We construct dual Type I' string descriptions to five dimensional
supersymmetric fixed points with $E_{N_f+1}$ global symmetry. The background is
obtained as the near horizon geometry of the D4-D8 brane system in massive Type
IIA supergravity. We use the dual description to deduce some properties of the
fixed points.Comment: 10 pages, LaTeX2e; improved references, corrected typo

Andreas Brandhuber

Bachas

Bergman

Ferrara

Gimon

Matalliotakis

Polchinski

Romans

van Nieuwenhuizen

Yaron Oz

Physics Letters B

English

arXiv

Crossref

We construct dual Type I' string descriptions to five dimensional supersymmetric fixed points with $E_{N_f+1}$ global symmetry. The background is obtained as the near horizon geometry of the D4-D8 brane system in massive Type IIA supergravity. We use the dual description to deduce some properties of the fixed points.We construct dual Type I' string descriptions to five dimensional supersymmetric fixed points with $E_{N_f+1}$ global symmetry. The background is obtained as the near horizon geometry of the D4-D8 brane system in massive Type IIA supergravity. We use the dual description to deduce some properties of the fixed points

Brandhuber, Andreas

Oz, Yaron

CERN Document Server

CERN-TH/99-135hep-th/9905148The D4-D8 Brane System andFive Dimensional Fixed PointsAndreas Brandhuber and Yaron OzTheory Division, CERNCH-1211, Geneva 23, SwitzerlandAbstractWe construct dual Type I’ string descriptions to ve dimensional supersymmetricxed points with ENf+1 global symmetry. The background is obtained as the nearhorizon geometry of the D4-D8 brane system in massive Type IIA supergravity. Weuse the dual description to deduce some properties of the xed points.May 19991 IntroductionThe consideration of the near horizon geometry of branes on one hand, and the low energydynamics on their worldvolume on the other hand has lead to conjectured duality relationsbetween eld theories and string theory (M theory) on certain backgrounds [1, 2, 3] 1. Theeld theories under discussion are in various dimensions, can be conformal or not, andwith or without supersymmetry. These properties are reflected by the type of string/Mtheory backgrounds of the dual description.In this note we will construct dual Type I’ string descriptions to ve dimensional su-persymmetric xed points with ENf+1 global symmetry2. These xed points are obtainedin the limit of innite bare coupling of N = 2 supersymmetric gauge theories with gaugegroup Sp(Q4), Nf < 8 massless hypermultiplets in the fundamental representation andone massless hypermultiplet in the anti-symmetric representation [5, 6]. The dual back-ground is obtained as the near horizon geometry of the D4-D8 brane system in massiveType IIA supergravity. The ten dimensional space is a bration of AdS6 over S4 and hasthe isometry group SO(2, 5)SO(4). This space provides the spontaneous compactica-tion of massive Type IIA supergravity in ten dimensions to the F (4) gauged supergravityin six dimensions [7].The paper is organized as follows. In section 2 we will discuss the D4-D8 brane systemand its relation to the ve dimensional xed points. In section 3 we will construct thedual string description and use it to deduce some properties of the xed points.2 The D4-D8 Brane SystemWe start with Type I string theory on R9S1 with N coinciding D5 branes wrapping thecircle. The six dimensional D5 brane worldvolume theory possesses N = 1 supersymme-try. It has an Sp(N) gauge group, one hypermultiplet in the antisymmetric representationof Sp(N) from the DD sector and 16 hypermultiplets in the fundamental representationfrom the DN sector. Performing T-duality on the circle results in Type I’ theory compact-ied on the interval S1/Z2 with two orientifolds (O8 planes) located at the xed points.The D5 branes become D4 branes and there are 16 D8 branes located at points on theinterval. They cancel the -16 units of D8 brane charge carried by the two O8 planes. Thelocations of the D8 branes correspond to masses for the hypermultiplets in the funda-1For a recent review of the subject see [4].2ENf+1 = (E8, E7, E6, E5 = Spin(10), E4 = SU(5), E3 = SU(3)  SU(2), E2 = SU(2)  U(1), E1 =SU(2)).1mental representation arising from the open strings between the D4 branes and the D8branes. The hypermultiplet in the antisymmetric representation is massless.Consider rst N = 1, namely one D4 brane. The worldvolume gauge group is Sp(1) ’SU(2). The ve-dimensional vector multiplet contains as bosonic elds the gauge eldand one real scalar. The scalar parametrizes the location of the D4 branes in the interval,and the gauge group is broken to U(1) unless the D4 brane is located at one of the xedpoints. A hypermultiplet contains four real (two complex) scalars. The Nf masslessmatter hypermultiplets in the fundamental and the antisymmetric hypermultiplet (whichis a trivial representation for Sp(1)) parametrize the Higgs branch of the theory. It is themoduli space of SO(2Nf) one-instanton. The theory has an SU(2)R R-symmetry. Thetwo supercharges as well as the scalars in the hypermultiplet transform as a doublet underSU(2)R. In addition, the theory has a global SU(2) SO(2Nf) U(1)I symmetry. TheSU(2) factor of the global symmetry group is associated with the massless antisymmetrichypermultiplet and is only present if N > 1, the SO(2Nf) group is associated with theNf massless hypermultiplets in the fundamental and the U(1)I part corresponds to theinstanton number conservation.Consider the D8 brane background metric. It takes the formds2 = H−1/28 (−dt2 + dx21 + . . . + dx28) + H1/28 dz2 , e−φ = H5/48 . (2.1)H8 is a harmonic function on the interval parametrized by z. Therefore H8 is a piecewiselinear function in z where the slope is constant between two D8 branes and decreases byone unit for each D8 brane crossed. Thus,H8(z) = c + 16zls−16∑i=1jz − zijls−16∑i=1jz + zijls, (2.2)where the zi denote the locations of the 16 D8 branes.Denote the D4 brane worldvolume coordinates by t, x1 . . . x4. The D4 brane is locatedat some point in x5 . . . x8 and z. We can consider it to be a probe of the D8 branesbackground. The gauge coupling g of the D4 brane worldvolume theory and the harmonicfunction H8 are related as can be seen by expanding the DBI action of a D4 brane in thebackground (2.2). We getg2 =lsH8, (2.3)where g2cl =clscorresponds to the classical gauge coupling. In the eld theory limit wetake ls ! 0 keeping the gauge coupling g xed, thus,g2 = xed, ) φ = zl2s= xed, ls ! 0 . (2.4)2In this limit we have1g2=1g2cl+ 16φ−16∑i=1jφ−mij −16∑i=1jφ + mij , (2.5)where the masses mi =zil2s. Note that in the eld theory limit we are studying the regionnear z = 0. The coordinate φ takes values in R+ and parametrizes the eld theoryCoulomb branch.Seiberg argued [5] that the theory at the origin of the Coulomb branch obtained inthe limit gcl = 1 with Nf < 8 massless hypermultiplets is a non trivial xed point. Therestriction Nf < 8 can be seen in the supergravity description as a requirement for theharmonic function H8 in equation (2.2) to be positive when c = 0 and zi = 0. At the xedpoint the global symmetry is enhanced to SU(2)ENf+1. The Higgs branch is expectedto become the moduli space of ENf+1 one-instanton.The generalization to N = Q4 D4 branes is straightforward [6]. The gauge group is nowSp(Q4) and the global symmetry is as before. The Higgs branch is now the moduli spaceof SO(2Nf) Q4-instantons. At the xed point the global symmetry is enhanced as beforeand the Higgs branch is expected to become the moduli space of ENf+1 Q4-instanton. Ourinterest in this paper will be in nding dual string (supergravity) descriptions of thesexed points.3 The Supergravity (String) DescriptionThe low energy description of our system is given by Type I’ supergravity [8]. The regionbetween two D8 branes is, as discussed in [9], described by massive Type IIA supergravity[10]. The congurations that we will study have Nf D8 branes located at one O8 planeand 16−Nf D8 branes at the other O8 plane. Therefore we are always between D8 branesand never encounter the situation where we cross D8 branes, and the massive Type IIAsupergravity description is sucient.The bosonic part of the massive Type IIA action (in string frame) including a six-formgauge eld strength which is the dual of the RR four-form eld strength isS =12κ210∫d10xp−g(e−2Φ (R + 4∂µ∂µ)− 12  6! jF6j2 − 12m2)(3.1)3where the mass parameter is given by3.m =p2(8−Nf)µ8κ10 = 8−Nf2pils. (3.2)The Einstein equations derived from (3.1) read (in Einstein frame metric)2Rij = gij(R− 12j∂j2 − e−Φ/22  6! jF6j2 − m22e5Φ/2) ++∂i∂j +e−Φ/25!Fim1...m5Fm1...m5j ,0 = rj∂j− 5m24e5Φ/2 +e−Φ/24  6! jF6j2 ,0 = ri(e−Φ/2Fim1...m5). (3.3)Except for these formulas we will be using the string frame only.For identifying the solutions of D4 branes localized on D8 branes it is more convenientto start with the conformally flat form of the D8 brane supergravity solution. It takesthe form [8]ds2 = Ω(z)2(−dt2 + . . . + dx24 + d~r2 + ~r2dΩ24 + dz2), (3.4)eΦ = C(32Cmz)− 56, Ω(z) =(32Cmz)− 16.In these coordinates the harmonic function of Q4 localized D4 branes in the near horizonlimit derived from (3.3) readsH4 =Q4l10/3s (~r2 + z2)5/3. (3.5)This localized D4-D8 brane system solution, in a dierent coordinate system, has beenconstructed in [11]. One way to determine the harmonic function (3.5) of the localizedD4 branes is to solve the Laplace equation in the background of the D8 branes.It is useful to make a change of coordinates z = r sin α, ~r = r cos α, 0  α  pi/2. Wegetds2 = Ω2(H− 124 (−dt2 + . . . dx24) + H124 (dr2 + r2dΩ24)),eΦ = C(32Cmr sin α)− 56H− 144 , Ω =(32Cmr sin α)− 16, (3.6)F01234r =1C∂r(H−14),3We use the conventions κ10 = 8pi7/2l4s and µ8 = (2pi)−9/2l−5s for the gravitational coupling and theD8 brane charge, respectively.4where C is an arbitrary parameter of the solution [8] anddΩ24 = dα2 + (cos α)2dΩ23 . (3.7)The background (3.6) is a solution of the massive Type IIA supergravity equations (3.3).The metric (3.6) of the D4-D8 system can be simplied tods2 =(32Cm sin α)− 13(Q− 124 r43dx2k + Q124dr2r2+ Q124 dΩ24)(3.8)where dx2k  −dt2 + . . . + dx24. Dene now the energy coordinate U by r2 = l5sU3. Thatthis is the energy coordinate can be seen by calculating the energy of a fundamental stringstretched in the r direction or by using the DBI action as in the previous section. In theeld theory limit, ls ! 0 with the energy U xed, we get the metric in the form of awarped product [12] of AdS6  S4ds2 = l2s(34piC(8−Nf ) sinα)− 13(Q− 124 U2dx2k + Q1249dU24U2+ Q124 dΩ24), (3.9)and the dilaton is given byeΦ = Q− 144 C(34piC(8−Nf) sin α)− 56. (3.10)The ten dimensional space described by (3.9) is a bration of AdS6 over S4. It is themost general form of a metric that has the isometry of an AdS6 space [13]. The spacehas a boundary at α = 0 which corresponds to the location of the O8 plane (z = 0). Theboundary is of the form AdS6  S3. In addition to the SO(2, 5) AdS6 isometries, theten dimensional space has also SO(4) isometries associated with the spherical part of themetric (3.9). In general S4 has the SO(5) isometry group. However, this is reduced dueto the warped product structure. As is easily seen from the form of the spherical part(3.7), only transformations excluding the α coordinate are isometries of (3.9). We are leftwith an SO(4)  SU(2) SU(2) isometry group.The two dierent viewpoints of the D4-D8 brane system, the near horizon geome-try of the brane system on one hand, and the low energy dynamics on the D4 branesworldvolume on the other hand suggest a duality relation. Namely, Type I’ string theorycompactied on the background (3.9), (3.10) with a 4-form flux of Q4 units on S4 is dualto an N = 2 supersymmetric ve dimensional xed point4. The xed point is obtainedin the limit of innite coupling of Sp(Q4) gauge theory with Nf hypermultiplets in the4The 4-form flux is just the dual of the 6-form in (3.6).5fundamental representation and one hypermultiplet in the antisymmetric representation,where m  (8 − Nf) as in (3.2). The SO(2, 5) symmetry of the compactication corre-sponds to the conformal symmetry of the eld theory. The SU(2) SU(2) symmetry ofthe compactication corresponds SU(2)R R-symmetry and to the SU(2) global symmetryassociated with the massless hypermultiplet in the antisymmetric representation.At the boundary α = 0 the dilaton (3.10) blows up and Type I’ is strongly coupled.In the weakly coupled dual heterotic string description this is seen as an enhancement ofthe gauge symmetry to ENf+1. One can see this enhancement of the gauge symmetry inthe Type I’ description by analysing the D0 brane dynamics near the orientifold plane[14, 15]. This means that we have ENf+1 vector elds that propagate on the AdS6  S3boundary, as in the Horava-Witten picture [16]. The scalar curvature of the background(3.9), (3.10)Rl2s  (C(8−Nf ))13Q− 124 (sin α)− 53 , (3.11)blows up at the boundary as well. In the dual heterotic description the dilaton is smallbut the curvature is large, too. For large Q4 there is a region, sin α  Q−3104 , where bothcurvature (3.11) and dilaton (3.10) are small and thus we can trust supergravity.The AdS6 supergroup is F (4). Its bosonic subgroup is SO(2, 5)  SU(2). Romansconstructed an N = 4 six dimensional gauged supergravity with gauge group SU(2) thatrealizes F (4) [7]. It was conjectured in [17] that it is related to a compactication of theten dimensional massive Type IIA supergravity. Indeed, we nd that the ten dimensionalbackground space is the warped product of AdS6 and S4 (3.9) (with Nf = 0). Thereduction to six dimensions can be done in two steps. First we can integrate over thecoordinate α. This yields a nine dimensional space of the form AdS6  S3. We can thenreduce on S3 to six dimensions, while gauging its isometry group. Roman’s constructionis based on gauging an SU(2) subgroup of the SO(4) isometry group.The massive Type IIA supergravity action in the (string frame) goes likel−8s∫ p−ge−2ΦR  Q5/24 , (3.12)suggesting that the number of degrees of freedom goes like Q5/24 in the regime where it isan applicable description. Terms in the Type I’ action coming from the D8 brane DBIaction turn out to be of the same order. Viewed from M theory point of view, we expectthe corrections to the supergravity action to go like l3p  l3seΦ  1/Q4, where lp is theeleven-dimensional Planck length. This seems to suggest that the eld theory has a 1/Q4expansion at large Q4. For example, the one-loop correction of the forml−8s∫ p−gl6sR4  Q1/24 (3.13)6is suppressed by Q24 compared to the tree-level action.According to the AdS/CFT correspondence the spectrum of chiral primary operatorsof the xed point theory can be derived from the spectrum of Kaluza-Klein excitations ofmassive Type IIA supergravity on the background (3.9). We will not carry out the detailedanalysis here but make a few comments. The operators fall into representations of theF (4) supergroup. As in the case of the six dimensional (0, 1) xed point with E8 globalsymmetry [18] we expect ENf+1 neutral operators to match the Kaluza-Klein reductionof elds in the bulk geometry, and ENf+1 charged operators to match the Kaluza-Kleinreduction of elds living on the boundary. Among the ENf+1 neutral operators we expectto have dimension 3k/2 operators of the type Trφk where φ is a complex scalar in thehypermultiplet, which parametrize the Higgs branch of the theory. Like in [18] we donot expect all these operators to be in short multiplets. We expect that those in longmultiplets will generically receive 1/Q4 corrections to their anomalous dimensions. Unlikethe hypermultiplet, the vector multiplet in ve dimensions is not a representation ofthe superconformal group F (4). Therefore, we do not expect Kaluza-Klein excitationscorresponding to neutral operators of the type Trϕk where ϕ is a real scalar in the vectormultiplet, which parametrizes the Coulomb branch of the theory. Among the ENf+1charged operators we expect to have the dimension four ENf+1 global symmetry currentsthat couple to the massless ENf+1 gauge elds on the boundary.AcknowledgementWe would like to thank N. Itzhaki for participating in the early stages of this work and formany useful discussions. We would also like to thank A. Zaaroni for a useful discussion.7References[1] J. M. Maldacena, \The Large N Limit of Superconformal Field Theories andSupergravity," hep-th/9711200.[2] S. 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B460, 506 (1996), hep-th/9510209 ; \Eleven-DimensionalSupergravity on a Manifold with Boundary," Nucl.Phys. B475, 94 (1996)hep-th/9603142.[17] S. Ferrara, A. Kehagias, H. Partouche and A. Zaaroni, \AdS6 interpretation of5-D superconformal eld theories," Phys. Lett. B431, 57 (1998).[18] E. G. Gimon and C. Popescu, \The Operator Spectrum of the Six-dimensional(1, 0) Theory," JHEP 9904, 018 (1999), hep-th/9901048.9

The D4-D8 Brane System and Five Dimensional Fixed Points

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The D4-D8 Brane System and Five Dimensional Fixed Points

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