Center symmetry and area laws

Abstract

SU($N_c$) gauge theories containing matter fields may be invariant under transformations of some subgroup of the ${\mathbb{Z}}_{N_c}$ center; the maximum such subgroup is ${\mathbb{Z}}_p$, with $p$ depending on $N_c$ and the representations of the various matter fields in the theory. Confining SU($N_c$) gauge theories in either 3+1 or 2+1 space-time dimensions and with matter fields in any representation have string tensions for representation $R$ given by ${\sigma }_R={\sigma }_f\frac{p_R(p-p_R)g\left(p_R(p-p_R)\right)}{(p-1)g(p-1)}$ with $p_R=n_R\mathrm{mod}(p)$, where ${\sigma }_f$ is the string tension for the fundamental representation, $g$ is a positive finite function and $n_R$ is the n-ality of $R$. This implies that a necessary condition for a theory in this class to have an area law is invariance of the theory under a nontrivial subgroup of the center. Significantly, these results depend on $p$ regardless of the value of $N_c$.