I read a physics paper, 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, which is not really comprehensible to me given my limited math background.

Now I'm interested in the classification(up to homotopy) 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.