A Few Comments on a Few Comments on Gauge/String Duality

A Few Comments on Gauge/String Duality Igor Klebanov

Department of Physics

`10 Years of AdS/CFT’ Buenos Aires, December 21, 2007 Stacking NonNon--DilatonicDilatonic D-D-BranesBranes

• A stack of N Dirichlet 33--branesbranes realizes N=4 supersymmetric SU(N) gauge theory in 4 dimensions. It also creates a smooth extremal background of 10-d theory of closed superstrings with a constant dilaton (artwork by E.Imeroni)

which for small r approaches

• As Juan Maldacena ppyg,ointed out 10 years ago, the small r limit is the lowlow--energyenergy limit where the N=4 SYM theory on D-D-branesbranes decouples from the bulk closed strings. This is AdS/CFT ìn a nutshell.’ • Now, 10 years later, Ànti-de Sitter ’ or ÀdS’ appears in a large (and growing?) ftifhfraction of hepep--th a bst rac ts. • Planar N=4 SYM is well on its way to becoming `the Harmonic Oscillator of 4-4-dd gauge theory .’ • Can we find `the Harmonic Oscillator of confining 44--dh’d gauge theory?’ Dynamical Confinement

• Need AdS5 space up to logarithms in the UV. Can get this by placing D3D3-- branes and wrapped D5D5--branesbranes at the tip of the conifold. • The 10-10-dd geometry dual to the gauge theory on these branes is the warped deformed conifold (IK, Strassler)

• is the metric of the deformed conifold, a simple CalabiCalabi--Yau space defined by the following constraint on 4libl4 complex variables: • The warp factor is finite at the `tip of the cigar’ t=0, as required for the confinement: h(t)= 2--8/38/3 γ I(t)

• The stringgpp() tension, is proportional to h(t) - 1/2 and is minimized at t=0. It blows up at largg(e t (near the boundary) where sp ace is `near`near--AdS.’AdS.’ • Dimensional transmutation in the IR . The dynamically generated confinement scale is • Comppparison of warp factors in the AdS, warp ed conifold, and warped deformed conifold cases. The warped conifold solution with e =0 has an unacceptable naked singularity where h=0 . • This is how string theory tells us that the chiral symmetry breaking and dynamical scale generation must take place through turning on the deformation e. The finiteness of the warp factor at r= 0 translates into confinement. 3 • The radiusradius--squaredsquared of the S at t=0 is gsM iiin string uni ts.

• When gsM is largg,e, the curvatures are small everywhere, and the SUGRA solution is reliable in `solvinggggg’ this confining gauge theory.

• Even when g sM is small , the curvature gets small at large t (in the UV). • In the d ual ga uge theo ry the co upling stays strong in the UV. It is not asympttillftotically free, b btut rath er un dergoes a `cascading’ logarithmic RG flow that prodSU(M)thithIRduces SU(M) gauge theory in the IR. Spectrum of Glueballs

• The confining string tension is

• The glueballs are the normalizable modes localized near small t. In the supergravity limit (at large gs M) their mass scale is

• The m2 of the nn--thth radial excitation scales as 2 n Ts/(gs M) (see, for example the recent plots from Benna,,y Dymarsky, y,, IK, Soloviev) . This is the behavior found in KaluzaKaluza--KleinKlein theory, but not in QCD. • Glueballs with spin > 2 have much higher masses: 2 m ~ Ts • This separation of scales is a new phenomenon found for theories with reliable gravity duals. • Need more models with dynamical confinement and SUSY breaking.

• Possible applications of these models to new physics include RS warped extra dimension models, KKLT moduli stabilization in flux compactifications, as well as warped throat DD-- brane cosmology (KKLMMT). Contact with the Observable World • Thermal gauge/string dualities are being used to model QCD above deconfinement temperature and to compare with HeavyHeavy--IonIon collider experiments. MhldbhidMany successes have already been achieved, e.g., the observed low shear viscosity appears naturally via AdS/CFT. Policastro, Son, Starinets • BottomBottom--upup model building in warped 5 dimensions ((/Q)pAdS/QCD) has provided new fruitful way s to look at lowlow--energyenergy hadron physics. Erlich et al, Da Rold, Pomarol, Brodsky et al, … • Large N M2 -brane theory at finite T and chemical potential has shed light on properties of superconductors near Tc. Herzog et al, Hartnoll et al. • More generally, there exists a large class of AdS4/CFT3 dualities that are not yet wellwell--understood.understood. Finding 33--dd fixed points in lattice//ycondensed matter systems that have AdS4 duals is a good goal. For large N Wilson-Fisher fixed points it was proposed that the dual is a higherhigher--spinspin gauge theory in AdS4. IK, Polyakov • Another developing contact between AdS/CFT and condensed matter physics is the Entanglement Entropy. Ryu, Takayanagi LLfthFt?essons for the Future ? • ItiI tried searchi ng f or th em i n th e past thit history… • The original spirit of DD--brane/blackbrane/black hole correspondence was: use DD--branesbranes to explain `black hole microstates.’ This led to quick success for some specially constructed black holes. Strominger, Vafa; Callan, Maldacena; … • But soon various puzzles appeared, like the ¾ reduction in the 33--branebrane entropy, or the N3 degrees of freedom for N coincident M5M5--branes.branes. TiThiATurning Things Around d! ! • Instead of using D -branes to explain black holes , use black holes to make predictions about theories on D-branes, i .e . strongly coupled gauge theories. • This has opened a whole new window on aspects of gauge theory that were previously inaccessible. In this picture the extra dimensions of string or M-M-theorytheory neatly fit with the observable dimensions to produce new geometrical pictures of various phenomena in strongly coupled gauge theory. • So, AdS/CFT really was a paradigm shift. • Yet, we have not been able to solve QCD or any other asymptotically free gauge theory. • For a simple reason: when the coupling in such theories becomes weak in the UV, curvature is expected to get large. The machinery of Einstein’s GR is not enough. We would need much more sophisticated string theoretic machinery to extract something precise this way. • This is an important project which may take a long time… Turn Things A round Y et A gai n ?

• Since QCD does not seem to want to be solved this way , why force it? We understand it pretty well anyway… • Rather, let us hope that Nature will present us with a gift of new physics that is `easy’ from the point of view of gauge/string dualities, i .e . is described by weakly curved string backgrounds. • FthFurther un ders tan ding of gener iftic features o f such theories is a good goal. • One such generic feature is the large separation of scales between bound states with spin up to 2 and spin > 2. This is found in the RS model (cut -off AdS) and extends to various other warped models. • Similarly, we can search for characteristic features of gauge/string dualities in the sky: in models of D-D-branebrane inflation, cosmic stringg(s (which are sometimes dual to confining strings), etc. • Let us follow the path where gauge/string dualities are taking us, and surely the wondflderful journey will continue. • Ten More Years! Ten More Years! …