Abstract
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.
Highlights
Lattice data for the heavy quark potential can be well described by the Cornell form, which is the sum of the linear term and the Coulomb one
Recent lattice data shows that such quadratic corrections exist above the deconfinement phase transition
The quadratic corrections of the thermal quantities and the Polyakov loop in the deconfined phase are studied with the gauge/string duality
Summary
Lattice data for the heavy quark potential can be well described by the Cornell form, which is the sum of the linear term and the Coulomb one. It is found that the lattice result for the trace anomaly Δ in SU(3) gauge theory, normalized in unit of T 4, is well described by a 1/T 2 term Such an observation is made manifest by plotting the ratio Δ/T 2 versus T in the deconfined phase [2]. In a large temperature region Tc T 4Tc, the ratio takes nearly a constant value Such behavior is further confirmed in SU(N) gauge theory for various N [3]. One would naturally expect that it achieves significant quadractic corrections in the deconfined phase It is first pointed out in [4] that the logarithm of the Polyakov loop in SU(3) gauge theory is dominated by a 1/T 2 term. The result is [11]
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