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 $g\geq2$.

Disclosure

“ly automatic, except for some initial human input on the setup of the mirror curve, a hint about its degeneration and a general proof strategy. From the generated knowledge graph, with the assistance of a Codex-based writ- ing agent and GPT-5.5 Pro verification, we wrote the main content of this paper block by block (mainly section by section) with extensive human–AI interaction. Afterwards, the authors manually proofread and edited the entire paper for math- ematical correctness”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-24T00:34:15.228Z

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