  We analyze the two main physical observables related to the momenta of
strongly correlated SU($N$) fermions in ring-shaped lattices pierced by an
effective magnetic flux: homodyne (momentum distribution) and self-heterodyne
interference patterns. We demonstrate how their analysis allows us to monitor
the persistent current pattern. We find that both homodyne and self-heterodyne
interference display a specific dependence on the structure of the Fermi
distribution and particles' correlations. For homodyne protocols, the momentum
distribution is affected by the particle statistics in two distinctive ways.
The first effect is a purely statistical one: at zero interactions, the
characteristic hole in the momentum distribution around the momentum
$\mathbf{k}=0$ opens up once half of the SU($N$) Fermi sphere is displaced. The
second effect originates from interaction: the fractionalization in the
interacting system manifests itself by an additional `delay' in the flux for
the occurrence of the hole, that now becomes a depression at $\mathbf{k}=0$. In
the case of self-heterodyne interference patterns, we are not only able to
monitor, but also observe the fractionalization. Indeed, the fractionalized
angular momenta, due to level crossings in the system, are reflected in
dislocations present in interferograms. Our analysis demonstrate how the study
of the interference fringes grants us access to both number of particles and
number of components of SU($N$) fermions.
