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 𝒳=[ℂ2/Γ]×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-Dl relativistic Toda curve of Brini--Ma--Strachan, and the Chekhov--Eynard--Orantin topological recursion is replaced by the ℤ2-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 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 a chosen simply connected domain; 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 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 𝒳 for g≥2. We also match the genus-one potential defined by the reduced calibrated tau function with the Gromov--Witten potential up to an additive constant.
Related
- Folding to the Bessel-Clifford system, and Equivariant Topological Recursion
- Universal x-y duality in equivariant topological recursion and quartic Kontsevich model
- Quantum modularity in refined topological recursion
- Eynard-Orantin Topological Recursion, Moduli-Space Virasoro Constraints, and Semisimple CohFTs
- Topological recursion for spin enumeration of hypermaps
BibTeX
@misc{fang2026involutionequivariant,
title = {{Involution-equivariant topological recursion and mirror symmetry for the affine binary dihedral Calabi--Yau threefold}},
author = {Bohan Fang and Zhuoming Lan and Jingxiang Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.07355},
eprint = {2607.07355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.07355},
}
RIS
TY - UNPB
TI - Involution-equivariant topological recursion and mirror symmetry for the affine binary dihedral Calabi--Yau threefold
AU - Bohan Fang
AU - Zhuoming Lan
AU - Jingxiang Ma
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2607.07355
UR - https://arxiv.org/abs/2607.07355
AB - We prove a closed-string remodeling statement for the affine binary dihedral Calabi--Yau orbifold threefold 𝒳=[ℂ^2/Γ]×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 relativistic Toda curve of Brini--Ma--Strachan, and the Chekhov--Eynard--Orantin topological recursion is replaced by the 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 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 a chosen simply connected domain; 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 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 𝒳 for g≥2. We also match the genus-one potential defined by the reduced calibrated tau function with the Gromov--Witten potential up to an additive constant.
ER -