N= 1 Theories, T-Duality, and Ads/CFT Correspondence

JHEP07(1999)005 July 2, 1999 = 1 fixed points in field theory and dis- N N Accepted: June 14, 1999, Received: = 1 superconformal field theory using branes of N ∗ 1/N Expansion, D-branes, Brane Dynamics in Gauge Theories, We construct an theories, T-duality, and AdS/CFT [email protected] [email protected] =1 HYPER VERSION On leave of absence from MIT, Cambridge, MA 02139, USA ∗ School of Natural Sciences, InstitutePrinceton, for NJ Advanced 08540, Study USA OldenE-mail: Lane, Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA E-mail: correspondence type IIA string theory.figuration The with IIA 3-branes construction inand is a related 7-branes. background via The generated T-dualitycompactification to IIB by previously a two studied background IIB by intersecting can Sen con- orientifold. O7-planes be in connection viewed We with discuss as the thescribe a Gimon-Polchinski deformations local a of new piece the supersymmetric ofthe IIA IIB configuration an and description with F-theory we IIB curving find constructions D6-branes. the supergravity and dual Starting de- of from the large Abstract: Anton Kapustin Martin Gremm N cuss the matching of operatorsexhibits and some KK interesting states. new The features.field matching We of theory also non-chiral under discuss primaries which a relevant itthe deformation infrared. flows of to For the a these line fixed of points strongly we coupled find aKeywords: partial supergravity description. Supersymmetry and Duality. JHEP07(1999)005 =4 N = 1 theories are not conformal N field theory and supergravity was N 3-branes, but subsequently more general 1 N limit of conformal field theories on branes and the near N limit 17 = 4 SYM realized on N N 2.1 The IIA2.2 brane configuration The conformal2.3 case: a The field non-conformal theory case analysis3.1 T-duality3.2 The seven-brane3.3 impurity theory A supersymmetric3.4 IIA configuration Comparison with with curving F-theory six-branes 15 5 4 7 11 16 8 4.1 The conformal4.2 case The non-conformal case 22 17 Brane constructions in string theory provide powerfulin tools diverse for dimensions analyzing and field with theories varying amountsand of references supersymmetry [1, see 2]. [3]. Forrelation More a review between recently the the large Maldacena conjecturehorizon geometry [4, of 5, the 6] corresponding added blackwas brane a stated solutions. new for The original conjecture 1. Introduction examples were discovered.configuration One [7, class 8] of and such anotherbackgrounds includes examples [9, theories are 10, on orbifolds 11]. 3-braneswith of in varying the amounts All nontrivial of F-theory of supersymmetry.at these A a third constructions conifold class singularity give of [12, riseat theories 13, all arises to 14, scales, on 15, conformal but 3-branes 16]. flow theories theories These to the a line correspondence of between conformalstudied the fixed in large points some in detail. thetype infrared. For branes IIA For on description all a these which conifold is it related turned to out the to IIB be configuration useful via to T-duality have [13]. a 3. The type IIB description4. The large 8 Contents 1. Introduction2. The IIA construction of the field theory 4 1 JHEP07(1999)005 ) gauge theory N Sp( × = 1 SUSY but breaks ) N N field theory, we know which N = 1 Sp( N 2 = 1 theories discussed in the AdS/CFT literature (see, e.g., [7, 12]) N Although there is no firm field-theoretical prediction for the dimensions of fields The type IIA construction involves D4-branes compactified on a circle as well as In this paper, we study a superconformal In most = 1 theory on the 4-branes with an exactly marginal parameter. We can also = 2 models [2, 17, 18, 11]. One advantage of the IIA description is that the moduli limit of the conformal field theory. The type IIA description, on the other hand, gauge boson on AdSfield is theory the operators superpartner withall of supergravity fields states, the and therefore graviton. we the can dimensions If determine of we the all are R-charges chiralin able primary the of infrared, operators. to for match thetion theory with is vanishing that beta function all thewill fields most be have natural born assump- canonical out dimensions,running by i.e., coupling the that constant, supergravity the analysis. the theoryto correct is guess. In charge finite. the Unfortunately assignment other This less the in case, solid supergravity the the footing analysis infrared theory and in with issuggests depends this a a harder on definite case circumstantial R-charge is assignment. evidence.explanation on It Nonetheless for would a be our it. considerably interesting analysis to find a field theory NS5-branes, D6-branes, and O6-planes.Our construction The is gauge very similar theoryN to lives the on brane the configurationsspace D4-branes. that of give the rise gauge theory tomotions is elliptic of realized the geometrically. 4-branes. The Similarly,masses flat relevant for directions perturbations the correspond matter of fields, to the are fieldThis also theory, realized allows such geometrically us as as to motions of identifyN the a 6-branes. 6-brane configuration that givesidentify rise relevant to perturbations a superconformal oftheories with the running superconformal coupling 4-brane constants. theory There is that one lead particular to perturbation that the R-current whichuniquely is by the field superpartner theorythe considerations. of case. For the There the is stress-energy a theorieswith one we tensor parameter vanishing discuss family can beta of here candidate function, be thispoints. R-currents, and both is fixed Since in in not the the R-charges its theory oftheory deformation the prediction fields which are for flows not the uniquely tohand, dimensions determined, a once there of line is we the no have of field chiral a fixed primary supergravity operators. dual of On the the large other with matter in the fundamental,We bifundamental, also and antisymmetric discuss representations. a specific deformation which preserves conformal invariance. The resultingto theory a has line a of runningwe superconformal gauge give a fixed coupling IIA and points brane flows description in construction as the allows well infrared. us as a toN For IIB both obtain orientifold of the construction. these supergravity Theprovides theories latter solution a that simple is waysuperpotential dual to of to determine the theories the the in gauge large question. group, the matter content, and the JHEP07(1999)005 /N =1 N 3 limit of the field theory along the lines of N charges of all fields in the conformal theory unambiguously. R . We interpret this as the evidence that at higher orders in 1 →∞ N In order to construct the supergravity duals we T-dualize the IIA configuration It should also be possible to find a supergravity description of the infrared limit gives rise to aThe theory moduli that space flows of to the a perturbed line theory of has conformal a Coulomb fixed branch. points A in generic the infrared. theory with a Coulomb branchover has the a moduli low space. energyfeature effective The that gauge theory the coupling low-energy we that effective areThis varies gauge will considering coupling be in does relevant this not when depend paper we on has discuss the the the moduli. special supergravityalong description the of compact these direction. theories. into This operation D7-branes turns and the D6-branesbackground. O7-planes. and Similar the probe O6-planes The theories wereto D4-branes studied supergravity in turn is [19, 20, into described 21,dual in D3-branes 22], to [9, and probing 3-branes their 10, probing relation this 11]. ais Our local IIA related piece of configuration to an turnsconsisting F-theory the out compactification to of Gimon-Polchinski [23, be 24] two model T- which each, intersecting [25]. corresponds O7-planes to with TheIn the four simplest the type IIA coincident such IIB construction 7-branes configuration, constructioncancelled the of on locally Ramond-Ramond the top by (RR) charges the superconformal of constant. of charges the 4-brane Since of 7-branes theory. the are thethe type orientifold supergravity IIB planes, dual description so[9, is the of 10]. a string the perturbative Matching coupling large the orientifold, is to spectrum we determine of the can primary U(1) find operators with the KK modes allows us The matching of non-chirala primaries short exhibits supergravity a multipletwhen new whose interesting field feature: theory counterpart We becomes find short only supersymmetry mixes one-particle and two-particle supergravity states. of the deformed theory. Althoughenergy this theory effective is coupling not along conformal,have it a the has constant Coulomb a dilaton. constant branch, low- backgrounds To so find in the this IIA dual supergravity and we dualbefore, IIB need this will and to is find study straightforward an the onmotions explicit deformations the of map of IIA the between the side, them. 6-branes. sinceanalyze the As the On deformations mentioned deformations correspond the on to The IIB the eight-dimensional side IIB theory the on side situationat the by is the 7-branes studying more has intersection the involved. six-dimensionalimpurity of theory matter theory We orthogonal on localized following can the 7-branes. [26], 7-branes.type and IIA We find deformations. analyze an Among explicit thethat supersymmetric map maps moduli type between to space IIB the athe of deformations type new background there IIB this is IIA of and one brane anto identify configuration NS5-brane. the which IIB involves Thewith configuration curving moduli-independent map that effective D6-branes between coupling. gives in deformations We rise do to also not have the allows a non-conformal complete us supergravity probe theory JHEP07(1999)005 2. π/ 6 R 3 , 2 π/ 6 R = 6 X direction. We put two 6 . Specifically we consider 6 X πR =4. = 2 configurations in [17] by rotating d N 4 = 1 supersymmetric field theory in four dimensions = 1 theory in circle is shown. N 6 N and an NS5-brane and its image at X 6 on a circle with circumference 2 D4-branes wrapping the compact 6 ,πR N X =0 6 6 X = 2 theories in four dimensions [17, 18, 11]. In fact, the configuration we Brane configuration: The vertical dashed lines are the O6-planes, the solid 7 N -planes at These brane configurations are very similar to the configurations that give rise In section 2, we discuss the type IIA construction of the probe theory and list − limit of our field theories and their supergravity duals. We discuss the matching the NS5-branes from the 45supersymmetries, directions giving into an the 89 directions. This breaks half of the study here can be obtained from one of the In order to cancel the totalAn RR example charge, we of place such four a physical D6-branes configuration on is the circle. shown into figure finite 1. configurations with O6 lines are the D6-branes, theNS5-brane. horizontal Only line are half the of D4-branes, the and the point represents the Figure 1: after compactifying 2. The IIA construction of2.1 the The field IIA theory brane configuration A configuration consistingplanes of extending D4-branes in extendingsupercharges. 0123789, in and We obtain 01236, NS5-branes an D6-branes extending and in O6- 012389 preserves four description of this theory,information but to a determine partial theuse description dimension field is of theory possible. all considerations chiral as It operators well. supplies in the enough infraredsome field if theory we results thatT-duality, we the need analysis of in the subsequentIIB 7-brane sections. impurity deformations. Section theory, We and 3 also the contains briefly mapthe the discuss between counterpart the IIA of exotic and an IIA ordinary deformationN deformation that in appears IIB. as In sectionof 4, operators we with analyze the Kaluza-Kleinanalysis large modes in the in non-conformal the case. conformal case and present a partial JHEP07(1999)005 6 × 1), X 2 , ) (2.1) 4 , N 1 ,where , d [29]. The ¾ Sp(2 d ). Following U(4) 2 gauge theory × =( ) 1 2 × q ) N ) u =2case,the SU(4) N N ˜ q. × N ˜ . Therefore, the beta Q u (Sp(2 q ˜ p Sp(2 γ 1 2 ) h × N 1 + ) p N Q ) under Sp(2 q ¯ 2 4 , h 1 , + ¾ ˜ Q , 2 J symmetry to SU(4) 2 5 , and fundamentals from the 4-6 strings. d A =(1 ˜ Q Q p , 1 h Q U(4) and all fundamentals have × u Q Q− 1 γ 1), and ˜ 2 in the antisymmetric representation of each of the J , , 1 ¯ 4 A , ˜ Q ¾ =1 1 , h ,i i = A =(1 W p . The superpotential reads d ), 4 , ) is the invariant antisymmetric tensor of Sp(2 1 2 SU(4) , J 1 ( , × 1 u ¾ J The matter content and the superpotential for a 4-brane probe in this back- Using standard techniques [3], we can determine the matter content and the , both bifundamentals have =( A ˜ q Here gauge groups, twoThe bifundamentals brane configuration, and consequentlyexchanges the the field two theory, Sp admit factors. athe symmetry To determine fundamentals which the we number need and toNote flavor understand representations that of the the classical worldvolume gauge ofbranes. theory the It on NS5-brane was the lies argued 6-branes. within in[28]). the [27] worldvolume that The of the gauge the 6-branes group D6- can on break on the the four NS5-brane 6-branes (see turns also out to be U(4) the two U(4) factorsOne-loop act effects on break the the upper U(4) and lower halfs of the 6-branes respectively. SU(4) with matter fields [30] it is easy to checkcoupling. that this The theory one-loop has beta afactors function line implies vanishes of that and fixed both the points passing symmetry antisymmetricγ through tensors between weak have the the gauge same anomalous dimension matter content of the 6-branestrings theory connecting includes upper a andabout bifundamental lower hypermultiplet the halfs from 6-brane of theory the whenour 6-branes. we present discuss We purposes the will deformations havetheory we of more is only this to the background. need say flavor For group to of know the that probe theory. theground gauge were group worked of out the in 6-brane [27]. The fundamentals transform as position of the D6-branesdistinguish will two play cases an that6-branes important are intersect role of the in interest NS5-brane, ourtwo for or to analysis. the the the analysis We left four need and ingive 6-branes right to this rise are of to paper. split the physically NS5-brane intoof Either (as inequivalent two fixed shown theories. all groups points in of (parametrized The figurezero by 1). former the coupling, configuration These dilaton while two yields expectation choices the aflows value) latter line that to corresponds a passes to line through of a strongly non-conformal coupled gauge fixed theory points. which 2.2 The conformal case: a fieldThe theory theory analysis on the 4-branes turns out to be an Sp(2 superpotential of the field theory on the 4-branes. Unlike the JHEP07(1999)005 Q and (2.3) (2.2) 4 X makes all flavors ˜ Q = 0. Note that requir- 2 . and h q γ = Q , q 1 +2 γ h Q = . It is a simple matter to show γ , +8 g D ˜ . Q ) Q ∼ † ˜ 2 Q N ˜ h Q Nγ ). Part of the bifundamentals are eaten − = 2 superconformal theory with gauge N ∼Q † +4 = 2 case. Moving the 4-branes in 6 5 A N . The effect of these motions on the 4-brane ,β γ N 7 iX A 1) X γ ∼QQ + − + 7 4 = 4 SYM in the infrared. Q X X N γ 2 2( N ∼ ∼ limit. 1 g h β N β direction, we can ignore the NS5-brane. The remaining branes 7 X directions as well as 5 ), one antisymmetric hypermultiplet, and four hypermultiplets in the X N = 4 SYM as expected. and N ). This theory flows to proportional to a unit matrix gives a mass to half of the fundamentals and 4 The moduli space of this theory includes subspaces where it flows to theories These field theory results are reproduced in the brane construction if we identify X N ˜ amounts to separating them from all other branes. The theory on the 4-branes direction. Turning on both bifundamentals corresponds to moving the 4-branes in Q 5 7 massive and breaks the gauge group to SU( Giving an expectation value toexpectation either value zero of corresponds the to bifundamentalsX moving while the keeping 4-branes in the the other positive or negative breaks the gauge group to the diagonal Sp(2 Setting all betacoupling functions constants. to The zero remainingconformal gives fixed coupling two points. constant independent parametrizes Sinceconstraints, constraints a this setting on line line all passes of the through anomalous the super- three dimensions free point to zero satisfies the that the resulting theory flows to an by gauge bosons, andSU( the rest give rise to three chiral superfields in the adjoint of group Sp(2 fundamental. Giving such expectation values to both X the with more supersymmetry.or For example, giving an expectation value to either ing the beta functionsThe to most vanish does natural not assumptionone fix is anomalous moves that along dimensions the unambiguously. the dimensionspergravity fixed of computation line. the in fields This the are last wouldthis unchanged section mean is as supports true that this in the conjecture the theory by large is showing finite. that The su- preserve eight supercharges, and standardand techniques gauge [3] group confirm stated the above matter for content the theory agrees with theNS5-branes in field the theory expectations. If we move the 4-branes off the functions of the gauge coupling and the Yukawa couplings in the superpotential are is then the positions of the D4-branes with the field theory moduli in the following way: JHEP07(1999)005 ), 0) 0) N 2 , , , 0 (2.6) (2.7) (2.4) (2.5) 1 , 2 , ¾ , ,m 1 1 m =( p ). . Actually, the 2 4 . In this case the p. ), , 4 1 1 = diag( , m ˜ 6-brane gauge group QQ 2 . ˜ p , =˜ SU(2) d q M global symmetry under 3 1 γ 3 h , 2 × , m + ¾ 1 + q Q γ ˜ =˜ q SU(4) γ =( ˜ 2 Q SO(4) × 1 2 , q +4 m Q SU(2) u × q ˜ transform as ( ), ˜ Q ˜ 3 p. = 1 M as follows 1 × . p † 1+ p h , 1 2 Nγ ˜ ˜ ˜ 2 ) ∼ + m M M M M , , 3 N 1 ˜ and ˜ Q +4 h − + , 2 p † M ˜ q A ¾ 7 J q γ 2 ∼M Sp(2 ˜ 5 A 1) =( M × Q ,β q − iX 1 1 = A in the bifundamental hypermultiplet of the (1 ) ) while h γ ∼MM + N 1 ˜ N 6 4 , W + M 4 , X X Q Q− 1. 1 γ M − J 4+2( 2 ). These bifundamental expectation values act as mass 1 4 = ˜ A ∼ ∼ m q ˜ , 1 g U(1). After integrating out the massive components of the Q 3 h 1 β ,γ ˜ β h m × , ) under Sp(2 2 0 = 2 =0 , , Q 1 W , transform as a ( γ SU(2) ¾ q directions. These brane motions are parametrized by expectation values = , 5 1 , × A 4 γ 1 and ˜ = diag(0 X =( q ˜ p M As in the conformal case we can analyze the RG flows both in field theory and An analysis along the lines of [30] shows that this theory also has a line of and 6 theory on the intersectionwe of relate the the 6-branes positions and of the the NS5-branes 6-branes [27]. to More precisely, and and ˜ The fundamentals now transform as Demanding that the betaconstraints functions are vanish, we independent, againhowever, leaving find the that us line two with out doesanomalous a of dimensions not the line must pass three of bealone through nonzero. fixed does weak not points. Again coupling, determine the thewill since vanishing In values argue of of at that this the anomalous supergravity least dimensions. beta case, considerationsand one allow functions In find us the of to last the fix section this we ambiguity for large To obtain the configuration shown in figure 1 we have to set superpotential, eq. (2.6), has an accidental SO(4) using the brane picture. From the brane construction it is clear that we flow to which superconformal fixed points. The beta functions are given by to SU(2) bifundamental expectation values break the SU(4) of the two complex scalars, 2.3 The non-conformal case We can deform the backgroundX for the 4-brane theory by moving the 6-branes in the terms in the 4-branetential theory. are The corresponding terms in the field theory superpo- We will be particularly interested in the case fundamentals, the superpotential of the 4-brane theory reads JHEP07(1999)005 , 5 , ˜ 4 Q X -planes − -planes. The (the centers). − 3 is not a Killing , and finally the R ˜ 6 Q ∂/∂X is larger than that of Q , fields get integrated out at a , while the coordinate along the .Since 7 ˜ 6 Q X ,X . Matching the gauge couplings at 5 ˜ )atascalesetby Q and ,X N Q 4 direction. Moving the 4-branes in 3 Q = 2 case by the orientation of the NS5- h X 7 8 N X = 2 case the T-duality maps the two O6 N = 2 supersymmetry on the 4-branes have appeared N is broken to SU( D . In the IIA configuration the orientifold projection ensures ) 7 so that its radius vanishes at two points on is parametrized by N 3 ,X 3 ) gauge group is broken to the diagonal group at a scale set by . The positions of the centers correspond to the positions of the 5 R 6 R directions. The T-dual of the two NS5-branes is a two-center N X ,X 5 , 4 4 = 4 SYM. The analysis in the field theory is a little more involved X orbifold acting on the 7-brane coordinates transverse to the D3-brane. Sp(2 X = 2 theory as in the conformal case if we move the 4-branes off the 2 N × Z ) N N Our first goal is to T-dualize the NS5-branes and the pair of O6 Our configuration differs from the . The diagonal Sp(2 branes. Since thishere. modifies the T-duality considerably we discuss it in some detail Q variety of scales. Assuming thatthe the Sp(2 expectation value of other branes can beimage added in later. the We begin by separating the NS5-brane and its NS5-branes in fundamentals are integrated out at scale that position of the physical NS5-brane and its image are related by a reflection of each of these scaleson we the find bifundamental expectation that values. thewill This low-energy be is effective important a coupling special later does feature on. of not this depend theory that and the four D6-branes tobecome an orientifold D3-branes 7-plane probing and four thisturn D7-branes. into background. a The D4-branes The NS5-brane and its mirror image vector, performing this T-duality is notconfigurations completely that trivial. preserve Similar T-dualities on IIA in the literature [11, 18]. In the In the present case The T-dual of the IIA configuration without NS5-branes was analyzed in [19, 22]. Taub-NUT space. Recall that theacirclefiberedover two-center Taub-NUT space can be thought of as 3. The type IIB description 3.1 T-duality In this section, wethe describe IIA the brane IIB configuration configuration which of is section obtained II by along T-dualizing again yields the same NS5-brane in the positive or negative in this case becausethere the will one-loop be beta thresholdgeneral function effects grounds does one in would not the expect vanish.size the matching of low-energy This of effective the implies coupling the bifundamental to that expectation runningarbitrary depend values gauge (nonzero) on in the coupling. expectation the field values On theory. to However, if we give circle is T-dual to JHEP07(1999)005 - ), ∼ − 0 ) Z (3.3) (3.2) (3.1) (3.4) ± is the , Y,Z b 0 , shrinks to + . 2 6 ) as expected X =0,whichwe 2   z 2 ALE space, also , Y X 1 , 2 − 1 iψ z , )  1 A θ − z  − . ( 2 cos( ± -planes with six common 2 Z exp arctan r − → φ, d +  , 8 1 in the IIA picture map into plane in the IIA description, ) ) for the angular coordinates. 2 θ 2   . 2 5 2 5 ψ , =2 = , Y − −  4 Y 4 ,z − 1 Z R dZ X + X z sin θ, 2 + + i − ,say 2 X + + 3 ± π ,Z ,σ ( Z R = R ) ) dY = 0 the Eguchi-Hanson metric becomes  ψ  ψ 2 ± ∼ a 2 + 9 iψ ) + , the two centers are located at (0 2  . The orientifold-induced projection ( )cos( )sin( dσ θ, ψ dX π θ θ   exp ,R 1 0  −  X, Y, Z sin( sin( Z − =0.For θ 2  1 4 4 R ± a a − a  1 . To make this explicit we can introduce two complex = 0 Eguchi-Hanson metric becomes flat. The identi- R Z + 2 − − 4 4 cos Z a + + = r r / 1 2 R has period 2 + ± √ √ requires that we identify ( 1  C Z 1 8 8 1 R -planes. To summarize, the NS5-brane together with two O6 + ψ 2 π − iφ 2 + = = 4 b ), and acting with both orientifold and orbifold identifications flips  2 2 4  and +2 -periodic coordinate on the circle fiber. The parameter Y b X z and one other coordinate of π 0 +  ψ − exp Z Z . The orientifold projections have two fixed planes, , r 1 1 = z z → ( 2 = . The additional orientifold identification acts on the new coordinates as =8 coordinates. The T-dual orientifold projection should therefore impose a 2 2 ), implies the identification ( ψ , base is parametrized by ds 2 1 → Z 5 is the 4 Z , a 3 z / 4 ) 2 − 2 R σ We will be interested in the limit when the asymptotic radius of the circle fiber, X C ,z Y, 1 , becomes infinitely large, while the T-dual circle parametrized by − z where The identify with two O7 known as Eguchi-Hanson space.the It metric is above useful into to the change Eguchi-Hanson coordinates form: [31] to transform directions. b where for In these coordinatesfication the zero. In this limit the two-center Taub-NUT space becomes an planes become, under T-duality, a pair of intersecting O7 the sign of ( The fixed locus of thisHanson identification is space a which two-dimensional submanifoldNS5-brane has of and the the its Eguchi- topology image back of to a the origin cylinder. of the Next we want to bring the asymptotic radius of the fiber. The reflection of and the coordinates reflections of which corresponds to setting ( similar constraint on themetric location has of the the following centers form of the Taub-NUT. The Taub-NUT an orbifold metric on JHEP07(1999)005 0 → (3.5) ,and 0 SO(8) -plane ,a − × →∞ D7-branes at b U(4) gauge group whole × ,

0 in the type IIA config- 4567 -plane with four coincident > − ,R 7 SU(4) by one-loop effects [29]. Ω ,X 67 × -planes and therefore have 7-brane R − L goes to zero consists of an O7 =0 F 6 5 1) X X − ( , = 10 = 0. In other words, they are wrapped on Ω 4 45 Z X R L = 0. Thus the upper halfs of D6-branes map to = F 1 z 1) Y = 0, while the lower halfs map to − = 0 in IIB. Similarly, the lower halfs of the 6-branes, ( 2 , 2 . Thus the four physical D7-branes must be located on z 2 z 1  Z / 2 = C 0, map to G < -branes. The orientifold group for this configuration is 7 0 =0in ,X 2 z =0 = 0. Under T-duality they become D7-branes wrapping the circle fiber of reflects the coordinates indicated and Ω is the worldsheet parity. 4. It follows that the 7-brane charge is cancelled between the D7-branes and 5 5 − R X D7-branes located at X strings. There are at least two such choices. One gives rise to an SO(8) The splitting of the D6-branes into half-D6-branes discussed in [27] becomes The theory on a 3-brane probe in the Sen model background was analyzed in [22]. Now let us put in D-branes. The four physical D6-branes in IIA are located at To specify the theory on the 7-branes completely we need to make a consistent = 0 = =0and =0. 4 4 -7 0 1 1 charge the orientifold planes, andD4-branes the into IIB D3-branes dilaton extending isconfiguration in constant. in 0123. the Finally, limit To T-duality when summarize, the turns radius the the of T-dual of the IIA with four coincident D7-branes in 01236789, another O7 D7-branes in 01234589 and 3-branesin in 0123. 01234589 as We will 7 refer to the 7-branes extending 7 the Taub-NUT and located at where obvious after T-duality. Indeed,location it of follows the easily upper from half the 6-branes, above formulas that the these planes. Recall that these planes are the O7 z X whole gauge symmetry [32],on and the classically 7-branes the [23, otherThe 24], which yields second is a case broken U(4) We is to will SU(4) related be to mainly interestedSen the in model. Gimon-Polchinski the [25] second Both orientifoldtotal orientifold, of which via of these we T-duality. four orientifolds will orientifoldsgauge refer were and to groups constructed listed as sixteen as here the physical compacta are 7-branes models 3-brane the of probe with parts near each of a one the kind. of total the 7-brane The intersections. group 7-brane The that gauge are group, visible matter to content,with and the the theory superpotential we are discussedration in in with complete all section agreement 6-branes 2.2. onmodel Thus top we [23, of conclude 24]. the that NS5-brane As the isspond in IIA T-dual the to to configu- IIA motions a description of local the piece the flat of 3-branes directions the in of Sen the the field 7-brane theory background. corre- Moving the 3-branes X uration maps to the locus the invariant cylinder of the orientifold projection. Taking the limit we find that thez invariant cylinder develops a neck and becomes a pair of planes choice for the action of the orientifolds on the Chan-Paton factors of the 7-7, 7-7 JHEP07(1999)005 . - 0 ), ¯ 6 2 + iA 6 + -branes 0 -branes. 0 1 A -plane in our gauge group. 2)( 0 − / 7 0) theory in six , =(1 U(4) ¯ -branes and the O7 z 0 × A 7 0) theory on the inter- , . Separating the 3-branes ˜ Q -plane with four coincident 7- − and , and moving the 3-branes off both ˜ Q -branes. The 7 Q 0 , Q . 2 11 ,A 1 on the plane and define A -branes. Focusing now on the 7-branes, we see -branes share corresponds to giving expectation 0 z 0 ) under the (classical) U(4) 4 , 4 strings are localized at the intersection of 7- and 7 0 = 1 SO(8) theory in eight dimensions. The bosonic degrees N are the two components of the SO(8) gauge field living on the 7-branes. strings. The 7-7 0 i -7 A 0 It is instructive to study the deformations of the Sen model and compare these Sen [23, 24] has studied the deformations of the compact model in great detail. In Before we launch into an analysis of the impurity theory we need to discuss the section, which transforms as a ( of freedom in thecomplex eight-dimensional scalar, vector both multiplet in the consist adjoint of of the a gauge vector group. field The and second O7 a in the direction which the 7- and 7 from 7 They yield a single hypermultiplet of the six-dimensional (1 branes gives rise to an values to the antisymmetric tensors configuration breaks half ofin the the supersymmetries vector multiplet. and Withsurviving imposes the constant projections modes projection of on matrices the for fields fields the are Sen a model vector [25, and a 23], complex the scalar in the These fields account for the 7-7 strings and there are similar fields on the 7 plane intersect this two-dimensional planewe in use a a point. complex To affine set coordinate up the impurity theory 3.2 The seven-brane impurity theory In this section,the we 7-branes analyze and theWe compare supersymmetric expect them the vacua with vacuum of fielddirections the configurations the common vacua to impurity to of be the theory translationally the 7- on invariant T-dual in and the 7 IIA six configuration. where off the intersection point alongpectation value either to of one the of O7-planes the bifundamentals corresponds to giving en ex- orientifolds gives an expectation value to both that we can capture thethe physics remaining by two studying directions the transverse dependence to of the 7 the 7-brane fields on to the deformations ofhas the the corresponding advantage that IIAthe all construction. 6-branes deformations or The of theare NS5-branes. IIA the geometric, construction background In others correspond the correspondIIB to IIB to deformations moving picture Wilson is only lines. established,(non-conformal) some we IIA Once of can configuration the the discussed also deformations map in find section between the 2.3. IIA IIB and descriptionthe compact of case the the field second theory on the 7-branes turns out to be a (1 dimensions. Since our IIBresults. configuration is In non-compact, fact, in we our cannotstead case simply it the is use theory an Sen’s on eight-dimensional the theoryhave 7-branes with been is six-dimensional not discussed impurities. even previously six-dimensional, Such in- theories [26, 33]. matter content of the 7-brane theory. A single O7 JHEP07(1999)005 ) ˜ =0 and and M (3.9) (3.7) (3.8) (3.6) , ¯ z z T A flatness M F , ˜ M and puts additional 0 M ) , D [25]. The impurity z 1 ( -plane, both of U(4) while δ − 0 P . − P ¯ 6  ) . In particular, at , can be interpreted as 2 1 ¯ z z + × − 0 A 2 − 6 . Φ= ( P 1 ) T ¯ s D 2 z z . These equations are known Φ × 8 matrices. Imposing the con- ¯ z − 2 PA 0 ( A 1 + × 0 − , − Pg )=  = 0. A very similar theory (without  ¯ , corresponds to the asymptotic (i.e., ∂ z 0 ( ˜ z = )= M = )=Φ † 4 z z ( ˜ ¯ g D M 12 Φ( -branes. transforms in the − 0 ,A † 1 0 ,P − -dependent, field configurations that satisfy the ]and ,  z P ¯ z 4 ) z T 8 matrices that commute with z P 0 MM ,A z −  − × = ( ) A 4 ¯ z T [ z 0 P ( A Φ δ −  P z = =  ¯ † ∂A are constant antisymmetric 8 , parametrizes a deformation of the shape of the 7-branes. )= P s z 8 matrices with certain constraints on the entries. This reflects Φ s − , Φ( ¯ × z Φ  strings is localized at the point ∂A − 0 ¯ z ,andΦ = z 0 F ¯ z Φ z ) positions of the 7-branes in the directions transverse to the O7-plane, while F T, We need to find all, possibly To make contact with the notation in [23, 24] we write all 7-brane fields as transform as adjoints. The background gauge field, = 0. The constant part of the scalar field, Φ →∞ s conditions in ordinarydescribes supersymmetric the field impurity theory theories. on the 7 A similar set of equations Here impurity equations, eq. (3.6),the modulo following gauge ansatz transformations. To this end we make equations are consistent if theof products eq. of (3.6) the are bifundamentals on antisymmetrized the in right-hand the side gauge indices. the gauge group reduces totals U(4). to be The arbitrary orientifold complex projections 8 allow the bifundamen- a flat connection thatz gives rise to a Wilson line around the intersection point at where Φ would transform in the adjoint of SO(8). Orientifolding with O7 straints, eq. (3.7), determines that Φ as Hitchin equations with sources. They are analogous to the z the singular part, Φ Φ constraints on these fields orientifold projections) wastheory described is in given by [26]. the solution The of the moduli equations space of the impurity The 7-brane theory alsodescribes contains a the complex transverse scalar, fluctuations Φ, of in the the 7-branes. adjoint of The SO(8) bifundamental that ( Orientifolding also breaks the gaugetinuous group SO(8)-valued functions from satisfying SO(8) down to the group of all con- the origin of the fields in the impurity theory. Without the O7 antisymmetric 8 where from the 7-7 JHEP07(1999)005 (3.12) .The (3.14) (3.13) (3.10) (3.11) ˜ , M      4 4 0 0 , for m i im , ) 3 m 3 4 ), and the right- 0 0 φ z m ( im δ 4 . 4 ) ∼ →− 0 0 2 im m ) 2 − ,φ 1 /z ,φ 3 3 replaced by ˜ 3 (1 ,φ φ 0 0 i ¯ 2 im ∂ m m − , =0. ,φ 1 0      . , →−  φ 2 2 ) | 1 i 2 = 0 , , parametrize the transverse position φ M ˜ Φ 2 ( 2 m , ), the singular fields above have the 1 , 1 1 z 1 0 ( φ =0 −| M δ = diag(  2 =Φ  | † 0 13 i 2 Φ = ,M m , Φ 0      Φ and Φ vanish. In that case eq. (3.6) reduces to M ∼| = diag(Φ 2  ,φ i 2 ¯ z s t in the gauge connection has the same structure as 0 0 , im  Φ A m 0 φ − T      2 φ 1 0 φ 1 0 − 0 0 im − m  1 − φ = 00 2 2 0 − 0 0 m im Φ 2 00 0 is replaced by 1 φ 1 i 0 0 1 . The matrix φ m 00 0 0 00 i im φ ˜ m           i = . The residue of Φ is given by = m 1 ˜ 1 ∼ M Φ M i φ The most generic expectation value of the bifundamentals for which the impurity The impurity equations, eq. (3.6), become inhomogeneous once we turn on an The moduli space of the impurity equations, eq. (3.6), has several branches with and , except that s and an expectation value of the same form, but with the condition Φ of two pairs of 7-branes.section. We For discuss the the remainder corresponding of this IIA section deformation we in set the Φ next with where right form to satisfy thevalues. impurity equations with nonzero bifundamental expectation equations have solutions reads hand side of eq. (3.6) is proportional to expectation value for the bifundamental fields. Since which is solved by rather different physics. The simplest situationvalues arises and if all all bifundamental singular expectation parts of As in ref. [23], the two complex parameters, impurity equations determine theM expectation values of the other fields in terms of JHEP07(1999)005 , 2 7- | 1 0 7 =0, m T ∼| U(4) are equal direction. 2 × i 5 completely. t , 7 4 s m = ,andallother X .Sincethe7- 4 4 1 -plane for small 0 t m with the motion of and Φ =˜ 1 3 T U(4) gauge symmetry. m m × -branes together. -branes contain the same 0 =˜ 0 =0,and 2 -branes together in the IIB 0 s we find U(1) m the 7-branes asymptote to the i = z m U(2). Note that this deformation 1 = 0 also corresponds to a simple × s m U(4) gauge symmetry on the 6-branes × position of the 6-branes in the IIA descrip- 6=0,Φ 14 6 T X )=0,the7-braneshavethesameshapeasinthe z direction. The classical gauge group on the 6-branes 6 , which determine the matrices -branes into a single smooth 7-brane that interpolates i 0 X m map into the , and for generic values of T and ˜ D -branes. This result agrees with the F-theory analysis in [24], 0 i U(4) gauge group to a diagonal subgroup. If all m . This is in complete agreement with the impurity analysis. Note 4 × , the right-hand side of the first impurity equation vanishes and . This bifundamental expectation value breaks the U(4) i 2 | m 1 m =˜ U(2) as expected from the IIB analysis. This brane motion also requires i = 0. We find one such solution if we set × m ∼−| s 4 t It is now a simple matter to identify these two singular solutions with the cor- There are also solutions of the impurity equations with nonzero gauge connection The second singular solution with It is straightforward to discuss more general choices for the bifundamental ex- Before discussing this general solution, we will focus on two special cases. If ) describes the shape of the 7-branes. For large 6= 0 corresponds to moving the 6-branes off the NS5-brane in the z = s . Thus we conclude that turning on this bifundamental expectation value deforms 3 information. Therefore it isΦ( sufficient to consider onlyO7-plane the as 7-brane theory. in The the field unperturbed case, while they approach the O7 components of the bifundamentals vanish.second For equation this in choice the eq. right-hand (3.6) side vanishes, of the which implies Φ The four parameters in this subgroup is U(4) t Φ case without any bifundamental expectation values. responding deformations in the IIA construction. The first solution with brane gauge group to a diagonallyis embedded purely U(2) non-geometric. Since Φ( and Φ between the 7- and 7 If none of the 6-branes coincide, the U(4) is broken to U(1) two pairs of 6-branes in the that a deformation that corresponds to fusing 7 and 7 only the residue ofbreaks Φ the is U(4) turned on. This expectation value of the bifundamentals description maps into a simplereconnecting brane the motion upper in and the lower IIA halfs construction, of which the involves 6-branes. brane motion in the IIA description. We identify turning on where this behavior was interpreted as fusing the 7- and 7 brane group is broken to7-branes a and diagonal the subgroup, corresponding the impurity impurity theory, theory eq. on (3.6), on the the 7 z pairs of intersecting 7- and 7 is U(2) that we reconnect the6-brane upper group and is lower a halfs diagonal of subgroup the of 6-branes, the original so U(4) that the resulting This is in perfect agreement with the analysis ofpectation the values. 7-brane impurity The theory. complex bifundamental expectation numbers, values are parametrized by eight we set JHEP07(1999)005 . The dot repre- 6 Type IIA configu- 4,5 7 sents the NS5-brane, theline thick corresponds to6-branes, four the half thinsponds to two line half 6-branes corre- and the curving linebrane. is another 6- Figure 2: ration for nonzerovalue expectation of the . 7 2 = positions. Thus we find complete X Z 2 5 / , z − 2 4 4 C X -branes 0 X 15 direction, it is 6 X -field produced by the NS5-brane stabilizes the curved -plane, and one pair of 0 correspond to the H s 4 = 0, i.e., a parabola in the / 2 c − 7 cX The deformations we discussed so far are rather From this picture one can see that turning on the constant complex scalar on the The effect of this deformation on the probe theory is what we expect from the − 2 4 agreement between the branesponding motions to in singular the fields IIA in description the and impurity3.3 the theory. moduli A corre- supersymmetric IIA configuration with curving six-branes straightforward to find thevolume of equation the for corresponding theX 6-brane. world- The result is are coincident with the O7 const. Starting fromthe this coordinate expression transformations that we took canTaub-NUT us space from reverse the to the flat coordinates on This provides an expressionthe for 7-brane the in world the volume Taub-NUT7-branes of coordinates. wrap Since the thefiber fiber T-dualizes of to the the Taub-NUT and compact the plane. Figure 2 showsT-dual the IIA to configuration the which following is IIB situation: All 7 complicated in the IIB picture andple correspond brane to sim- motions in thesimple IIA brane description. motions In incounted fact, the all for. IIA description However,motion there are in ac- is IIB, a namely veryimpurity simple the equations brane constant given solution inhave a of eq. counterpart the (3.11), in the thatdeformation IIA corresponds should description. to moving Since pairs this off of the 7-branes orientifold, wedescribing can the find position an ofthe explicit these coordinates equation branes. in eq. In (3.4) terms this of equation reads tion and the entries in Φ 7-branes is displaced from the O7-plane. 7-brane corresponds to fusingthem two off upper the halfs NS5-brane as ofdeformation shown the in preserves 6-branes the all figure. together supersymmetries. and OnIIA the moving This side. IIB is side Presumably it somewhatworldvolume the is harder of obvious to that the this see D6-brane. on the IIB picture. There we movesection two point 7-branes away of from the the orientifoldfrom 3-branes planes. sitting 7-3 at This strings. the gives inter- In a mass the to IIA half picture of the the deformation fundamentals accomplishes the same. In the JHEP07(1999)005 4 D there -plane 0 10 circle and ,X 6 6 X ,X 5 ,X direction by giving 4 7 X X SU(2) [23, 24]. Moving all × in the probe theory an expectation Q -plane and transverse to the O7-plane SO(8) gauge symmetry from the eight 0 × singularities. These fluxes give a mass 4 16 SU(4) symmetry and no tensionless strings. D × SO(8) gauge symmetry and contain tensionless × direction. All this agrees with the expectations from 7 singularities. In order to turn off the NS flux, we move 4 X D In the IIB description moving a pair of 7-branes away from the O7-plane breaks an expectation value to one4-branes of off the the bifundamentals NS5-brane (seewith and section the 2.2). towards curving the This 6-brane. intersection moves the of the lowerthe half-6-branes 7-brane gauge group from SU(4) down to SU(2) four 7-branes together breaks SU(4)gauge down group to on Sp(4). acurving This single branes implies curving that it 6-brane the should shouldfrom unbroken be be the enhanced IIA SU(2), to description. while Sp(4). for It two is coincident not at all3.4 clear Comparison how with to see F-theory this Sen argued [23, 24]F-theory that compactification the with certain T-dualcandidate fluxes version for through of such collapsed the an 2-cycles. GP F-theory The compactification model naive would [25] be is a related pair to of an intersecting the RR fluxFrom parametrizes the the IIB location pointat of of the view, the intersection they of NS5-branes are the on both the part M-theory of a circle. massless hypermultiplet living by giving an expectation valuethe to IIA one description. of the We can bifundamentals. move the This 4-branes is in also the reflected negative in the NS5-brane and itsO6-planes. image This configuration as has well anupper as SO(8) and all eight D6-branes lowerNS5-brane to halfs coincident of coincide with the with its 6-branes,corresponds one to as image of moving well [34]. the the as NS5-brane tensionless In off the strings addition orientifold from to in the the the hypermultiplet that F-theory. is now a tensorthe multiplet two whose NS5-branes scalar in expectation the value corresponds to separating corresponds to giving the bifundamental field strings, while the GP model has SU(4) singularities. However, thisit cannot be would directly give related rise to to the an GP SO(8) orientifold, since IIB picture moving the 3-branes along the O7 to 3-branes wrapping thisstrings. cycle, These thereby fluxes preventingmoduli are the in not our appearance IIA quantized ofconventionally description [23, identified tensionless that 24], with correspond the so to position we turning of should them the be off. NS5-branes able The on to NS the identify flux is value [22]. Thusthe it orientifolds is towards possible the to intersection move of the the 3-branes pair away of from 7-branes the with intersection the of O7 The difference is due2-cycle to at NS (and the possibly intersection RR) of 2-form the fluxes through two the collapsed JHEP07(1999)005 . , is 5 w × → X S 5 i =1 (4.1) (4.2) S z × × can be 5 5 N 5 ,and 2 ˜ S z , reflects the
parametrize , 3 3 1 2 are nonnega- i , is the angular z 1 [23, 24]. Thus i φ 5 dφ l φ ) ˜ S τ , associated with 2 i θ ( m 2 commutes with these 5 ,where 2 S +cos l 2 2 , +2 dφ 2 | ) 1 is the only thing which distin- SU(4) point, the Sen model is 2 l θ . The metric on 2 . The AdS masses of these states dw ( , × 5 | 5 1 2 ˜ S +2 S ]. The three angles φ + . Explicitly, the metric on | 3 3 2 ,π . As explained in the previous section, | are subject to the identifications 5 2 +sin m [0 | 2 S 2 2 dz | ∈ ,z + dθ 1 17 , is large. In this section, we will show how | 2 z + , 2 1 ) 2 N θ 1 | m . 1 θ | , is large there is a dual description of 2 ( 2 YM SU(4) gauge symmetry on the 6-branes is T-dual 2 dz g N + m | × | 1 = ], and m 2 π and +cos | 2 planes respectively. The periodicity of . In terms of the angular momenta, 2 1 1 ds , limit and argue that supergravity suggests a definite R- = m [0 w dφ is an orientifold of ) N k ,... 1 ∈ , the dilaton mode of the 1 θ X a a , and the variables ( limit . Since this periodicity of 3 and h 2 2 , φ 9 , the eigenvalues of the scalar Laplacian on the former can be 1 2 , 5 =0 z N ], 1 iX S z k +sin ,π subgroup of the SO(6) isometry group of + 2 1 = 4 case, the supergravity states with lowest mass squared come from [0 8 3 ,wehave i dθ X from φ ∈ N is an Einstein manifold (or orbifold) [4]. This dual description is valid 5 = 2 = , ˜ S 1 5 ˜ X 2 S φ w ds Let us denote the orientifolded five-sphere by In the +4),where .AU(1) i k z ( the KK reduction of identifications. It is convenient to take the generators that rotate − where identifications on where the angles tive integers. The orientifoldkeeping projection modes on with the even bulk supergravity states amounts to rotations in the Similar theories were analyzed in [9, 10, 11]. obtained from those on the latter. The eigenvalue of the scalar Laplacian on the AdS/CFT correspondence worksSU(4) for flavor symmetry the discussed conformal in section gaugeis 2.2, theory and finite. with provide evidence SU(4) thatsection We this 2.3 theory will in also the provide large a partial analysis of the non-conformal theory of charge assignment for all the fields in4.1 the infrared. The conformal case In the conformal case, separately as a basis in the Lie algebra of U(1) where when the t’Hooft gauge coupling, written as k When the number of D3-branes, 4. The large guishes part of the IIA configuration with SU(4) to a local piecea of perturbative the type Sen IIBthe orientifold model. near-horizon with geometry At constant of string the the coupling, SU(4) 3-branes is obtained by orientifolding AdS superconformal theory on the D3-branes in terms of a supergravity on AdS JHEP07(1999)005 × 3 and L 2 1 1 ,and 2and .The which L − − − 3 F charge, − =0the U(1) )inthe 4). We + ,w 0 should 2 2 1) 1 n − Q J z , − 2 2 ≤ 0 A , has R-charge 3 Ω( 2 charge z m + U(1) 3 . We will denote R Q R acts on 1 = )+Tr( is the U(1) 1 R 00 200 020 01 10 1 2 Charge assignments for 0and J U(1) γ 1 m U(1) 1). It follows that the 0), and (0 ≥ × A , − under these three U(1)’s 4 L 2 , SU(2) , 2 , ˜ 1 p 1 w with respect to this group. , × ˜ ,m ˜ Q q, A Q q,p 1 1 L Table 1: the matter fields. ) , the difference by U(1) m 2 2 charge, 2 0), (0 and for this tower of KK modes. The , , 1 , 1 0 k 1 , . =0, a a ,z h is given above. According to [6], the down to U(1) 2 2 l , is generated by z R k 2 while SU(2) = Z 18 ) is reducible and contains a singlet. 2), respectively. The supercharge 2 1 is the U(1) l , z × N 0 1 2 SU(2) , Z m charges by U(1) × L R ) transform as ( − acts on and (0 , N ,Q , which implies ∆ = 4) [35], where 2 + 0) ). Here 2 , 3 m Q − . The charges of 2 1 , and U(1) m k 2 ( charge. The normalization is chosen so that (0 L k , breaks SU(2) 3 4) = − is a positive integer [6]. We will see below that only for 2 2 . The field theory superpoten- 0) = n − γ , where U(1) ˜ is the R-charge which is part of the superconformal algebra. The R- Q m 0 as U(1) 2 , , N R Since ∆ = 1, this is a free field. For ∆ = 2 there are three chiral m Q N +2 , 1 2 1 . Orientifolding by the first generator breaks SO(6) down to SU(2) U(1) 2 ,where m is the U(1) z ,A n 1 1 × 3 . The results are summarized in the table 1. z = 1. This state corresponds to a chiral primary Tr( ,where ˜ + p A R R m 3)(2 R k k / With these charge assignments in hand it is now We discussed the identification of geometric mo- The simplest way to identify chiral primaries is to find all states for which ∆ = The antisymmetric representation of Sp( 2) = 1 ˜ q, p, / 2 a simple matter tothe match the chiral bulk primary KK operatorssupergravity modes spectrum in and contains the a singleton field chiral theory. primary with Let U(1) us give some examples. The R-charge which is inis the (1 same superconformal multiplet as the stress-energy tensor tial then fixesq, the R-charges of the fundamentals refer to U(1) 1. It follows that any KK mode with SU(2) Orientifolding by γ primary states with geometric U(1) charges (4 couple to a chiral primary operator in the boundary field theory. the sum of the U(1) ∆= are given by (2 field theory. survives the second orientifolding has U(1) charges (1 (3 current is a certain linearcombination combination we of the first three need U(1)projection. to currents. determine The To find which orientifold this group, supercharges linear survive the orientifold and 2 surviving supercharges ( tions of 3-branes with flatfield directions theory in in the 3-brane theThis previous allows us section to (see determinefields also the U(1) [22]). charges of the are given by decomposition of the other∆= supergravity fields yield towers of KK states for which AdS mass of aoperator KK by state ∆(∆ is related to the dimension of the corresponding boundary KK states couple to chiralto primary the operators. KK Therefore modes we from will the restrict decomposition our analysis of JHEP07(1999)005 . 0 7 6 + = = ) ¯ 6 2 2 2 2 | | J (4.3) down + w T w , | | 0) and 3 7 ˜ k , Q , S 1 0 + + k , 2 A 2 . It follows | | .Thechiral ˜ 7 1 2 Q 2 z z =1,itisodd ) | | which are odd 2 2 2) are complex J , 7 2 m 2 ], Tr[ This mode comes A 2 − J , Q 1 ,... +Tr( 2 J 2 1 , defined by ) -plane. 1 A 3 0 , which transform in the 1 J p S and one complex state in 1 J =1 2 . These modes couple to op- 2) and (0 T 3 A 6 k quantum numbers (0 − pJ S Q , 3 2 + , -branes wrapped on , 0 1 and k a J charge and does not correspond to a q Y . This projection breaks SO(8) T 1 k ,andTr( ,where 2 R 1 Q a γ N . They correspond to the KK states with 3. The state with ∆ = 2 transforms in qJ 2 J 3 , ˜ ) k A 19 6) respectively. Q 2 X 2 charges (0 J Q − U(1) J 2 , 3 = T 0 A × ˜ a , Q global symmetry group of the probe theory. The R A 0 7 couple to similar operators in the field theory that ,Tr +Tr( , while even states give adjoints of SU(4) 2 7 + 1 in the boundary field theory. For simplicity we 3 3 J ) -plane with four D7 SU(2) 1 k 0 S Q J SU(4) 1 gauge group on the 7-brane. Since it has 2), and (0 1 × J . 7 × 3 and is, therefore, a chiral primary. This state starts out in A − L 0 T 0) mode has no U(1) 7 , / 7 , 4 Q of SU(4) 0 , , ¯ 6 -th vector spherical harmonic on k + 2) state yields one complex state in . These modes couple to operators that are charged under the SU(4) 2), (0 6 3 − ], and Tr( S 1 , − , 2 J and decomposes into modes with U(1) , 0 ˜ is the Q , . As explained in [23, 24], states in the adjoint of SO(8) 0 yield 2 2). The (0 7 ) k a J 1 3 2 ∓ Y of the 7-brane group respectively. γ , , A 1 2 ¯ 2 6 The field theory also contains operators that carry charges under the 7-brane Reference [10] contains a detailed discussion of the 7-brane states and their multi- The KK reduction of the theory on an O7-plane with four coincident 7-branes J ± , T ˜ gauge groups. It was pointed outcoming in from [10] that the these KK operatorsO7-plane couple reduction to with of the AdS four the modes coincident 7-brane D7-branes fields. wrapping Our an configuration includes an These KK states correspond to operators conjugates of each other,first. so It has it R-charge is 4 the sufficient adjoint of to the consider SO(8) only one of them, e.g., the (0 that the (0 chiral primary. The states with U(1) primary operators with ∆ = 3 are Tr[ const., and similarly anconst. O7 The twofrom 3-spheres the first intersect over a circle. We can focus on the KK modes Q modes living on the other the ( erators of dimension ∆will only = consider operators with ∆ = 2 and where under plet structure. The lowest component of the multiplet is a real field in the ( under the additional orientifold projection was discussed in [10].ours. In The that simplest case way there toof compute were [10] the twice and KK as impose spectrum many the in supersymmetries our additional as case projection is in from to the use O7 the results transform under SU(4) representation of SU(2) subgroup of the SU(4) identify them with Tr charges (4 to SU(4) from KK reduction of the components of the 7-brane gauge field along the JHEP07(1999)005 - 0 × 2) L − , .The 4 ¯ 6 , charge. 1 + 6 and ∆ be the ,thisKKmode R R 2) / . representation of SU(2) p 2 +2 [10]. It is straightforward 2 J k ) + 2 contain no null states and 2 k | A 4) and their complex conjugates. , 2 R k | − , pJ 2) 2 / , (3 0) mode do not couple to chiral primary and , 2 q > 2) and their complex conjugates, as well as 20 1 , − , qA . The KK reduction of these states is straight- 4 5 representation and decomposes into even modes 0) and (0 , , S 0 2 ) , 4 , 2 4) mode couples to a chiral primary in the − 2) and (0 , 2 − , , charges (0 0 embedded in , 3 1 (chiral and anti-chiral representations), +2. S | | 2) mode and the odd (0 R R and couple to operators of dimension | | = 1 superconformal algebra [36]. For our purposes it is sufficient to − N , 2) 2) = 0 (the trivial representation), 0 / / N and ∆ may occur; the allowed possibilities are , charges (0 .Theodd(0 (3 R q 3 U(1) R 1 ≥ J × ˜ Q R 2 In the above analysis we have focused on chiral primaries. It is also interesting To answer this question we need to recall some facts about unitary representa- Representations of type (iii) with ∆ Finally, there are also states living on the intersection of the 7-branes and 7 Other scalars on AdS come from the decomposition of the complex scalar field The ∆ = 3 mode is in the ( pJ (i) ∆ = (ii) ∆ = (3 (iii) ∆ mode couples to a chiralas primary operator in the adjoint of SU(4) which we identify operators, because the R-charge does not match the dimension. The even (0 odd modes with U(1) The even (0 corresponding operators are given by consider multiplets whose primary states have zero spin. Let the forward, and we will not discuss it. to ask whether non-chiral statesof match the between non-chiral field scalars theoryof and we supergravity. complex have Some seen, scalars namely livingand the therefore on ones match the coming automatically. On from 7-branes,come the the from are other reduction the descendants hand, KK the ofmay non-chiral reduction ask the scalars whether of which chiral the the primaries superconformal gauge multiplet field they on live the intions is 7-branes of long are the or primary. short. One R-charge and the dimensionvalues of of the primary. Unitarity puts restrictions on which therefore are termed longnull multiplets. states at Chiral level one, and i.e., anti-chiral their representations primaries are contain annihilated by half of the supercharges. SU(2) branes which is an Since the R-charge anddoes the not dimension couple do tomodes a not of chiral satisfy the primary complex ∆ operator. scalar = field. (3 The same is true for the higher KK on the 7-branes. These KK modes are in the ( with U(1) to decompose and projectfield. these The ∆ modes = as 3 case we is did especially for simple, the since this KK mode modes carries of only U(1) the vector JHEP07(1999)005 . 0) N , 2 , 2 .For YM g charges ), i.e. it ∞ 3 pp .Evaluat- = p )(˜ † ˜ q ˜ N q Q q 2 and large h . We claim that it − 7 N p 2 J 2 . Its field theory counter- A 7 . An even more interesting † , where the flavor indices are p N 1 p † 2 h 2 YM g − pA 2 † . of SU(4) q h + 2 and therefore are in short multi- 1 N ¯ | 6 + J † 1 R + q . On the supergravity side this means | ˜ 6 qA Q 2) . The corresponding current is simply the 0) mode with ∆ = 2 we have found above 1 2 7 / , 21 h of order 1 the dimensions are also canonical, 0 pJ →∞ , 1 N h N and 1and q ∆=2. 1 , charges of these operators match those of the (0 J  † 1 3 ) imply that all chiral fields have canonical dimensions in 3 =0 A 1 2 YM m R g 2 qJ 2 − h is small. The supergravity analysis indicates that the operators 2 0) mode transforms in + , in the adjoint of SU(4) m N q 2 † 1 , 2) one-particle state is in a short multiplet only for J +2 2 YM pp − . It follows that this field theory operator lives in a long multiplet for ˜ 1 g Q , † − 0 N m descendant of this operator using the classical equations of motion, one p , q 1 † 2 h q the multiplet absorbs another short multiplet made of two-particle states ¯ 3)(2 , but is “close” to being in a short multiplet in the sense that its dimension D flavor current. The matching of non-chiral primaries with ∆ = 3 is a bit more 2) and ∆ = 3. This mode lives in the adjoint of SU(4) / N 7 N − , This concludes our analysis of the AdS/CFT correspondence for the Sen model. One can check that all non-chiral primaries coming from the reduction of the 0 , one needs to use the classical equations of motion. The manipulations one has to =(1 , 2 ¯ is in fact theoperator lowest component of a linear multiplet. It couples to a field theory mode. To show thatD these operators live in short multiplets, i.e. are annihilated by plets of type (iii). In particular, the (0 corresponds to the field theory operator antisymmetrized. The U(1) parts are (0 factorizes into a producting of at two large gauge-invariant operators and is therefore sublead- and becomes long. There is completefield agreement theory and between the the scalarcorrespondence Kaluza-Klein [4, spectrum states 5, on 6]. of AdS The primaryR as charge required assignments operators by in table the in 1 AdS/CFT together the with the formula it appears likely that the theory is finite for all the infrared. This isfunction, the but most as natural assumption weof for pointed this. a out theory The in with supergravity the vanishing computation beta introduction is there only is valid no for field large theory proof ing the However, given that for These representations are calledthe short. inequality are Representations also of short; the typeof null (iii) states a occur which at short saturate levelcorresponds two. multiplet to A is well-known the example a case linear multiplet which contains a conserved current. It gauge field on the 7-branes satisfy ∆ = (3 involved. The (0 situation arises when one tries to match the non-chiral primary with U(1) SU(4) finds that it does not vanish. Instead, the descendant has the form ( finite finite go through are very similar touse those of in the [37], classical and equations areregime of subject where motion to is the presumably same justified caveats. in The the weakly coupled in question belong to short multiplets even for large that the (0 approaches the unitarity bound as JHEP07(1999)005 2. / group 0 7 SU(4) × . Indeed, in sec- 7 τ 22 SU(2). To summarize, the deformation of the = 1 theory which flows to a line of conformal × N . N To find a supergravity dual for this non-conformal theory, we need to repeat the This is actually theshow lowest that dimension this for assignment thesupergravity of fundamental we dimensions, allowed would by or have unitarity. equivalentlywith to of the To show singular R-charges, flat that agrees connection thethe with switched on, KK chiral reproduces primaries reduction the that of expectedhow involve dimensions the to the of 7-brane analyze fundamentals. theory thewe Unfortunately excitations we cannot of do check the notprediction that know impurity for our theory the infrared around solution dimensionsthe nontrivial is of vacua, answer all consistent. fields. so by It directlytheory would Nevertheless, analyzing at be large we the interesting to perturbative get confirm expansion a of definite the non-conformal 7-brane background thatties leads of to the the theorysector. non-conformal on the theory 7-branes, changes but the it proper- appears notanalysis to above with change the theunchanged, new the closed spectrum 7-brane string of the background. bulkcontent modes of Since should the be the the conformal closed same andflavors as string before. the and sector non-conformal their The theory matter is coupling differsame to only spectrum the in of bifundamentals. the operators numberit Therefore that of appears both do that theories not the transform haveflavor dimensions the under group of the are all 7-brane the chiral groups. sametensors primaries as and uncharged Thus in with bifundamentals the respect have conformalthe to zero case, the beta-functions, anomalous i.e., dimensions canonical. eq. then (2.7), If the antisymmetric requires vanishing that of the fundamentals have dimension 1 to a diagonally embedded SU(2) 4.2 The non-conformal case Next we discuss the deformed fixed points inthough the the infrared Wilsonian (section gaugeenergy coupling 2.3). effective in gauge this We coupling theorythat have does depends the already not on corresponding pointed the vary over scale, IIB out the the that background moduli low- should al- space. have This constant implies tion 3, we showedilar that to the the 7-brane7-branes background background do for for this not the configuration bendthe conformal is 7-brane and theory. very is are sim- cancelledstring coincident As locally coupling with in by is the the the constant.cels O7-planes. conformal O7-planes, against Similarly, so case, The that the we RR the ofsector gravitational expect the charge is field that not orientifold of of the affected planes.mal the by type and this Thus 7-branes IIB the deformation. it can- non-conformalconnection appears The case on only that is difference the the in between 7-branes. closedconformal the the string case open confor- In string it the sector, is conformal namely case a in it flat the is gauge connection trivial, which while breaks in the the non- SU(4) JHEP07(1999)005 , = B 01 N . 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