The invariance of $\Sigma$ and the discriminant locus $D$ with respect to the duality between triangulated polytopes $\Delta$ and $\Delta^{\vee}$ is manifest. The vertices of $D$ change the type from $(k,l)$ to $(l,k)$. The nodes of the graph $\Gamma$ also interchange the $U$- and $V$-types. And if integral bases associated with vertices of $\Gamma$ are chosen to be dual to the original ones, then the transition matrices $f_{vw}$ will be replaced by the transpose inverses $(f_{vw}^{t})^{-1}$. Thus, on the same manifold $Y=\Sigma\backslash D$ the two dual to each other integral affine structures are realized.

Let us introduce notations for the following tori:

$$\mathbb{T}:=\mathbb{R}^{d}/\mathbb{Z}^{d},\qquad\mathbb{T}_{v}:=(\mathbb{R}^{d}_{v})/(\mathbb{Z}^{d}_{v}),\qquad\mathbb{T}/w:=(\mathbb{R}^{d}/w)/(\mathbb{Z}^{d}/w).$$

For $\langle v,w\rangle=1$, the transition isomorphism $f_{vw}\in\mathrm{Hom}(\mathbb{Z}^{d}_{v},\mathbb{Z}^{d}/w)$ induces an isomorphism of the tori, which we will denote by the same symbol

$$f_{vw}\colon\mathbb{T}_{v}\rightarrow\mathbb{T}/w.$$

The torus fibration over $Y=\Sigma\backslash D$ is constructed as follows. Using the affine integral structure on $Y$ one can choose a covariantly constant (with respect to the $\operatorname{SL}_{\mathbb{Z}}$-connection) lattice $T^{\mathbb{Z}}Y$ in the tangent bundle $TY$ and form the relative quotient $W\rightarrow Y$ with the fibers $W_{q}=T_{q}Y/T^{\mathbb{Z}}_{q}Y$. Thus, the fibers are $W_{q}=\mathbb{T}_{v}$ when $q\in U_{v}$, and $W_{q}=\mathbb{T}/w$ when $q\in V_{w}$, with the canonical identifications $f_{vw}:\mathbb{T}_{v}\rightarrow\mathbb{T}/w$ for $q\in U_{v}\cap V_{w}$.

Let $N(D)\subset\Sigma$ be a regular neighborhood of the discriminant locus. Let $W^{\epsilon}\to\Sigma\backslash N(D)$ denote the torus fibration associated to the original integral affine structure restricted to the complement of $N(D)$ in $\Sigma$. In the next section we will show that the torus fibration $W^{\epsilon}$ on $\Sigma\backslash N(D)$ embeds differentiably into $H_{s}$ for sufficiently large $s$. The dual torus fibration by symmetry embeds into the mirror hypersurface. This is the topological part of the mirror symmetry statement in the version of Kontsevich and Soibelman [KS01].