  Confinement in SU($N$) gauge theory is due to the linear potential between
colored objects. At short distances, the linear contribution could be
considered as the quadratic correction to the leading Coulomb term. Recent
lattice data show that such quadratic corrections also appear in the deconfined
phase, in both the thermal quantities and the Polyakov loop. These
contributions are studied systematically employing the gauge/string duality.
"Confinement" in ${\mathcal N}=4$ SU($N$) Super Yang-Mills (SYM) theory could
be achieved kinematically when the theory is defined on a compact space
manifold. In the large-$N$ limit, deconfinement of ${\mathcal N}=4$ SYM on
$\mathbb{S}^3$ at strong coupling is dual to the Hawking-Page phase transition
in the global Anti-de Sitter spacetime. Meantime, all the thermal quantities
and the Polyakov loop achieve significant quadratic contributions. Similar
results can also be obtained at weak coupling. However, when confinement is
induced dynamically through the local dilaton field in the gravity-dilaton
system, these contributions can not be generated consistently. This is in
accordance with the fact that there is no dimension-2 gauge-invariant operator
in the boundary gauge theory. Based on these results, we suspect that quadratic
corrections, and also confinement, should be due to global or non-local effects
in the bulk spacetime.
