GaugeGauge Theories,Theories, StringsStrings andand CCoossmmoollooggyy

Igor Klebanov

Department of Physics Princeton University

Talk at Conference Continuous Advanced in QCD Minneapolis, May 13, 2006 QCDQCD andand stringstring theorytheory

•• At short distances, must smaller than 1 fermi, the quark- antiquark potential is Coulombic, due to the Asymptotic Freedom. •• At large distances the potential should be linear (Wilson) due to formation of confining flux tubes. FFlluuxx TTuubbeess iinn QQCCDD •• These objectsobjects may be approximately described by the Nambu strings (animation from lattice work by D. Leinweber et al, Univ. of Adelaide) •• The tubes are widely used, for example, in jet hadronization algorithms (the Lund String Model) where they snap through quark-antiquark creation. LargeLarge NN GaugeGauge TheoriesTheories •• ConnectionConnection ofof gaugegauge theorytheory withwith stringstring theorytheory isis strengthenedstrengthened inin `t`t HHoooofftt’’ss generalizationgeneralization fromfrom 33 colorscolors (SU(3)(SU(3) gaugegauge group)group) toto NN colorscolors (SU(N)(SU(N) gaugegauge group).group). •• MakeMake NN large,large, whilewhile keepingkeeping thethe `t`t HHoooofftt couplingcoupling fixed.fixed.

•• TheThe probabilityprobability ofof snappingsnapping aa fluxflux tubetube byby quark-quark-aannttiiqquuaarrkk creationcreation (meson(meson decay)decay) isis 1/N.1/N. TheThe stringstring couplingcoupling isis 1/N.1/N. BBrreeaakkiinngg tthhee IIccee •• Diirichlet branes (Polchinski) led string theory back to gauge theory in the mid-90’s. •• A stack of N Dirichlet 3-branes realizes N==4 supersymmetric SU(N) gauge theory in 4 dimensions. It also creates a curved background of 10-d theory of closed superstrings (artwork by E.Imeroni)

which for small r approaches

•• Successful matching of graviton absorption by D3- branes, related to 2-point function of stress-energy tensor in the SYM theory, with a gravity calculation in the 3-brane metric (IK; Gubser, IK, Tseytlin) was a precursor of the AdS/CFT correspondence. TheThe AAddSS/CFT/CFT dualityduality Maldacena; Gubser, IK, Polyakov; Witten •• Relates conformal gauge theory in 4 dimensions to string theory on 5-d Anti-de Sitter space times a 5-d compact space. For the N=4 SYM theory this compact space is a 5-d sphere.

•• The SO(2,4) geometrical symmetry of the AdS5 space realizes the conformal symmetry of the gauge theory. •• The d-dimensional AdS space is a hyperboloid

•• Its metric is •• When a gauge theory is strongly coupled, the radius of curvature of the dual AdS5 and of the 5-d compact space becomes large:

•• String theory in such a weakly curved background can be studied in the effective (super)-gravity approximation, which allows for a host of explicit calculations. Corrections to it proceed in powers of

•• Feynman graphs instead develop a weak coupling expansion in powers of !. At weak coupling the dual string theory becomes difficult. •• Gauge invariant operators in the CFT4 are in one-to-one correspondence with fields (or

extended objects) in AdS5 •• Operator dimension is determined by the mass of the dual field; e.g. for scalar operators

•• Correlation functions are calculated from the dependence of string theory path integral on boundary conditions "0 in AdS5, imposed near z=0:

•• In the large N limit the path integral is found from the classical string action: SSppiinnnniinngg SSttrriinnggss vvss.. LLoonngg OOppeerraattoorrss •• VibratingVibrating closedclosed stringsstrings withwith largelarge angularangular momentummomentum onon thethe 5-sphere5-sphere areare dualdual toto SYMSYM operatorsoperators withwith largelarge R-charge.R-charge. Berenstein, Maldacena, Nastase •• Generally,Generally, semi-classicalsemi-classical spinningspinning stringsstrings areare dualdual toto longlong operators,operators, e.g.e.g. thethe dualdual ofof aa highhigh spinspin operatoroperator

isis aa foldedfolded stringstring spinningspinning aroundaround

thethe centercenter ofof AdSAdS5.. Gubser, IK, Polyakov •• The anomalous dimension of such a high spin twist-2 operator is •• AdS/CFT predicts that at strong coupling

•• A 3-loop perturbative N=4 SYM calculation gives Kotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, Smirnov

•• An approximate extrapolation formula works with about 10% accuracy:

•• Calculation of f(!) was recently addressed using integrability by Eden, Staudacher; Belitsky et al EExxaacctt IntegrabilityIntegrability •• Perturbative calculations of anomalous dimensions are mapped to inntegraable spin chains, suggesting exact integrability of the N=4 SYM theory. Minahan, Zarembo; Beisert, Staudacher •• For example, for the SU(2) sector operators Tr (ZZZWZW…ZW) , where Z and W are two complex scalars, the Heisenberg spin chain emerges at 1 loop. Higher loops correct the Hamiltonian but seem to preserve its integrability. •• This meshes nicely with earlier findings of integrability in certain subsectors of QCD. Lipatov; Faddeev, Korchemsky •• The dual string theory approach indicates that in the SYM theory the exact integrability is present at very strong coupling (Bena, Polchinski, Roiban). Hence it is likely to exist for all values of the coupling. FoldedFolded StringsStrings onon SS5 aanndd MagnonsMagnons •• Folded string spinning around the north pole has, for large R-charge J: Gubser, IK, Polyakov •• Recently this string was identified by Hofman and Maldacena with a 2-magnon operator where each magnon carries p=# . The exact energy of a free magnon is BPS protected Beisert •• Its dual is an open string spinning on S5 : •• Hence the 2-magnon operator dimension is

in exact agreement with the folded string result for large !. QuarkQuark Anti-QuarkAnti-Quark PotentialPotential

•• The z-direction is dual to the energy scale of the gauge theory: small z is the UV; large z is the IR. •• In a pleasant surprise, because of the 5-th dimension z, the string picture applies even to theories that areare conformal (not confining!). The quark and anti- quark are placed at the boundary of Anti-de Sitter space (z=0), but the string connecting them bends into the interior (z>0). Due to the scaling symmetry of the AdS space, this gives Coulomb potential (Maldacena; Rey, Yee) ExcitationsExcitations ofof thethe GGlluuoonniicc SSttrriinngg

Callan, Guijosa; Brower, Tan, Thorn; IK, Maldacena, Thorn •• TheThe spectrumspectrum ofof smallsmall oscillationsoscillations ofof thethe stringstring withwith DirichletDirichlet boundaryboundary conditionsconditions atat z=0z=0 isis known:known: •• ByBy conformalconformal invariance,invariance, allall energiesenergies scalescale asas 1/L.1/L. TheirTheir spectrumspectrum isis known,known, andand theythey correspondcorrespond toto gluonicgluonic excitationsexcitations atat strongstrong `t`t HHoooofftt coupling.coupling. •• ButBut atat weakweak couplingcoupling therethere areare nono suchsuch excitations:excitations: onlyonly thethe groundground statestate ofof energyenergy wherewhere •• A simplified model where only ladder graphs are summed indicates that the coupling where an infinite number of excitations appear is •• Their appearance is related to the fall-to-the center instability in the bound state equation

•• The near-threshold bound states are in exact agreement with the spectrum of a highly excited string containing a single very long fold: IsIs suchsuch aa `fall`fall toto thethe centercenter’’ transitiontransition possiblepossible inin QCD?QCD? •• Apparently not, since the asymptotic freedom makes the coupling weak when the ends of the string approach each other. •• A recent lattice calculation of the string excitation spectrum shows this explicitly. Only one level exists with energy much lower than others at short separation. Juge, Kuti, Morningstar WWhhyy aarree tthhee fflluuxx ttuubbee eexxcciittaattiioonnss stablestable inin aa CFTCFT ?? •• ForFor aa combinationcombination ofof energeticenergetic andand largelarge NN reasons!reasons! •• AfterAfter aa gluongluon isis emittedemitted thethe resultingresulting statestate wouldwould bebe inin aa colorcolor adjointadjoint,, andand wouldwould havehave positivepositive energy.energy. Hence,Hence, energyenergy conservationconservation forbidsforbids one-gluonone-gluon emissionemission fromfrom anan excitedexcited boundbound state.state. •• TheThe energyenergy conservationconservation doesdoes notnot forbidforbid gglluueebbaallll (closed(closed string)string) emission.emission. ButBut itit isis suppressedsuppressed atat largelarge N.N. StringString TheoreticTheoretic ApproachApproach toto CCoonnffiinneemmeenntt •• It is possible to generalize the AdS/CFT correspondence in such a way that the quark- antiquark potential is linear at large distance. •• A “cartoon’’ of the necessary metric is

•• The space ends at a maximum value of z where the warp factor is finite. Then the confining string tension is •• Several 10-dimensional backgrounds with these qualitative properties are known (the compact space is actually “mixed’’ with the 5-d space). •• Witten (1998) constructed a background in the universality class of non-supersymmetric pure glue gauge theory. While there is no asymptotic freedom in the UV, hence no dimensional transmutation, the background serves as a simple model of confinement. •• Many infrared observables may be calculated from this background using classical supergravity. The lightest glueball masses are found from normalizable fluctuations around the supergravity solution. Their spectrum is discrete, and resembles qualitatively the results of lattice simulations in the pure glue theory. CCoonnffiinneemmeenntt iinn SSYYMM tthheeoorriieess •• Introduction of minimal supersymmettry (N=1) facilitates construction of gauge/string dualities. •• A useful tool is to place D3-branes and wrapped D5-branes at the tip of a 6-d cone, e.g. the conifold. •• The 10-d 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 Calabi-Yau space defined by the following constraint on 4 complex variables: •• In the UV there is a logarithmic running of the gauge couplings. Surprisingly, the 5-form flux, dual to N, also changes logarithmically with the RG scale. IK, Tseytlin •• What is the explanation in the dual SU(kM)xSU((k-1)M) SYM theory coupled to bifundamental chiral superfields A1, A2, B1, B2 ? A novel phenomenon, called a duality cascade, takes place: k repeatedly changes by 1 as a result of thee Seiberg duality IK, Strassler (diagram of RG flows from a review by M. Strassler) •• Dimensional transmutation in the IR. The dynamically generated confinement scale is

•• The pattern of R-symmetry breaking is the same as in the SU(M) SYM theory: Z2M -> Z2. •• In the IR the gauge theory cascades down to SU(2M) x SU(M). The SU(2M) gauge group effectively has Nf=Nc. •• The baryon and anti-baryon operators Seiberg

acquire expectation values and break the U(1) ia -ia symmetry under which Ak -> e Ak; Bl -> e Bl. Hence, we observe confinement without a mass gap: due to U(1)baryon chiral symmetry breaking there exist a Goldstone boson and its massless scalar superpartner. •• The KS solution is part of a moduli space of confining SUGRA backgrounds, resolved warped deformed conifolds. Gubser, Herzog, IK; Butti, Grana, Minasian, Petrini, Zaffaroni •• This family of solutions is dual to the `baryonic branch’ in the gauge theory:

•• Using the SUGRA solutions, various quantities have been calculated as a function of the modulus U=ln |$|. •• Here isis aa plot of the string tension: aa fundamental string at the bottom of resolved warped deformed conifold is dual to an `emergent’ chromo-electric flux tube. Dymarsky, IK, Seiberg •• An interesting observable is the tension of a composite string connecting q quarks with q anti- quarks. In any SU(M) gauge theory it must be symmetric under q -> M-q. This is achieved through a stringy effect: q strings blow up

into a wrapped D3-brane. Herzog, IK •• Dashed line refers to being far along the baryonic branch (near the MN limit) where

•• Solid lineline refers to the KS background •• AllAll ofof thisthis providesprovides usus withwith anan exactexact ssoolluuttiioonn ofof aa classclass ofof 4-d4-d largelarge NN confiningconfining supersymmetricsupersymmetric gaugegauge theories.theories. •• ThisThis shouldshould bebe aa goodgood playgroundplayground forfor testingtesting variousvarious ideas.ideas. •• SomeSome resultsresults onon glueballglueball spectraspectra areare alreadyalready available,available, andand furtherfurther calculationscalculations areare ongoing.ongoing. Krasnitz; Caceres, Hernandez, … ItIt willwill bebe interestinginteresting toto looklook forfor asymptoticasymptotic degeneraciesdegeneracies inin thethe spectraspectra similarsimilar toto thosethose thatthat areare empiricallyempirically observedobserved inin hadronichadronic spectra.spectra. ApplicationsApplications toto D-D-branebrane InflationInflation • The Slow-Roll Inflationary Universe (Linde; Albrecht, Steinhardt) is a very promising idea for generating the CMB anisotropy spectrum observed by the WMAP. • Finding models with very flat potentials has proven to be difficult. Recent string theory constructions use moving D-branes. Dvali, Tye

• In the KKLT/KKLMMT model, the warped deformed conifold is embedded into a string compactification. An anti-D3-brane is added at the bottom to break SUSY and generate a potential. A D3-brane rolls in the throat. Its radial coordinate plays the role of an inflaton. Kachru, Kallosh, Linde, Maldacena, McAllister, Trivedi NewNew approachapproach toto D-D-bbrraannee iinnffllaattiioonn??

•• In a flux compactification, the U(1)b is gauged. Turn on a Fayet-Iliopoulos parameter % . •• This makes the throat a resolved warped deformed conifold. The probe D3-brane potential on this space varies very slowly, so seems favorable for slow-roll inflation. A. Dymarsky, IK, N. Seiberg •• No anti-D3 brane needed: in presence of the D3- brane, SUSY is broken by the D-term %. Related to the `D-term Inflation’ Binetruy, Dvali; Halyo •• Effects of compactification generically spoil the flatness of the potential. Some `fine-tuning’ seems to be needed, as usual. This is currently under investigation with D. Baumann, A. Dymarsky, L. McAllister, A. Murugan and P. Steinhardt. CCoonncclluussiioonnss

•• Throughout its history, string theory has been intertwined with the theory of strong interactions •• The AdS/CFT correspondence makes this connection precise. It makes a multitude of dynamical statements about strongly coupled conformal (non-confining) gauge theories. •• Its extensions to confining theories provide a new geometrical view of such important phenomena as dimensional transmutation and chiral symmetry breaking. This allows for calculations of glueball and meson spectra and of hadron scattering in model theories. •• This recent progress offers new tantalizing hopes that an analytic approximation to QCD may be achieved along this route, at least for a large number of colors. •• But there is much work to be done if this hope is to become a reality. Understanding the string duals of weakly coupled gauge theories remains an important open problem. Phenomenological AdS/QCD approaches to it appear to give nice results. •• Embedding gauge/string dualities into string compactifications offers new possibilities for physics beyond the SM, and for modeling inflation and cosmic strings. EEmmbbeeddddiinngg iinn FFlluuxx CompactificationsCompactifications •• A long warped throat embedded into a compactification with NS-NS and R-R fluxes leads to a small ratio between the IR scale at the bottom of the throat and the string scale. Randall, Sundrum; Verlinde; IK, Strassler; Giddings, Kachru, Polchinski; KKLT; KKLMMT

•• In the dual cascading gauge theory the IR scale is the confinement scale: confinement stabilizes the hierarchy between the Planck scale and the SM or the inflationary scale.

•• Cascading gauge theories dual to “standard model throats” may offer interesting possibilities for new physics beyond the standard model. Cascales, Franco, Hanany, Saad, Uranga, … AA differentdifferent suggestionsuggestion forfor D-D-bbrraannee iinnffllaattiioonn (A. Dymarsky, IK, N. Seiberg) •• In a flux compactification, the U(1)baryon is gauged. Turn on a Fayet-Iliopoulos parameter % . •• This makes the throat a resolved warped defoormed conifold. •• The probe D3-brane ppotential on this space is asymptotically flat, if we ignore effects of compactification. The plots are for two different values of U~%. •• No anti-D3 or D7-brane needed: in presence of the D3-brane, SUSY is broken by the D-term %. Related to the `D-term Inflation’ Binetruy, Dvali; Halyo •• StructureStructure ofof energyenergy levelslevels forfor thethe ladderladder sumsum problem.problem. AtAt strongstrong couplingcoupling thesethese levelslevels agreeagree withwith thethe semi-semi- classicalclassical quantizationquantization ofof aa stringstring withwith oneone longlong ffoolldd..