SU(N) is the N-fold cover of PSU(N). They share the same Lie algebra, so the Yang-Mills action would look identical locally. The center of SU(N) is just ZN. At the level of representations, the fundamental representation of SU(N) is a projective representation of PU(N), and only the adjoint ones are linear representations of PU(N).
If the matter fields all transform in the adjoint representation, then it makes sense to say that the gauge group is actually PU(N). A simple explanation is that by taking tensor product of adjoint representations you never get the fundamental ones, so the Hilbert space is restricted.
Because PSU(N)=SU(N)/ZN, the global topology of PU(N) is nontrivial. For example, the fundamental group π1(PU(N))=ZN, so there are nontrivial "vortex lines" in the scalar matter field, around which you pick up a holonomy in the center ZN. These topological excitations themselves are one-dimensional objects, and have "codimension" 2.
Quarks in SU(3) QCD transform as the fundamental representation.
This post imported from StackExchange Physics at 2015-03-23 13:03 (UTC), posted by SE-user Meng Cheng