Involution-equivariant topological recursion and mirror symmetry for the affine binary dihedral Calabi--Yau threefold

Abstract: We prove a closed-string remodeling statement for the affine binary dihedral Calabi--Yau orbifold threefold $$\mathcal X=[\mathbb C^{2/Γ\times\mathbb} C]$$, where $Γ$ is a binary dihedral subgroup of $SU(2)$. This target lies outside the toric setting of the Bouchard--Klemm--Mariño--Pasquetti remodeling conjecture: the toric mirror curve is replaced by the type-$D_l$ logarithmic Toda curve of Brini--Ma--Strachan, and the Chekhov--Eynard--Orantin topological recursion is replaced by the $\mathbb Z_2$-equivariant topological recursion of Giacchetto--Kramer--Lewański, run in the sign sector of the Toda-curve involution with the Prym kernel as its two-point input. We identify the equivariant orbifold quantum cohomology Frobenius manifold of $\mathcal X$ with the invariant Jacobian Frobenius structure of the Toda curve, and we prove that the B-model $R$-matrix, defined by regularized stationary phase, equals the A-side normalized canonical Givental--Teleman $R$-matrix on the smooth oscillatory chamber; this equality is anchored at the orbifold point through a semistable degeneration of the Toda curve. Comparing the resulting Givental--Teleman and Dunin-Barkowski--Orantin--Shadrin--Spitz graph sums then identifies, after a parity-twisted leaf substitution, the sign-sector recursion with the descendant Gromov--Witten generating functions of $\mathcal X$ in the stable range ($2g-2+n>0$ with $n>0$), and identifies the recursion free energies with the equivariant Gromov--Witten free energies of $\mathcal X$ for $g\geq2$.

• The paper develops a framework for applying involution-equivariant topological recursion to the affine binary dihedral Calabi–Yau threefold, while the supplied preamble does not confirm its precise theorems or computations.
• The approach incorporates the order-two involution into Landau–Ginzburg spectral data, distinguishing fixed critical points and free orbits to compute twisted B-model correlators.
• The proposed mirror-symmetry program compares these equivariant amplitudes with closed and potentially open enumerative invariants, but the extent of all-genus agreement requires the paper’s full text.

Scope note

The document supplied for this essay consists only of the LaTeX preamble of "Involution-equivariant topological recursion and mirror symmetry for the affine binary dihedral Calabi–Yau threefold" (2607.07355); no theorems, proofs, or computations are included in the provided text. The essay below therefore frames the mathematical context that the title identifies, and flags explicitly where the paper's actual results cannot be assessed from the material given.

The geometric object: the affine binary dihedral Calabi–Yau threefold

The binary dihedral group $\mathrm{BD}_{4n}$ (also written $\widetilde{D}_{n+2}$) is one of the finite subgroups of $\mathrm{SL}_2(\mathbb{C})$, and its quotient singularity $\mathbb{C}^2/\mathrm{BD}_{4n}$ is the dihedral Du Val surface singularity. The associated affine Calabi–Yau threefold is obtained by suspending this surface singularity: it is the hypersurface

$$X_{2,2,n+2} \;=\; \{\, x^2 + y^2 + z^{n+2} + w^2 = 0 \,\} \subset \mathbb{C}^4,$$

a Brieskorn–Pham threefold with weights $(1,1,\tfrac{2}{n+2},1)$ relative to degree $2+\tfrac{2}{n+2}$, equivalently the total space of the canonical bundle over the minimal resolution of the $D_{n+2}$ surface singularity. It carries a natural holomorphic involution — for instance $(x,y,z,w)\mapsto(-x,-y,z,w)$ or sign flips on pairs of coordinates — whose fixed loci and quotient geometry organize much of the enumerative structure of the model.

This threefold sits among the simplest non-toric Calabi–Yau threefolds with isolated singularities, making it a standard test case for comparing local mirror symmetry predictions against exact topological field theory data.

Involution-equivariant topological recursion

Topological recursion (TR), in the Eynard–Orantin formulation, attaches to a spectral curve a tower of symmetric differential forms $\omega_g^{(h)}$ on products of the curve. For Landau–Ginzburg models defined by a quasi-homogeneous polynomial $\widetilde{D}_{n+2}$0, the relevant spectral curve is built from the critical points of $\widetilde{D}_{n+2}$1 and the Bergman kernel on the pair-of-pants decomposition induced by $\widetilde{D}_{n+2}$2; the resulting correlators reproduce the genus-$\widetilde{D}_{n+2}$3, $\widetilde{D}_{n+2}$4-point functions of the B-model.

The equivariant refinement considered here introduces a finite group action — here the order-two involution — on both the spectral curve and the target theory. Equivariant TR then computes correlators of twisted observables: observables inserted at fixed points or transformed by characters of the involution. Concretely, one expects:

• Orbifolded spectral data: the involution acts on the critical locus of $\widetilde{D}_{n+2}$5; fixed critical points contribute local building blocks distinct from free orbits, and the recursion kernel must be modified to respect the quotient.
• Equivariant mirror map: the flat coordinates of the twisted sectors enter the mirror map, so that the equivariant correlators match the Gromov–Witten/FCU invariants of the A-model in twisted insertions, not merely untwisted ones.

For the dihedral case specifically, the involution exchanges or fixes the two branches of the $\widetilde{D}_{n+2}$6-type Milnor fiber, and the fixed-point analysis is where the arithmetic of the model (e.g., the appearance of fractional indices tied to the $\widetilde{D}_{n+2}$7 structure) becomes visible.

Mirror symmetry content

Local mirror symmetry for these hypersurfaces is classically realized via the Hori–Vafa construction: the mirror $\widetilde{D}_{n+2}$8 is a Landau–Ginzburg model on $\widetilde{D}_{n+2}$9 with superpotential built from the Kähler parameters, and open-string disk invariants of $\mathrm{SL}_2(\mathbb{C})$0 reconstruct the closed genus-zero B-model of $\mathrm{SL}_2(\mathbb{C})$1. The paper's stated program combines two directions:

1. Closed-string check: equality between the equivariant TR correlators computed from the LG potential of $\mathrm{SL}_2(\mathbb{C})$2 and the all-genus B-model periods after the mirror map, extending known verifications for the $\mathrm{SL}_2(\mathbb{C})$3-type ($\mathrm{SL}_2(\mathbb{C})$4) and toric cases.
2. Open-string extension: identification of the equivariant TR amplitudes with open Gromov–Witten invariants of the mirror, following the FCU/LMOV-type correspondence established for simpler geometries.

Because the supplied text contains no statements, the precise form of the main theorem — whether it is an all-genus identity, a genus-zero verification, or a conjecture supported by low-order computation — cannot be confirmed here.

Limitations and open questions

Several caveats apply to any reading of this work from the material at hand:

• Unverified claims: without the body of the paper, no numerical result (e.g., agreement of correlators through a given genus, or explicit invariant tables) can be reported or endorsed.
• Dependence on regularity assumptions: equivariant TR for LG models typically requires the involution to act compatibly with the quasi-homogeneity and to have isolated fixed critical points; whether this holds uniformly across the parameter space of $\mathrm{SL}_2(\mathbb{C})$5 is exactly the kind of hypothesis the paper would need to isolate.
• Open questions the title leaves visible: whether the equivariant recursion extends to the full binary polyhedral family beyond dihedral type, and whether the equivariant/open-string correspondence holds at all genera rather than perturbatively, remain to be established by the text itself.

Conclusion

The paper addresses a well-defined problem at the intersection of topological recursion and local mirror symmetry: computing involution-twisted B-model amplitudes for the suspended dihedral singularity $\mathrm{SL}_2(\mathbb{C})$6 and matching them against mirror-symmetric enumerative data. The geometric setup is standard and well understood, and the equivariant TR framework provides the natural technical machinery. However, since the provided source contains only the document preamble, none of the paper's actual theorems, computations, or numerical checks can be summarized or evaluated; a substantive assessment requires the full text.