Center symmetry and area laws

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 $σ_R =σ_f \, \, \frac{p_R (p-p_R) \, \, g\left (p_R (p-p_R) \right )}{(p-1) \, \, g(p -1 )}$ with $p_R={n_R \, \rm mod}(p)$, where $σ_f$ 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 $N_c$.

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