Classical Yang-Mills vacua on $T^3:$ Explicit constructions

Flat 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.