Let $B$ be an $n$-dimensional manifold. An affine structure on $B$ is given by an atlas $\{(U_{i},\psi_{i})\}$ of coordinate charts $\psi_{i}:U_{i}\rightarrow\mathbb{R}^{n}$, whose transition functions $\psi_{i}\circ\psi_{j}^{-1}$ lie in ${\rm Aff}(\mathbb{R}^{n})$. We say the affine structure is tropical if the transition functions lie in $\mathbb{R}^{n}\rtimes GL(\mathbb{Z}^{n})$, i.e., have integral linear part. We say the affine structure is integral if the transition functions lie in ${\rm Aff}(\mathbb{Z}^{n})$.

If an affine manifold $B$ carries a Riemannian metric $g$, then we say the metric is affine Kähler or Hessian if $g$ is locally given by $g_{ij}=\partial^{2}K/\partial y_{i}\partial y_{j}$ for some convex function $K$ and $y_{1},\ldots,y_{n}$ affine coordinates.