Given an integral affine structure on a topological manifold $M$, there is a monodromy representation

$$T:\pi_{1}(M)\rightarrow\textrm{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$$

defined by covering a loop in $M$ by affine charts and composing the corresponding transition functions. See [KS06, 2.2] for more details.