Then we need to consider We have been able to pick out the smallest energy eigenvalue E$_{0}$ in (4.9) because in the T → ∞ limit, it will dominate all the other terms. All we have left to do to demonstrate (4.7) is to show that lim T →∞ Ω( T, R ) ∼ lim T →∞ W ( C ). However, the Wilson loop corresponds to an extended ‘source’ in the fundamental representation (ie. quarks), separated by a distance R , being created at T = 0 and annihilated at some general T , so we are done. Let us apply equation (4.6) to the Constable-Myers geometry considered in this chapter. The Nambu-Goto action of the fundamental string is [109]: with G$_{MN}$ denoting the string frame metric, which is related to the Einstein metric by a scaling factor: It will be useful to recast (4.11) with an explicit 5-sphere symmetry: 62 ds 2 = $_{H}$− 1 / $^{2}$f (Δ+ δ ) / $^{4}$dx2 4 + R $^{2}$H1 / $^{2}$f (Δ+2 − δ ) / $_{4}$w 4 − b 4 w 4 ( dr 2 + r $^{2}$d Ω 2 5 ) (4.12) ds 2 string = e φ/ $^{2}$ds2 Einstein (4.11) = $_{H}$− 1 / $^{2}$f (Δ+ δ ) / $^{4}$dx2 4 + R $^{2}$H1 / $^{2}$f (Δ+2 − δ ) / $_{4}$w 4 − b 4 w 4 6 ∑ i =1 dw 2 i S = 1 2 πα ′ ∫ d $^{2}$σ √ − det G$_{MN}$∂$_{a}$x $^{M}$∂$_{b}$ x N (4.10) lim T →∞ Ω( T, R ) ≡ lim T $_{→∞}$〈 q ( T, 0)¯ q ( T, R ) | ¯ q (0 , 0) q (0 , R ) 〉 = lim T $_{→∞}$〈 $^{$_{Γ}$†$_{(}$$_{T, R}$$_{)Γ(0}$$_{, R}$$_{)}$}$〉 = lim T →∞ ∑ n |〈 $^{$_{Γ}$†$_{(0}$$_{, R}$$_{)}$}$〉 | 2 $_{e}$− E$_{n}$T ∼ $_{e}$− E 0 ( R ) T (4.9)