monge–Ampère Domain and Hessian Structures

The construction is deeply rooted in the geometry of a real Monge–Ampère domain, denoted $\mathbb{C}^{n+1}$2, consisting of probability densities $\mathbb{C}^{n+1}$3 parameterizing torus fibrations. For smooth strictly convex domains, Brenier–Caffarelli theory guarantees the existence and uniqueness of convex potentials $\mathbb{C}^{n+1}$4 solving

$\mathbb{C}^{n+1}$5

These endow $\mathbb{C}^{n+1}$6 with a Hessian metric

$\mathbb{C}^{n+1}$7

and a totally symmetric cubic tensor

$\mathbb{C}^{n+1}$8

In the mirror LG framework, both the original and mirror varieties are fibered in Lagrangian tori over $\mathbb{C}^{n+1}$9, which carries a pre-Frobenius manifold structure (Combe, 2024).

3. Berglund–Hübsch–Krawitz Duality and State-Space Mirror Symmetry

The BHK construction starts from an invertible polynomial $(w_0, \ldots, w_n; d)$0 and its diagonal symmetry group $(w_0, \ldots, w_n; d)$1, selecting a subgroup containing the exponential grading element. The dual pair $(w_0, \ldots, w_n; d)$2 arises by transposing the exponent matrix and constructing the dual group: $(w_0, \ldots, w_n; d)$3 Chiodo–Ruan proved that the Chen–Ruan orbifold cohomology and state spaces of the LG orbifold models $(w_0, \ldots, w_n; d)$4 and $(w_0, \ldots, w_n; d)$5 are isomorphic, with corresponding Frobenius manifold structures on the respective state spaces (Combe, 2024).

4. Pre-Frobenius Manifold and Frobenius Loci

On $(w_0, \ldots, w_n; d)$6, the connection $(w_0, \ldots, w_n; d)$7 (flat from affine charts), together with the Hessian metric $(w_0, \ldots, w_n; d)$8 and cubic tensor $(w_0, \ldots, w_n; d)$9, satisfies the potential pre-Frobenius axioms:

1. Flat, torsion-free connection,
2. $$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$0-parallel Hessian metric,
3. Symmetric exact cubic tensor,
4. Commutative multiplication defined via $$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$1,
5. Frobenius compatibility: $$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$2.

On loci where the curvature vanishes (“flat locus”), the product is associative and $$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$3 supports Frobenius manifold structures, aligning with the expectations from the WDVV (associativity) equations (Combe, 2024).

5. Torus Fibrations and Homological Mirror Symmetry

The quantum-mechanical (Koopman–von Neumann) enhancement realizes $$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$4 as a principal torus bundle over $$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$5. Fibers over each $$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$6 in $$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$7, $$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$8, and their intersections correspond to Lagrangian tori, resulting in dual torus fibrations

$$W(s^{w_0} x_0, \ldots, s^{w_n} x_n) = s^d W(x_0, \ldots, x_n), \qquad \gcd(w_0, \ldots, w_n, d) = 1,$$9

Via the SYZ principle and the Orlov–Seidel approach, such correspondences yield derived equivalences: $$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$0 where $$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$1 is the wrapped or compact Fukaya category and $$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$2 the category of matrix factorizations (Combe, 2024).

6. Explicit Example: The Symmetric Cone $$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$3

For the symmetric cone $$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$4, the Monge–Ampère potential

$$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$5

renders $$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$6 a non-compact symmetric space and pre-Frobenius domain, with a geodesic flat submanifold $$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$7 (the Cartan subgroup), isomorphic to an algebraic torus. On $$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$8, the pre-Frobenius multiplication is strictly associative, so $$X_W = \{ W(x) = 0 \} \subset \mathbb{P}_{(w_0,\ldots,w_n)}.$$9 inherits a genuine Frobenius manifold structure, concretely illustrating these general constructions (Combe, 2024).

7. Synthesis and Theoretical Implications

The mirror Landau–Ginzburg paradigm as developed in (Combe, 2024) unifies optimal transport theory, Monge–Ampère/Hessian geometry, and the BHK construction. The Monge–Ampère domain $(X_W, W)$0 organizes mirror pairs as dual torus fibrations, with Frobenius (and pre-Frobenius) structures encoding the deformation, enumerative, and categorical invariants underlying homological mirror symmetry. All key features—state-space duality, categorical equivalence, and the geometry of torus fibrations—are expressed and connected within this analytic and algebraic framework.