8 821 F2008 Lecture 25 Thermal aspects of N 4 SYM Lecturer McGreevy Scribe Thomas Faulkner 12 10 08 The N 4 plasma at large 1 Previously in lecture 23 we gave evidence that the Black Hole thermodynamics of the AdS Black Brane non extremal solution was dual to the thermal ensemble of N 4 SYM on R3 at large N and Thus the gauge theory provides the microstates that are being coarse grained by the Bekenstein 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 duality In the long wavelength and small frequency limit both are consistent with the hydrodynamics of a relativistic CFT Additionally the duality allows one to compute various transport coefficients of the gauge theory at large such as the shear viscosity R charge conductivity etc See the review 1 1 2 Thermal Screening At finite temperature T correlators die off exponentially at large separation r 1 T even if in vacuum there are massless fields which mediate long range interactions This is because such particles 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 large distances as Vqq r e mth r 1 r 1 T 1 T T T T Figure 1 A particle receives multiple kicks from its thermal surroundings This generates a thermal contribution to the particles mass where mth is the thermal mass The q q potential can be calculated using a Wilson loop I A hWCr iT exp iVqq r T P exp i Cr 2 T where the contour Cr is a rectangle with spatial separation r and temporal separation T not to be confused with temperature The expectation value of the Wilson loop is taken in a thermal ensemble 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 using the duality One must find the minimal action string in the bulk which approaches Cr at the boundary Then the expectation of the Wilson loop is hWCr i exp iSmin Cr There are two stable saddle points to this problem1 The two string configurations are shown in Fig 2 For r r the curved string has the smallest action Smin VU r T while for r r the two straight strings win Smin V T which is independent of r Where the point of cross over is r 1 T zH z 0 z 0 0 2 0 4 0 2 zH 0 4 r z 0 0 2 0 4 0 2 zH 0 4 r 0 2 0 4 0 2 0 4 r zH Figure 2 The two string configurations as a function of r increasing from left to right with the minimum action configuration highlighted in red For large enough r the curved string solution ceases to exist its action becomes complex Both actions diverge at the boundary To yield a finite result for the q q potential we must renormalize by subtracting V 1 There is a third unstable saddle point which one can think of as coming from a local maximum of some fiducial potential 2 After renormalization the potential looks like Vqren q r Vqq r V VU r V r r 0 r r 3 4 For small r 1 T the potential approaches the T 0 result Vqren q r 1 c rT See Fig 3 Since the potential vanishes for r r there is no trace of the interaction and the gluons mediating the interaction are screened more than the expected exponential fall off 2 Note that the kink is an artifact of the large approximation Vqq HrL 0 0 0 2 0 4 0 6 0 8 1 0 1 2 r 0 5 1 0 1 5 Figure 3 The red line follows the actual potential The other curves represent string configurations with higher action In particular we have included a third configuration the highest curve which corresponds to the unstable solution mentioned in footnote 1 1 3 Polyakov Susskind Loop The Polyakov loop is an operator defined in the Euclidean theory by I A U P exp i 4 S 1 where the contour S 1 is around the Euclidean time circle The expectation value of this operator gives the free energy in the presence of an external charge in representation R htrR U i e Fq T T 5 This gives an order parameter for confinement at finite T confinement htrU i 0 Fq finite deconfined htrU i 6 0 2 6 From this one may conclude that the screening length is r However there is another candidate for the screening length which comes from the next order correction to V ren in the 1 expansion the two long strings may still interact via exchange of a massive glueball in the high T effectively 3 0 confining gauge theory This results in a potential e mgball r where mgmall is the lightest mass glueball of this 3d theory See 2 3 In the gravity dual we can compute htrR U i using a string which ends on S 1 The 5 dimensional Euclidean AdSBH has the topology R3 D where D is a disk with boundary S 2 at the boundary of AdS as in Fig 4 z 0 z z H D S 1 Figure 4 The Euclidean time circle at the boundary S 1 closes off smoothly in the bulk Euclidean black hole solution A string is begging to be placed here This geometry follows from the form of the metric L2 f z d 2 7 2 z Since f z vanishes at some z zH the Euclidean time circle which is periodically identified shrinks to zero size at this point 3 So S 1 is contractable in the full geometry and we may wrap a string world sheet on the disk D It follows that the expectation value of the Polyakov loop is non zero ds2 htrU i e AD 6 0 8 where AD is the area of the disk D Note actually AD is infinite coming from the usual UV divergence associated with an infinite quark mass probe This can be regulated in the usual way to yield a finite result We thus conclude that N 4 SYM at strong coupling is not confining on R3 for any temperature T 2 A large N deconfinement phase transition We now want to consider the more interesting case of N 4 SYM on S 3 which does indeed exhibit a confinement to deconfinement phase transition Along the way we will see that the AdS CFT dual of this phase transition implies a dramatic consequence for Quantum Gravity we must sum over geometries and topologies consistent with certain boundary conditions 2 1 N 4 …