With a given affine structure on $Y$ we can associate a flat affine connection $\nabla^{aff}$ (see [KN]). The corresponding parallel transport acts on tangent spaces by affine transformations. For a ${\bf Z}$-affine structure the monodromy of $\nabla^{aff}$ belongs to $GL(n,{\bf Z})\ltimes{\bf R}^{n}$, i.e. $\forall y\in Y$ we have a monodromy representation

$$\rho:\pi_{1}(Y,y)\to GL(n,{\bf Z})\ltimes{{\bf R}}^{n}\,\,.$$

Alternatively, we can define the monodromy representation by covering a loop in $Y$ by ${\bf Z}$-affine coordinate charts and composing the corresponding transition functions.

Notice that a ${\bf Z}$-affine structure on $Y$ gives rise to a class

$$[\rho]\in H^{1}(Y,T^{{\bf Z}}\otimes{{\bf R}})=H^{1}(Y,T_{Y}^{\nabla})\,\,\,,$$

where $T_{Y}^{\nabla}\subset T_{Y}$ is the subsheaf of $\nabla$-flat sections¹¹ 1 Here we slightly abuse notations because $Y$ is not necessarily connected.. De Rham representative of class $[\rho]$ is given by a differential $1$-form $\theta\in\Omega^{1}(Y,T_{Y})$ such that $\theta(v)=v$ for any tangent vector $v$. In affine coordinates one has $\theta=\sum_{i}\partial/\partial x_{i}\otimes dx_{i}$. Clearly $\nabla(\theta)=0$.

We will need later an explicit formula for the ${\bf R}$-valued pairing of $[\rho]$ with a closed singular 1-chain with coefficients in the local system $(T^{\ast})^{\bf Z}=(T^{*}Y)^{\bf Z}$, the dual covariant lattice in $T^{\ast}Y$. With any singular $1$-chain $c$ with values in $(T^{\ast})^{\bf Z}$ we associate a real number $j(c)$ in the following way. Suppose that $c$ is given by a continuous map $\gamma:[0,1]\to Y$ and a section $\alpha\in\Gamma([0,1],\gamma^{\ast}(T^{\ast})^{\bf Z})$. Parallel transport via the connection $\nabla^{aff}$ gives rise to a map $\overline{\gamma}:[0,1]\to T_{\gamma(0)}Y,\,\,\,\overline{\gamma}(0)=0$. Let $\alpha_{0}=\alpha(0)\in(T_{\gamma(0)}^{\ast})^{{\bf Z}}\subset T^{*}_{\gamma(0)}Y$. We define $j(c)=\langle\alpha_{0},\overline{\gamma}(1)\rangle$. We extend $j(c)$ to an arbitrary singular $1$-chain $c$ by additivity. Then the class $[\rho]$ can be calculated as $\langle[\rho],[c]\rangle=j(c)$ for any closed $1$-chain $c\in C_{1}(Y,(T^{\ast})^{{\bf Z}})$.