Classical Yang-Mills Vacua on $T^{3}$ : Explicit Constructions

20 pages, 1 figure, corrected some typosFlat connections for unitary gauge groups on a 3--torus with twisted boundary conditions as well as recently discovered periodic nontrivial flat connections with ``nondiagonalizable'' triples of holonomies for higher orthogonal and exceptional groups are constructed explicitly in terms of Jacobi theta functions with rational characteristics. The (fractional) Chern-Simons numbers of these vacuum gauge field configurations are verified by direct computation