Consider a triangulated orientable surface with the following data: on each edge a vector with integer coordinates is written so that for each triangle the sum of the vectors corresponding to three edges of its boundary is $0$.


May such an object be interpret as a discrete version of some topological construction, say, flat connection? 


Since for each triangle sum of the vectors corresponding to its boundary is $0$, it is possible to assign to each triangle its "area"- the determinant of the matrix formed by any two vectors of the boundary of the triangle.
I also wonder, if there exists some interpretation of determinants of these triangles. Maybe, the 2-cochain formed by them is representing some characteristic class?



Discrete version of some topological object.
Let $\Sigma$ be your surface.  

I'll construct a new surface $\tilde \Sigma$ as follows.
Take $\Sigma$ apart into individual triangles, and reglue them with a twist at each edge (the adjacency graphs for the faces of $\Sigma$, and for the faces of $\tilde\Sigma$ are the same). Note that the surfaces $\Sigma$ and $\tilde\Sigma$ could have different genus, and that $\tilde\Sigma$ could be non-orientable.


Your data is equivalent to having a piecewise linear map (not necessarily an embedding) of the **universal cover** of the new surface $\tilde\Sigma$ into $\mathbb R^n$, so that the vertices of the triangulation map to $\mathbb Z^n$.  
 That map is well defined up to an overall translation.


