What's the symmetry group $SU(N)/Z_N$?

Question

I'm trying to understand David Tong's  http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html, specifically the discussion around page 92 where he's arguing that a different symmetry group may the group of QCD, namely $G'=SU(N)/Z_N$ instead of $G=SU(N)$. 

I understand that having $G'$ as a group leads to a different quantization of the $\theta$ term, but I want to know if this is really the crucial aspect? Namely, how does the cyclic group $Z_N$ act on the fundamental fields (quarks, gluons)? Can anybody show this invariance to me at the level of the Lagrangian? I'm a bit confused so even the most pedestrian computation would help :)

  

Answer 1

$\mathbb{Z}_N$ is realized in $SU(N)$ as the subgroup of scalar matrices which are diagonal matrices with a $N$-th root of the unity ($e^{2i\pi k/N}$) on the diagonal (exercise: check that these matrices are in $SU(N)$). These matrices are in the center of $SU(N)$ so they act trivially in the adjoint representation (where gluons are living). But they act non-trivially in the fundamental representation (where quarks are living).