Let $\mathbb{A}^{n}$ be $n$-dimensional affine space. An integral affine structure on an $n$-dimensional manifold $Y$ is given by an open covering $\{U_{\alpha}\}$ of $Y$ together with coordinates $\phi_{\alpha}:U_{\alpha}\to\mathbb{A}^{n}$ such that the transition maps $\phi_{\alpha}\circ\phi_{\beta}^{-1}$ are in $\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$ on the non-empty overlaps $U_{\alpha}\cap U_{\beta}$. An integral Kähler affine structure on $Y$ is a Riemannian metric $g$ which is potential in local affine coordinates, i.e. $g_{ij}=\frac{\partial^{2}K_{\alpha}}{\partial y_{i}\partial y_{j}}$ for some local potentials $K_{\alpha}$.

The dual Kähler affine structure on the same Riemannian manifold $(Y,g)$ is defined as follows (cf [KS01]). We use the same covering $\{U_{\alpha}\}$. The new affine coordinates are $\hat{y}_{i}=\frac{\partial K_{\alpha}}{\partial y_{i}}$ which take values in the dual affine space $({\mathbb{A}^{n}})^{*}$ (the underlying vector spaces for ${\mathbb{A}^{n}}$ and $({\mathbb{A}^{n}})^{*}$ are naturally dual). The new local potentials $\hat{K}_{\alpha}$ are defined by the Legendre transforms of the old ones:

$$\hat{K}_{\alpha}(\hat{y})=\max_{y\in U_{\alpha}}\{\langle\hat{y},y\rangle-K_{\alpha}(y)\}.$$

Here one needs to choose origins in ${\mathbb{A}^{n}}$ and $(\mathbb{A}^{n})^{*}$ to define the pairing. Different choices give rise to equivalent Kähler affine structures. The dual affine structure is integral iff the original one is.

Given an affine structure on $Y$ one can consider its monodromy representation $\pi_{1}(Y)\to SL(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$. Two equivalent affine structures have conjugate monodromies.

For a Kähler affine manifold $(Y,g)$ one can define (cf. [KS01]) a characteristic class $[g]$ of the metric, which is an analog of the Kähler class in complex geometry. Let $\mathcal{A}f\negmedspace f_{Y}$ be the sheaf of locally affine functions. The metric is given by local potentials in affine coordinates: $g_{ij}=\frac{\partial^{2}K}{\partial y_{i}\partial y_{j}}$. Then the differences of the potentials on the overlaps will define a Čech cohomology class $[g]\in H^{1}(Y,\mathcal{A}f\negmedspace f_{Y})$.

It is more natural to combine the monodromy representation and the metric class into one class, which we will call the class of affine polarization. It can be represented by a Čech 1-cocycle with values in the semi-direct product $(\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n})\ltimes\operatorname{Af{}f}_{n}$, where the affine transformations $\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$ act on the affine functions $\operatorname{Af{}f}_{n}$ from the right.

The natural projection onto the normal component $\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$ in the above semi-direct product gives the monodromy representation. To recover the metric class, however, one needs to fix a splitting of the natural map $(\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n})\ltimes\operatorname{Af{}f}_{n}\to\operatorname{Af{}f}_{n}$. Different splittings will give conjugate metric classes.

For the purposes of this paper we consider a convenient $(n+2)$-dimensional faithful representation of the group $(\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n})\ltimes\operatorname{Af{}f}_{n}$. Let us choose $q$ – an integral vector in $\mathbb{R}^{n+2}$, and $p$ – an integral vector in the dual space $(\mathbb{R}^{n+2})^{*}$. Then this representation provides an isomorphism of $(\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n})\ltimes\operatorname{Af{}f}_{n}$ with the following subgroup of $\operatorname{GL}_{n+2}(\mathbb{R})$:

$$G_{n}(p,q):=\{g\in\operatorname{GL}_{n+2}(\mathbb{R})\ :\ g(q)=q,\ g^{*}(p)=p,\ g|_{\{\langle p,x\rangle=0\}/q}\text{ is integral}\},$$

where $g^{*}:(\mathbb{R}^{n+2})^{*}\to(\mathbb{R}^{n+2})^{*}$ is the adjoint linear transformation. The affine space $\mathbb{A}^{n}$ can be identified with $\{\langle p,x\rangle=1\}/q$, and the $G_{n}(p,q)$ action on it gives the corresponding affine transformation of $\mathbb{A}^{n}$. To recover the affine function $f:\mathbb{A}^{n}\to\mathbb{R}$ one needs to fix an integral linear functional $l\in(\mathbb{Z}^{n+2})^{*}$, such that $l(q)=1$. Then $f(x)=l(g(x))-l(x)$ is a function, well defined on the quotient $\{\langle v,x\rangle=1\}/w$. Only the representing Čech cocycle depends on the choice of $l$, not the metric class itself.

For the (mirror) symmetry sake we also choose an integral element $k\in\mathbb{R}^{n+2}$ with $p(k)=1$. The vector $k$ defines an origin in $\mathbb{A}^{n}$, hence it allows to recover the translational part of the affine transformation. Again, the class of this translational part, called the radiance obstruction (cf. [GH84]), is independent of the choice of $k$ (different $k$’s give rise to conjugate monodromies). As was noted in [GS02] the radiance obstruction class is dual to the linear part of the metric class under the duality between the Kähler affine structures.

More generally, it is also clear that the full polarization class for the dual Kähler affine structure can be represented by the adjoint inverse transformations for each $U_{\alpha}\cap U_{\beta}$, with the rôles of $q,k$ and $p,l$ exchanged. Given a basis $\{e_{i}\}$ of $\mathbb{R}^{n+2}$ such that $\langle p,e_{i}\rangle=0,\ i=1,\dots,n+1$, $e_{n+1}=q$ and $e_{n+2}=k$, the group $G_{n}(p,q)$ can be represented by non-degenerate matrices in the form

$$\left(\begin{array}[]{ccc}A&0&b\\ a&1&c\\ 0&0&1\end{array}\right),$$

where $A$ and $b$ represent the linear and translational parts of the affine transformations, and $a,c$ are the linear and constant parts of the affine function, respectively.

All of the above (including the affine structure itself) can be defined even if we do not require the affine charts to be maps into the same affine space. We won’t have groups anymore, but in all cocycle conditions the compositions still make sense. In particular, to specify an integral affine structure we would need continuous maps $\phi_{\alpha}:U_{\alpha}\to\mathbb{A}^{n}_{\alpha}\cong\{\langle p_{\alpha},x_{\alpha}\rangle=1\}/q_{\alpha}$ together with integral elements $q_{\alpha},k_{\alpha}\in\mathbb{R}^{n+2}_{\alpha}$ and $p_{\alpha},l_{\alpha}\in(\mathbb{R}^{n+2})^{*}$ with $\ p_{\alpha}(k_{\alpha})=1,\ l_{\alpha}(q_{\alpha})=1$, such that the transition maps $\phi_{\alpha\beta}:{\mathbb{R}^{n+2}}\to{\mathbb{R}^{n+2}}$ satisfy the corresponding invariance, coinvariance and integrality conditions.

The cohomological information, such as monodromy, radiance obstruction and the metric class, is encoded in the transition maps $\phi_{\alpha\beta}$.