I read a [physics paper](http://arxiv.org/abs/1212.3324), of which the main idea is based on the topological classification of maps from 3-torus to the space of $N\times N$ unitary matrices. To quote their equation (4), which gives a winding number:
$$W(U)=\frac{1}{8\pi^2}\int_{T^3} dtdk_xdk_y U^{-1}\partial_t U\cdot[U^{-1}\partial_{k_x} U,U^{-1}\partial_{k_y} U],$$ 
where $(t,k_x,k_y)\in S^1\times S^1\times S^1\cong T^3$, and $U:=U(t,k_x,k_y)$ taking values of N-dimensional unitary matrices. The formula comes from [this math paper](http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.cmp/1103904396&page=record ), which is not really comprehensible to me given my limited math background.

Now I'm interested in the classification of maps from 2-torus to the space of $N\times N$ unitary matrices, is this already well developed? Any explanation or reference will be appreciated. 

PS. I'm familiar with the language of fundamental groups(but not many sophisticated theorems) and a little bit homology. Also some basics on differentiable geometry. If possible, please explain in a way accessible to me, but if this means too much compromise, just go ahead and shower me with the proper mathematics.