I have been reading about higher-form symmetries, particularly how they are applied to non-abelian gauge theories. I have come across the claim that pure $SU(N)$ Yang Mills (i.e. with no quarks) possesses a discrete global $\mathbb Z_N$ symmetry (see e.g. page 112 of https://arxiv.org/abs/1810.05338). By contrast, ordinary E&M with a $U(1)$ gauge field possesses a continuous $U(1)$ one-form symmetry
Question 1: Why does the $SU(N)$ gauge theory give rise to a discrete one-form symmetry, while the $U(1)$ gauge theory gives rise to a continuous one-form symmetry?
Question 2: If I have a gauge theory with generic gauge group $G$, is there a general rule for figuring out what the corresponding global one-form symmetry group is?
I would greatly appreciate an answer to one or both of these questions.
