Landau--Ginzburg mirror symmetry for ADE singularities

Technical Summary: Landau–Ginzburg Mirror Symmetry for ADE Singularities

Problem Statement
The paper addresses the Landau–Ginzburg (LG) mirror symmetry conjecture for ADE singularities. Specifically, it seeks to establish an isomorphism between two distinct mathematical theories associated with a quasihomogeneous singularity $f$ and a group of diagonal symmetries $G$:

1. The B-side: The "orbifold Saito theory" of the pair $(f, G)$, which generalizes K. Saito's theory of primitive forms to include symmetry groups, resulting in a Dubrovin–Frobenius manifold.
2. The A-side: The Fan–Jarvis–Ruan–Witten (FJRW) theory of the Berglund–Hübsch–Henningson (BHH) dual pair $(\tilde{f}, \tilde{G})$, which constructs a cohomological field theory (CohFT) and its associated genus-zero Dubrovin–Frobenius manifold.

While the conjecture was previously proven for ADE singularities with the trivial symmetry group $G=\{id\}$, this work extends the proof to specific cases where $G$ is a nontrivial group of diagonal symmetries. The paper focuses on the specific pairs $(f, G)$ listed in Table 1, which represent the non-trivial admissible symmetry groups for ADE singularities in three variables as identified in prior work [BTW23].

Methodology
The author employs a two-step strategy to prove the mirror isomorphism for the specific nontrivial symmetry groups listed in Table 1. The proof relies on the classification of Dubrovin–Frobenius manifolds via the reconstruction lemma (Lemma 2.2), which states that for ADE singularities, the manifold structure is uniquely determined by the pairing, three-point correlators, and four-point correlators.

1. Step (i): Equivalence of Orbifold Saito Theory to Classical Saito Theory

   • The paper utilizes the concept of "orbifold equivalence" between pairs $(f, G)$ and a classical polynomial $\bar{f}$ (where $G$ is trivial). This equivalence is established via an equivalence of categories of matrix factorizations, $MF_G(f) \cong MF(\bar{f})$.
   • By invoking the Morita invariance of negative cyclic homology and the categorical formulation of Saito theory (Theorem 4.3), the author demonstrates that the Variations of Semi-Infinite Hodge Structures (VSHS) for $(f, G)$ are isomorphic to those of the classical Saito theory of $\bar{f}$.
   • Consequently, the Dubrovin–Frobenius manifold of the orbifold Saito theory for $(f, G)$ is isomorphic to the classical Saito theory of $\bar{f}$.

2. Step (ii): Equivalence of FJRW Theory to Classical Saito Theory

   • The author computes the FJRW theory for the dual pairs $(\tilde{f}, \tilde{G})$ case-by-case (corresponding to the entries in Table 1).
   • The computations utilize the "sum of singularities" axiom to decompose complex polynomials into simpler components (e.g., $A_k$ or $D_k$ types) whose FJRW theories are known or easier to compute.
   • For cases involving non-trivial groups where direct decomposition is insufficient, the author applies the WDVV (Witten–Dijkgraaf–Verlinde–Verlinde) equation and the concavity axiom to compute specific four-point correlators.
   • By identifying the primitive vectors and calculating the necessary correlators, the author reconstructs the Dubrovin–Frobenius manifold for each dual pair and shows it is isomorphic to the classical Saito theory of the corresponding $\bar{f}$.

Key Results
The paper proves Theorem 1.1, stating that the LG-LG mirror symmetry conjecture holds for the specific ADE singularities and nontrivial symmetry groups enumerated in Table 1. The proof is broken down into Theorem 1.2, which asserts that for every pair $(f, G)$ in this specific list:

• (i) The orbifold Saito theory of $(f, G)$ is isomorphic to the classical Saito theory of a specific polynomial $\bar{f}$.
• (ii) The FJRW theory of the dual pair $(\tilde{f}, \tilde{G})$ is isomorphic to the same classical Saito theory of $\bar{f}$.

The specific cases covered include various $A_k$, $D_k$, and $E_6$ singularities with nontrivial diagonal symmetry groups (e.g., groups generated by elements like $\frac{1}{2}(0, 1, 1)$ or $\frac{1}{3}(1, 2, 0)$) as detailed in Table 1.

Significance and Claims
The author claims that this work provides the first proof of the LG-LG mirror symmetry conjecture as an isomorphism of Dubrovin–Frobenius manifolds where the B-model is equipped with a nontrivial symmetry group, specifically for the cases listed in Table 1.

• Previous results by Krawitz [K10] established the isomorphism at the level of vector spaces.
• Subsequent work by Francis et al. [FJJS] established the isomorphism of Frobenius algebras.
• This paper extends these results to the full geometric structure of the Dubrovin–Frobenius manifold for nontrivial groups within the scope of the enumerated cases.
• The paper notes that this is the second known example of such a mirror symmetry isomorphism with a nontrivial B-model group, following the work of [BT22].

The author explicitly states that the cohomological field theory for the orbifold Saito theory is not defined in the standard sense within this note but can be reconstructed via Givental's formalism, implying that the mirror symmetry holds at the level of cohomological field theories as well, consistent with Teleman's results. The work relies heavily on the classification of admissible symmetry groups for ADE singularities provided in [BTW23].