Discrete symmetry breaking and baryon currents in U(N) and SU(N) gauge theories

In SU($N$) gauge theories with fermions in the fundamental or in a two-index (either symmetric or antisymmetric) representation formulated on a manifold with at least one compact dimension with non-trivial holonomy the discrete symmetries $C$, $P$ and $T$ are broken at small enough size of the compact direction(s) for certain values of $N$. We show that for those $N$ in the broken phase a non-zero baryon current wrapping in the compact direction exists, which provides a measurable observable for the breaking of $C$, $P$ and $T$. We prove that in all cases where the current is absent there is no breaking of those discrete symmetries. This includes the limit $N \to \infty$ of the SU($N$) gauge theory with symmetric or antisymmetric fermions and U($N$) gauge theory at any value of $N$. We then argue that the component of the baryon current in the compact direction is the physical order parameter for $C$, $P$ and $T$ breaking due to the breaking of Lorentz invariance.