8.821 F2008 Lecture 25: Thermal aspects of N = 4 SYMLecturer: McGreevy Scribe: Thomas Faulkner12/10/081 The N = 4 plasma at large λPreviously in lecture 23 we gave evidence that the Black Hole thermodynamics of the AdS BlackBrane (non-extremal) solution was d ual to the thermal ensemble of N = 4 SYM on R3at largeN and λ. Thus the gauge theory provides the microstates that are being coarse grained by theBekenstein-Hawking entropy of the black hole SBH.A few comments on this observation:1.1 Hydrodynamics• Perturbing the equilibrium of the boundary theory with a kick will result in thermalization -relaxation back to equilibrium.• In the bulk the response to such a kick is for the energy of the kick to fall into the black hole.The above two statements are related by the du ality. In the long wavelength and small frequencylimit both are consistent with the hydrodynamics of a relativistic CFT. Additionally the dualityallows one to compute various transport coefficients of the gauge theory at large λ, such as theshear viscosity, R-charge conductivity, etc. See the review [1].1.2 Thermal ScreeningAt fi nite temperature T correlators die off exponentially at large separation r ≫ 1/T , even ifin vacuum there are massless fields which mediate long range interactions. This is because suchparticles develop a thermal mass from continuously interacting with the thermal bath.For example the force between two external charges in a gauge theory should behave at largedistances as,Vq ¯q(r) ∼r≫1/Te−mthr(1)1TTTTFigure 1: A particle receives multiple kicks from its thermal surroundin gs. This generates a thermalcontribution to the particles mass.where mthis the thermal mass.The q¯q potential can be calculated using a Wilson loop,exp−iVq ¯q(r)˜T=P expiICrAT≡ hWCriT(2)where the contour Cris a rectangle with spatial separation r and temporal separation˜T not tobe confused with temperature. The expectation value of the Wilson loop is taken in a thermalensemble. The above formula is defined for˜T ≫ r large.Of course as we learned in a previous lecture we can compute Wilson loops at large λ using theduality. One must find the minimal action string in the bulk which approaches Crat the boundary.Then the expectation of the Wilson loop is hWCri = exp(−iSmin(Cr)).There are two (stable) saddle points to this problem1The two string configurations are shown inFig. 2. For r < r∗the curved string has the smallest action Smin= VU(r)˜T while for r > r∗thetwo straight strings win Smin= V||˜T which is independent of r. Where the point of cross over isr∗∼ 1/T = πzH.-0.4-0.20.20.4rzHz=0-0.4-0.20.20.4rzHz=0-0.4-0.20.20.4rzHz=0Figure 2: The two string configurations as a function of r (increasing from left to right) with theminimum action configuration highlighted in red. For large enough r the curved string solutionceases to exist (its action becomes complex!)Both actions diverge at the boundary. To yield a finite result for the q ¯q potential we must renor-malize by subtracting V||.1There is a third (unstable) saddle point, which one can think of as coming from a local maximum of some fid ucialpotential.2After renormalization the potential looks like,Vrenq ¯q(r) = Vq ¯q(r) −V||=VU(r) −V||r < r∗0 r > r∗(3)For small r ≪ 1/T the potential approaches the T = 0 result; Vrenq ¯q∼√λr(1 + c(rT )4+ . . .). SeeFig. 3. Since the potential vanishes for r > r∗there is no trace of the interaction, and the gluonsmediating the interaction are screened (more than the expected exponential fall off.2)Note that the kink is an artifact of the large-λ approximation.0.20.40.60.81.01.2r-1.5-1.0-0.50.0VqqHrLFigure 3: The red line follows the actual potential. The other curves represent string configurationswith higher action. In particular we have included a third configuration, the highest curve, whichcorresponds to the unstable solution mentioned in footnote 1.1.3 Polyakov-Susskind LoopThe Polyakov loop is an operator defined in the Euclidean theory by,U = P exp iIS1τA!(4)where the contour S1τis around the Euclidean time circle.The expectation value of this operator gives the free energy in the presence of an external chargein representation R;htrRUi = e−Fq(T )/T(5)This gives an order parameter for confinement at finite T ,Fq=∞ confinement, htrUi = 0finite deconfined htrUi 6= 0(6)2From this one may conclude that the screening length is r∗. However there is another candidate for the screeninglength which comes from the next order correction to Vrenin the 1/√λ expansion: the two long strings may stillinteract via exchange of a massive glueball in the high-T effectively 3+0 confining gauge theory. This results in apotential ∼ e−mgballrwhere mgmallis the lightest mass glueball of this 3d theory. See [2].3In the gravity dual we can compute htrRUi using a string which ends on S1τ. The 5 dimensionalEuclidean AdSBH has the topology R3× D where D is a disk with boundary S2τat the boundaryof AdS, as in Fig. 4.Sz=0z=zHDττ1Figure 4: The Euclidean time circle at the boundary S1τcloses off smoothly in the bulk Euclideanblack hole solution. A string is begging to be placed here.This geometry follows from the form of the metric,ds2=L2z2f(z)dτ2+ . . .(7)Since f(z) vanish es at some z = zHthe Euclidean time circle which is periodically identified shrinksto zero size at this point.3So S1τis contractable in the full geometry, and we may wrap a stringworld sheet on the disk D. It follows th at the expectation value of the Polyakov loop is non zero,htrUi ∼ e−AD/α′6= 0 (8)where ADis the area of the disk D. Note actually ADis infinite, coming from the usual UVdivergence associated with an infinite quark mass probe. T his can be regulated in the usual wayto yield a finite result.We thus conclude that N = 4 SYM at strong coupling is not confining on R3for any temperatureT .2 A large-N deconfinement phase transition.We now want to consider the more interesting case of N = 4 SYM on S3which does indeed exhibita confinement to deconfinement phase transition. Along the way we will see that the AdS/CFTdual of this phase transition implies a dramatic consequence for Quantum Gravity; we must sumover geometries and topologies (consistent with certain bound ary conditions.)2.1 N = 4 SYM on S3- Kinematic ConfinementAs opposed to the theory on R3this theory has a unique vacuum, |Ω >:3Recall in a previous lecture we used this fact to find the period with