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

We prove a closed-string remodeling statement for the affine binary dihedral Calabi--Yau orbifold threefold $\mathcal X=[\mathbb C^2/\Gamma\times\mathbb C]$, where $\Gamma$ 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 $g\geq2$.