  We classify $su(N_c)$ gauge theories on $\mathbb R^3\times \mathbb S^1$ with
massless fermions in higher representations obeying periodic boundary
conditions along $\mathbb S^1$. In particular, we single out the class of
theories that is asymptotically free and weakly coupled in the infrared, and
therefore, is amenable to semi-classical treatment. Our study is conducted by
carefully identifying the vacua inside the affine Weyl chamber using Verma
bases and Frobenius formula techniques. Theories with fermions in pure
representations are generally strongly coupled. The only exceptions are the
four-index symmetric representation of $su(2)$ and adjoint representation of
$su(N_c)$. However, we find a plethora of admissible theories with fermions in
mixed representations. A sub-class of these theories have degenerate
perturbative vacua separated by domain walls. In particular, $su(N_c)$ theories
with fermions in the mixed representations adjoint$\oplus$fundamental and
adjoint$\oplus$two-index symmetric admit degenerate vacua that spontaneously
break the parity ${\cal P}$, charge conjugation ${\cal C}$, and time reversal
${\cal T}$ symmetries. These are the first examples of strictly weakly coupled
gauge theories on $\mathbb R^3 \times \mathbb S^1$ with spontaneously broken
${\cal C}$, ${\cal P}$, and ${\cal T}$ symmetries. We also compute the fermion
zero modes in the background of monopole-instantons. The monopoles and their
composites (topological molecules) proliferate in the vacuum leading to the
confinement of electric charges. Interestingly enough, some theories have also
accidental degenerate vacua, which are not related by any symmetry. These vacua
admit different numbers of fermionic zero modes, and hence, different kinds of
topological molecules. The lack of symmetry, however, indicates that such
degeneracy might be lifted by higher order corrections.
