Flat connections for unitary gauge groups on a 3-torus with twisted boundar
y conditions as well as recently discovered periodic nontrivial flat connec
tions 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 numb
ers of these vacuum gauge field configurations are verified by direct compu
tation.