JHEP07(2000)038 July 23, 2000 Accepted: = 4 super-Yang-Mills theory are N June 21, 2000, ∗ [email protected] , Received: D-branes, AdS-CFT Correspondance. States on the Coulomb branch of [email protected] [email protected] [email protected] HYPER VERSION Theory Division, CERN, CH-1211 Geneva 23, Switzerland ∗ Department of Physics andLos Astronomy, Angeles, University CA of 90089-0484, SouthernEmail: USA California Lyman Laboratory of Physics,Cambridge, Harvard University MA 02138, USA Email: Department of Mathematics andMassachusetts Center Institute for of Theoretical Physics Technology Cambridge, MA 02139-4307, USA Email: Abstract: Krzysztof Pilch and Nicholas P. Warner Steven S. Gubser studied from thesupersymmetric point of solutions view provide examples ofsupergravity of in gauged consistent ten supergravity truncation dimensions. inand from five perfect A type-IIB dimensions. screening mass forWe gap offer These external an for quarks interpretation states of arisedistributions. created these in surprising by the features local in supergravity terms operators approximation. of ensembles of brane Keywords: Daniel Z. Freedman Continuous distributions of D3-branesgauged and supergravity JHEP07(2000)038 6 R near N (1.1) 5 AdS , then the given by L L correspond to ` e supersymmetries ) gauge symmetry has been N , 2 / N 5 10 π 2 κ 1 = between them is much less than 4 in the interior. Sometimes a simpler picture of ` L 5 AdS . Each factor has a radius of curvature 5 2) by scalar vacuum expectation values (VEV’s). This S × N/ 5 SU( AdS × D3-branes in the six transverse dimensions. These configurations 2) N N/ = 4 super-Yang-Mills theory. However, it also includes other states in N is the ten-dimensional gravitational constant. If the branes are not co- 10 κ At the origin of moduli space, where all the branes are coincident, the near- preserve sixteen supersymmetries, as appropriate since the Poincar´ the vacuum state of the gauge theory whereconfiguration the has been SU( studiedtribution in of [4, the 5]. More generally, one could considerof any dis- the gauge theoryHiggsing. are maintained The but space of superconformal possible invariance is distributions is broken precisely by the the moduli space Sym the Hilbert space ofgravity the field equations theory; in these which correspond the bulk to space-time certain geometry solutions approaches of the super- where broken to SU( states in the gauge theorytwo emerges equal from the clusters ten-dimensional of geometry. coincident For instance, D3-branes separated by a distance the boundary, but differs from incident, but the average distance horizon geometry is of the gauge theory.which It remain is massless known mediate as long-range Coulomb the interactions. Coulomb branch because the gauge bosons 1. Introduction The AdS/CFT correspondence [1,vacuum 2, of 3] has been primarily studied in the conformal Contents 1. Introduction2. Supergravity solutions3. A two-point function4. Wilson loops5. Discussion 3 1 7 11 12 JHEP07(2000)038 . 5 5 , S 4 , (1.2) 3 × , 5 2 esuper- , of SO(6). 0 AdS =1 =4. n = 4 geometries N 2. Also, Wilson n , and so we recover =1Poincar´ 5 n> N . l =2and AdS X n k X tr  kl δ ij δ 6 1 2 − USp(8), of which one field is a component of l j = 1 flows lie in two-dimensional submanifolds δ / 2 for quark-anti-quark separations larger than k i δ N ≥ 6(6)  n E 10 representation. The = . There is a privileged member in each class which ) 5 ⊕ j X ) of the SO(6) local gauge symmetry for i ( n AdS X − tr = 8, five-dimensional supergraviton multiplet. More specifically, SO(6 -dimensional ball. Next, we investigate the behavior of two-point n = 2 and a completely discrete spectrum for × N geometry with scalar profiles in five dimensions, it is natural to ask ) n n 5 10 component vanishes and supersymmetry is enhanced to AdS and the other of the 10 ⊕ 0 This study is an outgrowth of the work of [6], in which it was found that there are From the point of view of supergravity, the two-center solution is complicated All the geometries we consider preserve sixteen supercharges, and this allows loops exhibit perfect screening for the inverse mass gap. Weof suggest an a average tentative over interpretation positions of of these branes results within in the terms soliton distribution. solutions of the supergravity theory which preserve (as usual SO(1) isten the dimensions trivial which group). leadsdistribution We to is identify a each the of distributioncorrelators the of and privileged D3-branes Wilson geometries: in correlator loops. in for each Surprisingly, case we the find a mass gap in the two-point symmetry. The scalar fields inof these the 42-dimensional scalar coset the usual picturevacuum of state, states specified in bytotically a the distribution gauge of theory branes in in ten AdS/CFT. dimensions Given or a an particular asymp- These operators and their dual fields in supergravity transform in the 20 From the five-dimensional perspective,arise through the non-zero deviations background from valueslinear for this order scalars limiting in these geometry thethe background supergravity scalars values theory. with are At regular solutions behavior of near the the free boundary wave-equations for of we will investigate geometriesthe involving operators profiles for the supergravity modes dual to because infinitely many scalar fieldsThe are present involved in paper the isare five-dimensional therefore description. simple concerned from the within point states of the on view massless the of Coulomb supergravity: branch they that will involve only the scalars what predictions the correspondenceloops. makes regarding Green’s functions and Wilson us to reduce theseparate into field five universality equations classes, to identifiedfar according a from to the first-order the asymptotic system. boundary behavior of The geometries naturally geometry will still have a near-horizon region which is asymptotically the 20 preserves SO( considered below correspond to specialthe solutions 10 of the flow equations of [6] in which JHEP07(2000)038 of 0 (2.3) (2.2) (2.1) .The 5 S = 8 gauged -dependence . is the metric M       N 2 K 3 4 5 1 2 SO(6). Choosing ds α α α α α has the form of a / )       to be diagonal: 2 R ˆ s coordinates, and it is . ,          d M 5

3 3 6 S 6 6 6 6 / / β 2 2 2 √ √ √ √ / / / / ,e p p 5 .Therecanbe β − − , 2 5 (see [7] and references therein), 2 K S 7 ,e 1 4 S ds β − 2 ) as a maximal subgroup. The 20 + R ,e 3 , 20 1 20 1 2 M β 201 201 2 √ √ ds √ √ / / 3 ,e 3 / / 1 1 2 SL(2 / 2 β − − 2 − × ) ,e 1 R 2 21 β 2 21 , =∆ 2 √ √ e √ √ 2 / /  ˆ / / s 1 1 USp(8). An important ingredient in consistent trun- d / − − ) for a specified element of the coset, we can form the . The theory depends only on this combination, and the , although a formal proof has not been given. Consistent R T 5 , 6(6) 2 21 S 21 2 E =diag SS √ √ √ √ 000 0001 / / / / SL(6 = 1 1 . The identity element is associated to the round M 1 1 contains SL(6 5 . The warp factor ∆ depends on the − − 5 ∈ S M S          6(6) S = 8 supergravity in five dimensions [9, 10, 11] involves 42 scalars E = N USp(8). The resulting ten-dimensional metric = 8 gauged supergravity in four dimensions is a consistent truncation of          is the metric on the five noncompact coordinates and / 1 2 3 4 5 6 β β β β β β N 2 M 6(6) sum to zero, and we take the following convenient orthonormal parametriza- but not vice versa.          ds E i 2 K β Gauged The group ds ,of symmetric matrix in general form of the mapV was essentially given in [12], in terms of the scalar 27-bein, roughly the local dilation of the volume element of The truncation means that fieldsan of Ansatz the such parent that any theorylifts solution and unambiguously of its to the a truncation equations solution of are of motion related the of by the parent truncated theory. theory parametrizing the coset scalars which we want to consider parametrizes the coset SL(6 SO(6) symmetry acts on it by conjugation. So we may take where and the same hastheory recently been in demonstrated seven for dimensionssupergravity the [8]. in maximal five gauged There dimensions supergravity is istype-IIB on supergravity likewise little a doubt consistent truncation that of maximal ten-dimensional warped product: on the deformed Maximal 2. Supergravity solutions eleven-dimensional supergravity compactified on cation arguments is aformed map metric that on takes any element of the coset to a particular de- tion: arepresentative JHEP07(2000)038 µ a far are are γ =0, (2.9) (2.4) (2.7) (2.8) (2.5) (2.6) a a is the ∞ = γ β 0 abc a + 20 β δχ O → =2sothat 2 µ γ M. ,where =0and tr which controls the i 0 4 1 2 µa ` 20 − δψ signature, is hO = 1). This is the known unstable W, , which enters the gravitino 3 g . 1 W . Asymptotically they must ,  ρ − 1 ab P, 2 b , −−−  . 1 W = − M , µ 2 1 γ 2 , ) dρ 5 i ab dρ dA 2tr is a fixed vector, with − − W ( ∂α a where − 6 g matters because we take ( 2 µ γ 2 12 1 2 ) a log 3 − dx , γ ) − 4 2 a 5 M ρ =1 (  i and X = µ A W 2 a where (tr + 2 D e and is proportional to β  3 g µ in finite or infinite 5 2 R a = = at i g 32 1 4 γ − 2 fields of the theory. It can be shown that sixteen 2 except for the global maximum when all the V 2 µa / − − ∂α ∂W AdS ds →  δψ 2 g = i W = a = 8 lagrangian [11], in + . It is straightforward to verify that the possible β →±∞ L P 5 = ∂α ∂W a i . The Killing spinor conditions are N β  ab dρ dα AdS 5 =1 i X Wδ 2 . The symmetry groups preserved by the privileged flows are also 8 g = µ , there is a privileged flow determined by the condition that ab = a γ W P = 0 are the 48 spin 1 near the boundary of is the radial coordinate of a metric of the form abc µ ρ χ For each There are no extrema of All flows have some There is however an extremum of 1 is canonically normalized. The sign of 1 from the boundary of approach a fixed direction: SO(5) invariant critical point of the theory [9, 13]. where has the form where operator dual to where size of 0. those listed in table 1. transformations as µ exactly all along the flow,one rather than parameter just of asymptotically. freedom This condition to leaves determine just the flow: a quantity In analogy with the results of [6] it is possible to show that It is also straightforward to show that the tensor The relevant part of the supercharges are preserved if and only if JHEP07(2000)038 5 σ (2.10) (2.11) discrete 2pt fnct . are listed in continuum in (2.11) one |  5 ) ~ w 2 σ ) discrete . The analogous | 2 ) gapped 2 1 , w 2 2 σ w w 2 i = w ), depends only on w − − − ~ 1 dy w 2 w − − of the six dimensions ( 2 5. The distribution ` 2 ( ` , 2 2 ` σ 6 =1 ( δ ` ` 4 i n X √ ( δ 2 , σ 5. . √ θ 2 2 w H 2 2 ` ` in 1 1 2 2 1 2 =3 √ − ` 1 π` π` π π n< indicates an object of negative 2 n w>` − ` 5
√ σ 2 3 for 1 dx . − )+ as functions of SO(2) SO(3) SO(2) 4 n 2 − | n σ w 2 2 ~ × × × w σ 4 5 − L dx − 2 and = 4 geometries and their properties. It is helpful y ~ ` − | ( n ) 2 1 θ symmetry N σ ~ 2 w dx / ( 3 ) − 1) SO(5) eqn. (2.11) discrete 2 wσ 2 − w vanish by definition for n 1 , 1) SO(4) dt d 1 σ − ` 1)− SO(3) is not uniformly positive. This is a deep pathology which 2 , − |≤ 2)− SO(4) ` , 1 5 H 5) SO(5) ~ ( , w 1 1 | − σ 1 − − √ 2 1 Z , , , − , 2 − , n 1 1 − , = = 1 σ , − 1 −  , 1 2 , − , 2 1 − H , 1 ` , , 1 a ds 1 1 3 , 1 1 − and compresses it to a line segment by projecting the position of each , , , − 1 π 2 , , 1 1 2 , , , σ (5 (1 = 5 case has such pathological ten-dimensional origins: in five dimensions )= (1 (1 (2 A summary of the privileged 6 3 6 15 15 n w 1 1 1 1 1 ( √ √ √ √ √ 5 -dimensional integral is over a ball of radius σ n and is normalized to 1. The various 3 4 5 1 2 nγ | ~ w | would require infinitely manythat O3-planes, the which again seemsits senseless. naked singularity It is is of curious much the same type as for projection relations obtain between The transverse to the D3-branes. The distribution of branes, brane perpendicularly onto one axis, the result is the distribution leads us to conclude thatin this terms geometry of is anti-D3-branes: unphysical. negative It “charge” cannot in even be interpreted has the form Table 1: listed in table 1.geometry, which in Each each of case them can can be written be in lifted the unambiguously form to a ten-dimensional to note that the distributions tension as well as oppositeobjects Ramond-Ramond in charge to string the theory D3-brane. are The orientifold only planes, such but to make up the table 1. Itspecified is by amusing to note that if one starts with the uniform disk of branes Unlike all the other JHEP07(2000)038 , , in , to + =1 M ;and =3 Q (2.12) (2.13) =2 µ n n n → for for for →− M µ    ` L ` ` L L    limit of a doubly-R- + +log +log +log 2)

µ √ µ = 2 but with µ . Next map the matrix 6 15 6 µ 4 12 = 4 geometry is precisely the µ W, √ µ n µ √ √ Q/ 3 15 6 2 e 3 g ( e e 2 2 n √ √ √ e − e e − − − →− → 1 1 1 = arose in [14] in this context.

µ 4 M σ dρ dA log log log 2 1 2 1 2 1 6 , )= )= )= µ µ µ ( ( ( ∂µ ∂W 2 g = ,A = 1 but again with ,A µ dρ dµ µ ,A 15 n 10 3 √ 2 µ was considered previously in [14] in connection with a zero = 2 can be shown to be precisely the extremal limit of this √ 6 − 4 2 − √ n ) are the same functions as for e e σ − µ 4 1 4 3 ( e 2 1 − A − µ µ − 3 15 2 2 limit, but it is seen as a benign effect of the Kaluza-Klein reduction: µ ) is the integration constant for the first order differential equation. For √ √ 6 2 e e √ − )and 4 4 3 5 e µ Q `/L limit of R-charged black holes whose mass is less than their charge. These ( − − − − W → Q = 5, the same as for )= )= )= It is in principle straightforward to derive all the information in table 1 by the The distribution M µ µ µ n → ( ( ( = 4 geometry can also be obtained as a =4, W W W M appropriate five-dimensional units. Amusingly, the black hole geometry where the mass approaches the charge from above: temperature, zero angular momentum limit oflar a momentum spinning in D3-brane a metric single with planethe angu- perpendicular Kaluza-Klein reduction to of the the branes. spinning braneonly As geometry shown the to in five fields [15, dimensions of 16, involves 17], theR-charged black gauged hole supergravity of multiplet, the and typeetry in discussed corresponding in fact to [18]. it is Indeed the a five-dimensional non-extremal geom- the ten-dimensional geometry has onlysame a sort null of singular naked horizon.in singularity Geometries in [20, with five 21, the dimensions havegeometry 22] been provides in studied the connection ingeometries [19] first with and must clear-cut confinement. also have evidence The an that well-defined role such ten-dimensional in singular the five-dimensional correspondence. It should be noted that the black holes have nakedsolution, timelike and they singularities are like usuallythe the deemed unphysical. negative mass The naked Schwarzschild singularity remains in charged black hole correspondingorthogonal to planes, D3-branes and with that two the equal distribution angular momenta in following strategy. First integrate thefive supersymmetry special conditions (2.8), flows which become for our to obtain where log( n for JHEP07(2000)038 , - i s and (0) (3.2) (3.1) 4 0. In  (2.14) 2 2
O →−∞ ωL 2 1 e ) Ω x Ω A transverse = 1 cases, in (2.1) to ( z> θd 4 i θd 2 n 2 K y 2 hO sin ds sin 4 can be arbitrary , 6 λ where 2 , 5 ) . λ ζ 2 1 λ 2 4 + ζ , ζ λ λ 4 + -wave dilaton. By . 2 2 Ω =2and = s R 2 3
Ω = = ω`/L n 2 θd 3 Ω / 3 3 2 θd 2 / / dz 2 2 θd 2 − − − 2 − ∆ ∆ ∆ 2 3 +cos coordinates. In ten dimensions 2 one has the behavior +cos dx 5 +cos 2 θ, 5 dθ S θ, θ, − 2 2 2 2 2 dθ 2 2 ζ . 2 dθ ( ζ cos 2 AdS 1 = 0. Solutions exist which are exactly ω dx 3 cos cos ζ  2 2 λ ζ ` r 2 2 2 2 2 φ 2 − ` ` r r L 2 λζ  ˆ L ζ is the largest positive entry in the vector 2 1 7 L L − − ζ dx −   =1+ , and only the leading term in small  2 −  2 6 =1+ =1+ 6 2 2 λ ω ` 2 6 λ dr . dr 2 2 λ 2 dr 4 4 4 4 dt ` r L 4 r 4 L = 0 in the near-horizon geometry. If we restored the r L ) ,ζ ,where φ ) − ,ζ ,ζ z − 2 ( 2 2 = 5 can be obtained from the ,and − ` r 2 2 2 2 , then this equation would only hold in the near-horizon ζ 2 µ A 2 µ →− ` ` r r 6 L 2 2 µ n 2 H , in the manner described in [12], and use − e dx dx , ` (1 2 K dx `   is the operator which couples to the = A  2 λ e 2 ds 2 2 =1+ L coordinates (not just asymptotically), and these obey the five- 2 2 2 3 =1+ =1+ L ζr L ··· 5 λ ζr ζr ds 12 6 4 S λ λ λ =4and + = = = 2 n 2 2 2 6, the geometry is conformal to the half of ˆ ˆ ˆ (const.) s s s F metric, √ d d d 5 / 1 → S ( ( ( is available via the AdS/CFT correspondence. =tr ≤ 4 is the energy of a radially infalling dilaton. The ratio ζ O =2: =3: =1: `/L ω dA/dz The properties of the five-dimensional wave equation will be most transparent if n n n .If a wave we mean asymptotically independent of the and The metrics for in the limitcomplicated indicated function in of (3.1).small The absorption cross-section for thethe dilaton metric is is of a the form One can show that far from the boundary of to the brane sotedious, that but the metric it assumes isproduct the possible form form to are (2.10). show as that The follows: the details ten-dimensional are metrics somewhat in their warped where extract the full ten-dimensional metric. Finally, introduce coordinates the dilaton obeys the free wave equation, Here independent of the 3. A two-point function Usually the simplest two-point function to compute in supergravity is a deformed 1 in the harmonic function region, and only in the limit where dimensional laplace equation γ respectively, by replacing JHEP07(2000)038 5. , =1 =1, (3.4) (3.3) n n =4 at some n : it consists 6 for . V √ 2 / ) : a) Vanishes; b) 1 →−∞ . Note that the z 5 ( 4 0 z z A A /L ζ< AdS 4 9 2 ` ) exhibits four different z )+ ( 5 the spectrum is discrete z 6, then , ( V 4 00 √ , / A = 0 reduces to 1 3 2 , φ ) =3 = 3; and the fourth for z ( n n ζ> )= R z 2 ( is on the order / ) .If V z s ( 2. A is determined by the form of 3 8 − 2 →∞ z z e p n> x where ) far from the boundary of · z z = ip ( − s V e 6 for R, = 2 √ p / convention. The potential φ 1 = c) d) a) b) = 2 flow; the third for R ζ>  n ) z −−−− ( V -axis: there is a mass gap! For s = 1 the spectrum is continuous, and it covers the whole positive = 2 the spectrum is also continuous, but it covers only the interval + VV VV 2 z =2,and n n ∂ n −  The various behaviors for )onthe 6 for ∞ = 5, though pathological in ten-dimensional origin, is stable with respect to √ , , and there is a naked timelike singularity there. We have 4 -axis. For / ∗ n s z Setting /L 2 = =1 ` and positive, and the lowest eigenvalue for As usual we work in + behaviors, which are illustrated in figure 1. The first is encountered for the one finds that the five-dimensional wave-equation z fluctuations of a minimally coupledof perceiving scalar. the It pathology would ofcharge be “ghost” directly interesting D3-branes in with to negative five find tension dimensions. some and way negative case Asymptotes to a finite value; c) Increases without bound; d) Decreases without bound. flow; the second for the The spectrum of possible values for ζ general, curvatures are unbounded as Figure 1: of discrete points forcorresponding solutions to of solutions (3.4) which which arewaves almost are normalizable are. normalizable in and/or the For a same continuum sense as plane real ( JHEP07(2000)038 =2 (3.5) (3.9) (3.6) (3.7) (3.8) n (3.10) =2and , . 2 n 2 p 2 , p 4 2 ` 4 ` L L !# − s 2 4 1 ` 1+ =2,therelevant L r n r 2 1 2 1 ) is analytic except at , 1+ s + ( + ! 4 r 2 1 i 2 1 s 2 1 2 4 − ,... . -wave dilaton falls into the ` − 4 (0) L s − 4 , = = 3 1 2 -axis extends over the interval − O a , ) s a 2

1 x , ( ψ 2 4 r and 2 1 and + =1 6 hO 6 1 x j λ + ! · λ ip =0is 2 1 s . The function Π = 2 e 4 = 2 9 ` φ 4

v L p ) u p ψ ) can be determined from solutions to (3.4) 4 ), and intervals in the continuous spectrum π = s s 2 d = 4 arose in the study of euclideanized D3-branes with ( ( (2 s s 1+ 4 4 n 2 =0is 0, using the prescription of [2, 3]. In practice one + 1) for 2 π r where Z where j φ N 2 1 ( → 32 j , 2 , z , and this same replacement is necessary in Wick rotating + 4 )= − =0for ) ` 2 ) i` L 2 1 v 4 φ p u ; (

→ 4 a )= = s ` ). The cut across the real ;1; Π ψ ( 2 z a 4 " m 2 -axis which are included in the spectrum: points in the discrete Π Γ( − where s s / ;2+2 2 1 ) 2 2 π z = 4. To compute the two-point function for − ( m ) is the hypergeometric function. The two-point function is -plane, where again a, a 0 N v 64 ( n s a, ; = ( F c − s a ; F v x · x )=Γ = 4, the relevant solution to · ), and this is indeed the spectrum of (3.4). The discontinuity across the )= ip a, b z ip s ( ( n − ( ∞ − e ψ F 4 , e =2and 4 Π = For The spectrum of (3.4) determines the analyticity properties of the two-point = n An equation equivalent to /L φ 2 2 φ = 4 metrics are related via ` solutions to the wave equation in the complex has poles at where where needs an explicit solutionfor to make much progress, and so far we have results only which approach a constant as The two-point function, cut is related tobranes an from absorption asymptotically flat cross-section infinity where (far an from the D3-branes). correspond to cuts. In principle, Π ( spectrum correspond to poles in Π the points along the function a single large imaginaryn angular momentum [23, 24].to This euclidean is signature. notwith The a discrete (3.10), surprise, “glueball” since but spectrum the VEV’s it computed are appears responsible there for (numerically) tofor the coincides have useful energy a discussions scale, regarding not rather discrete confinement. different spectra S.S.G. in interpretation would similar like in contexts. to this thank M. context: Cvetic the Higgs JHEP07(2000)038 4 O (3.11) (3.12) . The average ` , ) ˆ e m )is2 0 y ~ , are for us side issues, α ( 4 σ O + =0,whichis , y ~ i ( m W σ ) m m y ~ h ( yσ log . , n , 3 3 2 d i i i i 2 2 W W W W Z m 1 dimensions. In all cases the maximum m m 1 m m m m − h h h h 10 n n up to a factor of order unity. In (3.11) we ) ∼ ∼ ∼ ∼ 0 are hypergeometric polynomials. m ) ) ) ) 0 ) = 1. The qualitative features follow from the 2 = 2 it does not even effect the numerical value of α as an angular momentum quantum number. The m m m m `/α m ( m ( ( ( ( n ( 1 j ρ ρ ρ ρ , because the diameter of − 0 n ), and the behavior near 0 S dm ρ can create two gauge bosons, and the two-point func- `/α R `/α 4 =4: =2: =3: =1: Vol ,isalso 2 0 O i , and for , n n n n α V W m h )= so that m ρ ( ρ ,whichis(0 ρ is an arbitrary unit vector in e = 0. We will not go into detail here, but only state that it does not affect the θ The distribution of masses of gauge bosons follows from the distribution of branes It is straightforward to extend the analysis to two-point functions of operators In weakly coupled gauge theory, the behavior of the two-point function is very have normalized support of where ˆ gauge boson mass, insertions. In theindicate conformal that vacuum of the one-loop super-Yang-Millsgives theory, graph, exact the agreement with with results only the strong of gaugeof coupling [25] bosons result. this running agreement The around subsequent from understanding theand the loop, the point role of of view lower of spin non-renormalization fields and theorems on-shell [26]–[30], ambiguities in These are precisely thesee excited any state obvious interpretation energy of levels of a rigid rotator, but we do not wave functions for these values of because none of the non-renormalizationorigin theorems of is expected the to moduli holdpartners away space. are from the protected, The and massesnot this imply of is that individual because interactions gauge they cannot bosons correct are or the BPS one-loop their excitations; graph. super- through but the this formula does tion at zero ’t Hooft coupling can be evaluated from a one-loop graph with two corresponding to partial waves ofwith the dilaton whosequalitative angular behavior momenta of are in planes mass for a gauge boson is 2 the gap. Partial waves withlead angular in momentum general not perpendicular towe to non-separable expect the partial the D3-branes same differential qualitative equations conclusions in to our stand. variables,different. but The operator JHEP07(2000)038 r 2 u π/ (4.1) ,then (3.13) = . Also i 2 θ W , 3, in the s/` 4 m xx ˆ g πL , with correc- tt − h n> 2 3and ˆ g e i 4 are in contrast − m = 1, and perfect , , which is a much ≤ W 2 1. m n n . for )= =3 2 δ `/L u n ( s m , each end of the string = 0 for h n> 1+ = 2 case, one can show 4 5 i  s θ s n W − = 0 for 1 AdS m for θ h ,f geometry controls the small 2 / r 2 δ ) s as in the weakly coupled gauge  5 3and 2 m m ( ∂x ∂u N /` ≤ 0 ∼ AdS  . This is in contrast with the super- α ) ∼ 2 n i dm ρ m ) ( ; also, in the 2 )+ s W . Most of the trajectories do not lie in a ). The total discontinuity has the approx- ρ ( u s/ 5 s N 4 11 ( m ( √ S are suppressed by powers of 2 for f 4 0 .If 2 YM Z 2 g s π/ s 2 h /s 2 s 2 p 2 s i dx = N / i W N C , rather than θ Z ∼ W for . Near the boundary of . The deviations from the Coulomb law become im- m ¯ ≈ q 0 /` h ) q δ m 2 ) s ) exhibits nearly conformal power-law behavior down to V 1 s L ( = 2 and the discrete spectrum for πα 1+ s ( N/r 4 ( i 2 4 ∼h n 4 which contains the brane distribution, or in the orthogonal W 2 YM 2 = 6 g m ¯ q R h q 1. Then the distance between the quark and anti-quark and the p / `/L δ Disc Π − ∼ 5+ = s ) r √ ( uu ¯ q ˆ g q ∼ tt dx , V V g ) C 3. The analysis proceeds most straightforwardly with a radial variable s Z ( 4 = . We will come back to this conundrum in section 5. Because there is no dilaton profile, the ten-dimensional string metric and Einstein r n> n coordinate systems used in (2.14); the second case is can be constrained to lie anywhere on that corrections to Disc Π lower energy since plane, and are difficult toin analyze. the The hyperplane simple of cases are where the string stays either a much lower energy scalethe than physics the is typical radically gaugeto different boson from mass. gauge theory Below expectations, this and low scale very sensitive tions suppressed by powers of theory. Beyond thisscreening point, (with one a caveat sees which a we stronger will come power to law shortly) for for imate form hyperplane. The first case is gravity results: there the conformal limit is recovered for behavior: portant on a length scale the gapped spectrum for with the expectation basedthe on two-point function the Π continuous distribution of branes. In summary, 4. Wilson loops To compute the quark-anti-quarkside, potential we from Wilson follow loops [31, on 32]. the supergravity The asymptotically In the weakly coupled gaugetion theory, threshold each to massive the species discontinuity contributes in a Π pair-produc- potential energy between them are Disc Π One recovers the conformal limit Disc Π metric are the same,D1-branes and will “Wilson” show loops the (moreto same properly the behavior ’t overall as Hooft coefficient those loops) of built built from from fundamental strings, up for such that ˆ JHEP07(2000)038 2  )is =0 π/ u (4.2) m ( u 4 = f 2, then 2 )which θ /r for 1 m π/ . ( δ i i ρ ∼ γ> would even- = 1+ W ¯ 2, then there W q 1 1 i r q θ m m h h V W with m h   γ / r r u δ = 0 plane and enjoy <γ< m θ ∼ geometries with profiles for for 5 ) 2 ∼ u = 0 in supergravity. ( ) δ /r . f 1 θ plane. By assumption, r with 0 AdS 2+ m r ( =0 u δ γ N ∼ ∼ ρ – u ;if θ ¯ 1+ q ) x i 2 YM r q N r g ( V W ∼ r ¯ q 2 YM m q Quark/anti-quark interactions derived ) symmetry breaking associated with ) g h = 4 super-Yang-Mills theory as vacua V u n = 0 case we find that perfect screening ( ( θ f N ∼ 1)-sphere of (2.14). This is the caveat we 12 =1 = 2 perfect= 3 screening perfect= perfect 4 screening screening perfect= perfect 5 screening screening perfect screening “confinement” “confinement” − =2, mr r n n n n n − n n Table 2: from Wilson loops withrelative to two distribution different of orientations branes. e ) , just as we saw for 2 α m ( r /r = 4 corresponds to the Coulomb law), and if and has the behavior (2 + ∼ SO(6), and in ). Namely, if γ rather i r dm ρ / α/ tt ( ` ) W ¯ g q up to a factor of order unity. Interestingly, the infrared =0.The q Z = 0) at sufficiently large R 0 m = V , r h =4 ¯ q N q r ), one can proceed as in [22, appendix A] to determine the . We have put “confinement” in quotations in table 2 because V γ `/α u = ` in each of the ( 2 YM =1caseis1 2 f g u / = m n (note that 2 i ) 3 does exhibit an area law, a physical string at large γ )= πL W ,then − r β ( m before performing the (2 = ¯ q r h / q 2 ` r n> V r is the trajectory of the Wilson loop in the ∼ + = 0, so one should replace ∼ C ). A subtlety arises when rr r ,is 3: in these cases the distri- r ) β i g The weak coupling gauge theory expectation, given a distribution It is straightforward to trans- r ( W + → ¯ q q m results are quoted in tablesets 2. in For the at scaling analysis around h plane for tually find it energetically favorable to creep up toward the it is really a fake: while it is true that a Wilson loop constrained to lie in the than r ten cases we consider,the or general to derive result that if ˆ V As before, is perfect screening ( for some scalars in SL(6 perfect screening. There is a spontaneous SO( the orientation of the string in the ( mentioned in the first section. is cut off around n> bution of branes is at α 5. Discussion We have constructed and studiedfive-dimensional geometries gauged which supergravity as have asymptotically simple descriptions both in and ˆ form to variables where qualitative behavior of is the location ofAssuming the convex branes (in our cases, it is where curvatures become infinite). bounded below, then one obtains an area law power law in the JHEP07(2000)038 perpen- 6 R parallel D3- N in the flat metric on the 0 N /α i 2 YM y g = p i `/ U , so this is the energy scale at 0 `/α space perpendicular to their world- 6 = where supergravity is valid and gives , inside which curvatures are stringy. i 0 R 6 W is substantially reduced. R `/α m h N 13 ambiguity. in the gauge theory: one can construct color 2 YM 2 N g pertaining to stretched strings differ by a factor 2 YM `/L E g i 2 W p as the scale at which conformal invariance is substan- m `/L h N 2 YM g p / i W m h in the ’t Hooft coupling from the conversion appropriate to supergravity probes. In the present = 2 case. There is a “halo”, of thickness can be generalized to a coordinate system = . At this energy scale, one may argue that the gauge theory is no longer 2 N n 2 , energies such as N U U , surrounding the disk of branes in 2 YM `/L 2 i `/L g In this picture, a natural expectation would be that the gap, the discrete spec- A feature that all our geometries share is that curvatures become large close to Before suggesting a possible resolution, let us re-examine the energy scales in- dy = p i c volumes, leaves little doubtor of discrete the spectra Coulomb in branchWilson the interpretation; loops two-point do but not function the seemthere and gapped compatible just the with isn’t perfect gauge a theory screening mass expectations. observed gap In of in particular, size tially lost and thetakes interesting place. dynamics (e.g. In screening, ascales sense observed gaps, in this and [33]: is discretedinate precisely spectra) when the converting discrepancy an in energy normalization into of a energy value of the radial coor- which one would expect deviationstwo-point from function conformality to and become Wilson important.E loop But the calculations identify the much smaller energy trum, and perfect screeningthe will low-energy all weakly be coupledpower washed law out gauge behaviors in we theory the described description,possible at process resolution the to of ends of be matching of our onto sectionsis replaced difficulties. 3 still with and the It 4. the region is of This not energies indeed is a one complete resolution because there P of context, the brane distribution.Polchinski This correspondence raises principle the [34] possibilityonly is that the at an analog work. of For the specificity Horowitz- let us consider nearly conformal predictions at oddsIf with it the is Higgs taken mass scale seriously, in then the the gauge results theory. of [20, 21, 22] regarding confinement from singlet states of lower mass by putting two lightvolved. gauge A bosons typical far gauge apart. boson has mass dicular to the branes.gap This in does an not absorptionbrane seem calculation excitations a without is satisfactory any something resolution because that a can mass be compared to masses of strongly coupled because mostthe of large the degrees of freedom have been integrated out: on the Coulomb branch. The ten-dimensional geometry, composed of Outside this halosuper-Yang-Mills supergravity dynamics applies, at and highexpect it energies; that keeps inside, a track orrunning direct of at from gauge the the lower theory disk strong energies,order to description coupling one a becomes may test practical. D3-brane on the An edge open of the string halo has a mass on the branes in some continuous distribution in the JHEP07(2000)038 ), ~ w ( (5.1) σ ): the sparser ~ w ( σ , is controlled by how λ ). It is possible that if ··· s ( + 4 2 are large, they may change ) 0 2 20 ) , the discrete spectra, and/or 0 2 O Typically one expects such mass 20 ( is excluded on the grounds that λ 3 operator whose VEV characterizes O `/L 2 i ) + j X 2 ] i X branes at our disposal. It seems more i j ( P X N ,X i 14 leads to color-independent mass corrections X branes are allowed to move slightly relative to 2 ) N 0 + tr[ 20 2 = 4 gauge dynamics alone does not seem likely to O ) i . λ is a traceless combination of mass terms for the scalars, N ∂X 0 20 O = tr( ) as specifying not a single distribution of branes, but an limit, the wave-function overlap of energy eigenfunctions with ~ w eff ( σ L N is the dimension two tr 0 2 YM 20 g O , large contribute corrections to the two-point function Π 2 N ) 0 As another possible resolution, we would like to suggest that the gauge theory The double trace form of ( ). If this integration is done first, its effect is to induce extra interaction terms We thank D. Kabat for a useful discussion regarding the use of a similar mechanism in another 20 3 ~ w ( O in the lagrangian.trace With structure. regard Keeping towe only expect color the for indices, lowest the these dimension lagrangian is operators, terms the do schematic not form have a pure where for simplicity weThe keep operator only the scalar fields and work in euclidean signature. which can only be approximated by the ensemble of distributions where the natural to regard one another. One shouldintegral: then rather include taking an integration aintegrate over specific over the the point ensemble region in inσ of moduli the moduli path space space as which the is vacuum, one consistent should with the distribution the screening observed inthis sections 3 scenario. and 4. However, the We emphasize the speculative nature of encompass the variety ofa physical clue behaviors that the thatradical deviations we as from have the the branes seen, expected become and more weak we sparsely coupling take distributed. behavior it become as more the state. The SO(6) singlet operator tr the physics enough to induce the mass gap at AdS/CFT predicts a large dimension for it [2, 3]. The size of the region of stringythe or most Planckian one would curvatures expect is is controllably a small. slight broadening In ofphysics the such might eigen-energy a not delta limit functions. beThe as supergravity featureless solution as specifies the only usual a Coulomb continuous branch distribution analysis of implies. branes, the mass corrections or the interactions due to ( similarly singular supergravity geometriesseems must possible be to regarded argue, aslarge both suspect. in our However it case and in [20, 21, 22], that in an appropriate at least some of( the mass corrections are positive. Also, bubble graphs built using corrections to be negative becausetion they theory, but come because from a second order effect in perturba- through diagrams shown schematically in figure 2. context [35]. the distribution, the larger is densely the branes are packed in the distributions approximating JHEP07(2000)038 , a ...... (1998) 253 + + 2 ) . 2 ab m + 2 i 0 p 20 )( ]. ab hO includes a positive or negative 2 (0) λ 0 hep-th/9802109 m 20 O + Gauge theory correlators from non- 2 Adv. Theor. Math. Phys. 4 p , ( ... (1998) 105 is represented as a dotted line connecting the + hep-th/9711200 ∓ ∓ 15 i 2 0 ) 0 20 20 O B 428 ab hO λ 2 (0) ± limit of superconformal field theories and supergravity m 1 (1998) 231 [ ab + N 2 2 2 (0) p sign is chosen according as m Phys. Lett. ± , . The shaded circle indicates the full dressed propagator, and i = = = ]. X The large- = 2 ab 2 ab . i m i m 0 Anti-de Sitter space and holography 1 X 20 + Self-consistent treatment of leading order color-independent corrections to 2 hO p insertions. The 0 a b 20 are color indices. The operator ( hep-th/9802150 Adv. Theor. Math. Phys. critical string theory [ The research of D.Z.F. was supported in part by the NSF under grant number b O Note added: These results have subsequently appeared in [36]. 4 [1] J. Maldacena, [2] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, [3] E. Witten, mass term for two and References masses of the scalar Acknowledgments We would like to thankand M. particularly Grisaru, J. E. Polchinski Martinec,cations for H. with useful Saleur, K. L. discussions Sfetsos, Susskind, and wewhich E. commentary. have have Witten, learned some In that overlap communi- with he the has present independently work. obtained results Figure 2: PHY-97-22072. The researchFellows, of and S.S.G. also in wasDOE supported part grant by by DE-FGO2-91ER40654. the thepart The NSF Harvard by work under Society funds of grant provided of K.P. by number and the PHY-98-02709, DOE N.W. and under was by grant supported number in DE-FG03-84ER-40168. JHEP07(2000)038 , , ]. B Nucl. ]. ]. , reduction ]. Nucl. Phys. (1998) 005 supergravity supergravity , Phys. Lett. KK ]. supergravity in , 06 =5 =11 hep-th/9902195 . =8 D supergravity in five- D N hep-th/9810206 hep-th/9902170 =8 Charged AdS black holes ]. =8 (1986) 598. N hep-th/9810227 N Renormalization group flows (1999) 024 [ hep-th/9811120 04 truncation in B 272 hep-th/9904017 Consistent nonlinear 7 Gauged (1999) 1405 [ , Gauged S Compact and noncompact gauged su- 2 (1999) 064018 [ (1999) 317 [ J. High Energy Phys. hep-th/9903109 (1999) 003 [ ]. Non-extreme black holes of five dimensional , New vacua of gauged The Coulomb branch of gauge theory from D60 16 Nucl. Phys. 03 B 553 , Quark potentials in the Higgs phase of large- ]. and self-duality in odd dimensions . 4 ]. (1999) 22 [ S (1985) 268. J. High Energy Phys. , Phases of R-charged black holes, spinning branes and AdS/CFT correspondence and symmetry breaking On asymptotic freedom and confinement from type-IIB × Phys. Rev. 7 Non-supersymmetric conformal field theories from stable , The consistency of the Nucl. Phys. B 456 hep-th/9905104 B 154 , AdS c and W.A. Sabra, Adv. Theor. Math. Phys. , (1987) 211. (1985) 460. hep-th/9903214 ]. hep-th/9812035 Embedding AdS black holes in ten and eleven dimensions J. High Energy Phys. hep-th/9905075 , , (1999) 89 [ Phys. Lett. Phys. Lett. B 281 B 259 , , c and S.S. Gubser, cetal., (1999) 96 [ AdS supergravity B 556 (1999) 96 [ supersymmetric Yang-Mills theories =2 hep-th/9805104 of 11d supergravity on Nucl. Phys. N dimensions Nucl. Phys. pergravity theories in five-dimensions 469 [ Phys. from holography supersymmetry and a c-theorem five dimensions anti-de Sitter spaces rotating branes and catastrophic holography strongly coupled gauge theories B 558 N supergravity [8] H. Nastase, D. Vaman and P. van Nieuwenhuizen, [4] J.A. Minahan and N.P. Warner, [9] M. Gunaydin, L.J. Romans and N.P. Warner, [6] D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, [5] I.R. Klebanov and E. Witten, [7] B. de Wit and H. Nicolai, [10] M. Pernici, K. Pilch and P. van Nieuwenhuizen, [11] M. Gunaydin, L.J. Romans, and N.P. Warner, [12] A. Khavaev, K. Pilch and N.P. Warner, [13] J. Distler and F. Zamora, [14] P. Kraus, F. Larsen and S.P. Trivedi, [15] A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, [16] M. Cvetiˇ [18] K. Behrndt, M. Cvetiˇ [17] M. Cvetiˇ [19] A. Kehagias and K. Sfetsos, JHEP07(2000)038 , 05 Nucl. Phys. Phys. D59 (1998) , , , J. High , (1998) 4859 (1998) 7570 gauge theory B 526 80 N ]. Yang-Mills Phys. Rev. . , D57 Positivity constraints =4 ]. Nonperturbative formu- N J. High Energy Phys. Nucl. Phys. . , , ]. Confinement and condensates Phys. Rev. Lett. Phys. Rev. QCD from rotating branes , hep-th/9708005 , hep-th/9906201 ]. N ]. , ]. hep-th/9903132 , talk on joint work with G. Lifschytz at ]. Large- ]. hep-th/9902155 ]. (1997) 41 [ , 3-point functions in field theories 17 hep-th/9612146 N A correspondence principle for black holes and Absorption by branes and Schwinger terms in the (1999) 010 [ B 413 Wilson loops from multicentre and rotating branes, hep-th/9803001 ]. hep-th/9906030 UV/IR relations in AdS dynamics ,[ 07 hep-th/9902074 ]. A non-renormalization theorem for conformal anomalies Thermodynamic stability and phases of general spinning hep-th/9810186 (1997) 6189 [ hep-th/9702076 hep-th/9808162 Macroscopic strings as heavy quarks in large- ]. Phys. Lett. , (1999) 100 [ D55 ]. ]. (1999) 007 [ World volume approach to absorption by nondilatonic branes hep-th/9809022 Asymptotic freedom and confinement from type-0 string theory Wilson loops in large- , Montreal, June 1999. 04 (1997) 231 [ Dilaton-driven confinement (1998) 341 [ B 561 hep-th/9903026 (1999) 065012 [ Gaussian matrices and black holes c and S.S. Gubser, Phys. Rev. J. High Energy Phys. , , B 496 B 444 D59 hep-th/9708042 hep-th/9711035 hep-th/9803002 Energy Phys. without fine tuning in supergravity duals of gauge theories (1999) 026 [ Rev. branes Phys. las for central functions of supersymmetric gauge theories 543 [ on anomalies in supersymmetric gauge theories [ world volume theory Lett. Nucl. Phys. and anti-de Sitter supergravity [ strings (1999) 065011 [ Black Holes II mass gaps and phase structure in gauge theories [20] J.A. Minahan, [21] S.S. Gubser, [22] L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, [23] C. Csaki, Y. Oz, J. Russo and J. Terning, [24] M. Cvetiˇ [25] I.R. Klebanov, [26] D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen, [27] D. Anselmi, J. Erlich, D.Z. Freedman and A.A. Johansen, [28] S.S. Gubser and I.R. Klebanov, [30] A. Petkou and K. Skenderis, [29] P.S. Howe, E. Sokatchev and P.C. West, [31] S.-J. Rey and J. Yee, [32] J. Maldacena, [33] A.W. Peet and J. Polchinski, [34] G.T. Horowitz and J. Polchinski, [35] D. Kabat, [36] A. Brandhuber and K. Sfetsos,