MIT OpenCourseWare http ocw mit edu 8 821 String Theory Fall 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 8 821 F2008 Lecture 21 Con nement continued Lecturer McGreevy January 30 2009 Con ning geometry and mass gap main lesson mass gap smooth minimum of the wrap factor in the dual geometry Black hole mechanics Last time we were talking about attempting to nd a gravity dual of a deformation to 3D YM theory which we get from the N 4 on a thermal circle1 In these compacti cations the 3D YM coupling g3 is of order of the mass of the scalars g3 mX g4 N Ry 1 where g4 N is the 4D t Hooft coupling The 3D theory is not just YM but it also has a bunch of scalars and this obscures our intuition about the theory We can do a similar thing to get 4D YM theory by putting ND4 branes at low energies on a thermal circle with a radius Ry and again Fermions will have APBCs The ND4 theory is a gauge theory in 5D with 16 supercharges and its well de ned at low energies The boundary conditions on the thermal circle will break supersymmetry and the fermions will get 4D masses of order 1 Ry while bosons will get masses at one loop of order m2X g42 N m2 4 Ry2 At energies E 1 Ry the 4D and 5D YM couplings are related by 1 g42 Ry g52 which implies that mX 4 Ry The relation between the the 4D and 5D YM couplings comes from integrating over the extra dimension in the compacti ed theory The nice thing about the above construction is that if at the UV scale E 1 Ry the t Hooft coupling 4 1 then the theory is asymptotically free and we can calculate the function perturbatively There will be a dynamically genrated mass scale such that e 1 4 1 4 QCD mX 2 Ry Ry Ry Notice that 1 Ry plays the role of a UV Its like putting the 4D YM theory on a lattice where Ry plays the role of a lattice spacing If we probe the theory with energies of order 1 Ry we will see a 5D theory the UV completion 1 Remember that thermal circle means periodic boundary conditions for Bosons and antiperiodic boundary condi tions for Fermions 1 Notice also that when E 1 Ry we will have a pure YM theory The previous analysis was for 4 1 the gravity dual has a curvature which goes like 1 4 and we can t trust the supergravity description The gravity dual is valid when QCD UV i e when 4 is large At large 4 the QFT isn t weakly coupled at the cuto so there is never a regime where the usual perturbative calculation of asymptotic freedom is valid Back to the 3D case Despite that the 4D YM theory is a little bit cleaner in the weak coupling limit than the 3D case but the details of the gravity dual in the 3D is simpler To nd a gravity dual we would like some solution which satis es Einstein equations in the bulk with a negative cosmological constant 1 Rab gab gab 2 3 which asymptotes at the boundary to R2 Sy1 There are two solutions which satisfy these require ments y NOTHING zm z 0 a Compactify a circle on AdS b The other solution which is slightly more interesting is ds2 L2 dz 2 2 2 2 dt d x f dy z2 f where f 1 z4 4 zm y y 2 Ry 4 5 Note that f has a zero at zm where the radius of the y circle shrinks to zero For a random value of z there will a canonical de cit angle For z zm there is NOTHING i e when the circle shrinks the space ends We can make the solution much more well de ned if we demand that there is no singularity at z zm Locally near z zm ds2 2 z dy 2 d 2 where zm z zm 2 2 zm Surface gravity 6 7 This form of the metric makes it clear that the shrinking of the y circle is like a cone where the y direction is bred over the direction which shrinks to zero at z zm To avoid the conical de cit we chose the periodicity of y such that y y y 2 1 2 8 ds2 2 d 2 d 2 9 and the metric becomes and 0 is the origin of the polar coordinates If we chose a di erent periodicity there will be a conical de cit angle To match the boundary conditions and since we know that y is periodic with a period 2 1 where 2 zm then zm 2Ry The topology of the boundary is R2 1 S 1 and the topology of the bulk is R2 1 D2 z y where D2 z y is a disk with angular coordinate y and the y circle is contractible in the bulk which implies APBCs on the bulk Fermions Side Comments The importance of APBCs on the circle for the existence of a solution where the geometry ends is quit reminiscent of the instability of the Kaluza Klein vacuum 1 to the appearance of a bubble of nothing which can only exist if we put APBCs on the Fermions and is similar to the region z zm here We should think of the two solutions which we described as two saddle point contributions to the path integral of gravity with the speci ed boundary conditions The solution which dominates is the one which has a smaller action when evaluated on the saddle point John claims that solution b which has a mass gap dominates over a and its free energy wins for all values of Ry 1 2 X t looks like a boundry if we ignore the y direction zm z 0 f 1 AdS 2 The warp factor in b has a minimum Wmin z 1 zm ds2 W z 2 dx dx dz 2 z2 10 Heuristically the reason why there is a mass gap is that if we have a particle propagating in this space it will fall to this minimum value of the warp factor which acts like a gravitational potential i e lowest energy states are localized around z zm unlike the Poincare AdS case where particles fell to the Poincare horizon This statement will be made more precise next time 3 The spectrum of the 2 1 theory from the 2 point function Lets think about the two point function in momentum space of some gauge invariant operator such as the glueball operator O TrF F In a unitary eld theory the two point function has the spectral representation 1 11 G2 k hO k O k i d 2 k 2 2 0 where is the density of states in the eld theory which couples to the operator O kh O k 0ik2 12 and …