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Warped Compactifications and Ads/CFT TMR9: Quantum aspects of gauge theories, supersymmetry and unification PROCEEDINGS Warped Compactifications and AdS/CFT∗ Yaron Oz Theory Division, CERN CH-1211, Geneva 23, Switzerland Abstract: In this talk we discuss two classes of examples of warped products of AdS spaces in the context of the AdS/CFT correspondence. The first class of examples appears in the construction of dual Type I’ string descriptions to five dimensional supersymmetric fixed points with ENf +1 global symmetry. The background is obtained as the near horizon geometry of the D4-D8 brane system in massive Type IIA supergravity. The second class of examples appears when considering the N =2 superconformal theories defined on a 3 + 1 dimensional hyperplane intersection of two sets of M5 branes. We use the dual string formulations to deduce properties of these field theories. 1. Introduction of N = 2 supersymmetric gauge theories with gauge group Sp(Q4), Nf < 8 massless hypermul- The consideration of the near horizon geometry tiplets in the fundamental representation and one of branes on one hand, and the low energy dy- massless hypermultiplet in the anti-symmetric rep- namics on their worldvolume on the other hand resentation [8, 9]. These theories were discussed has lead to conjectured duality relations between in the context of the AdS/SCFT correspondence field theories and string theory (M theory) on in [10]. The dual background is obtained as the certain backgrounds [1, 2, 3, 4]. The field theo- near horizon geometry of the D4-D8 brane sys- ries under discussion are in various dimensions, tem in massive Type IIA supergravity. The ten 4 can be conformal or not, and with or without su- dimensional space is a fibration of AdS6 over S persymmetry. These properties are reflected by and has the isometry group SO(2; 5) × SO(4). the type of string/M theory backgrounds of the This space provides the spontaneous compactifi- dual description. cation of massive Type IIA supergravity in ten In this talk we discuss two classes of exam- dimensions to the F (4) gauged supergravity in ples of warped products of AdS spaces in the con- six dimensions [11]. text of the AdS/CFT correspondence. The warped The second class of examples [12] appears product structure is a fibration of AdS over some when considering the N = 2 superconformal the- manifold M [5]. It is the most general form of ories defined on a 3 + 1 dimensional hyperplane a metric that has the isometry of an AdS space intersection of two sets of M5 branes. The N =2 [6]. supersymmetry algebra in four dimensions con- The first class of examples [7] appears in tains a central extension term that corresponds the construction of dual Type I’ string descrip- to string charges in the adjoint representation tions to five dimensional supersymmetric fixed of the SU(2)R part of the R-symmetry group. points with ENf +1 global symmetry, ENf +1 = This implies that N = 2 supersymmetric gauge (E8;E7;E6;E5 = theories in four dimensions can have BPS string × Spin(10);E4 =SU(5);E3 =SU(3) SU(2);E2 = configurations at certain regions in their moduli × SU(2) U(1);E1 = SU(2)). These fixed points space of vacua. In particular, at certain points in are obtained in the limit of infinite bare coupling the moduli space of vacua these strings can be- ∗Talks presented at the TMR conference in Paris, come tensionless. A brane configuration that ex- September 99. hibits this phenomena consists of two sets of M5 TMR9: Quantum aspects of gauge theories, supersymmetry and unification Yaron Oz branes intersecting on 3 + 1 dimensional hyper- plet in the antisymmetric representation is mass- plane. The theory on the intersection is N =2 less. supersymmetric. One can stretch M2 branes be- Consider first N = 1, namely one D4 brane. tween the two sets of M5 branes in a configu- The worldvolume gauge group is Sp(1) ' SU(2). ration that preserves half of the supersymme- The five-dimensional vector multiplet contains as try. This can be viewed as a BPS string of the bosonic fields the gauge field and one real scalar. four dimensional theory, and we will study such The scalar parametrizes the location of the D4 brane configurations using the AdS/CFT corre- branes in the interval, and the gauge group is bro- spondence [1, 4]. ken to U (1) unless the D4 brane is located at one The talk is organized as follows. In the next of the fixed points. A hypermultiplet contains two sections we will discuss the first class of ex- four real (two complex) scalars. The Nf mass- amples. In section 2 we will discuss the D4-D8 less matter hypermultiplets in the fundamental brane system and its relation to the five dimen- and the antisymmetric hypermultiplet (which is a sional fixed points. In section 3 we will construct trivial representation for Sp(1)) parametrize the the dual string description and use it to deduce Higgs branch of the theory. It is the moduli space some properties of the fixed points. In the follow- of SO(2Nf ) one-instanton. The theory has an ing two sections we will discuss the second class SU(2)R R-symmetry. The two supercharges as of examples. In section 4 we will construct the well as the scalars in the hypermultiplet trans- dual string description of the four dimensional form as a doublet under SU(2)R. In addition, theory on the intersection and discuss the field the theory has a global SU(2)×SO(2Nf )×U(1)I theory on the intersection. In section 5 we will symmetry. The SU(2) factor of the global sym- use the dual string description to deduce some metry group is associated with the massless anti- properties of these strings. We will argue that symmetric hypermultiplet and is only present if from the four dimensional field theory viewpoint N>1, the SO(2Nf ) group is associated with the they are simply BPS string configurations on the Nf massless hypermultiplets in the fundamental Higgs branch. and the U(1)I part corresponds to the instanton number conservation. Consider the D8 brane background metric. 2. The D4-D8 Brane System It takes the form −1 2 9 1 2 = − 2 2 2 We start with Type I string theory on R × S ds = H8 ( dt + dx1 + :::+dx8) 1=2 2 with N coinciding D5 branes wrapping the cir- + H8 dz ; cle. The six dimensional D5 brane worldvolume −φ 5=4 e = H8 : (2.1) theory possesses N = 1 supersymmetry. It has an Sp(N) gauge group, one hypermultiplet in the H8 is a harmonic function on the interval parametrized antisymmetric representation of Sp(N) from the by z. Therefore H8 is a piecewise linear function DD sector and 16 hypermultiplets in the funda- in z where the slope is constant between two D8 mental representation from the DN sector. Per- branes and decreases by one unit for each D8 forming T-duality on the circle results in Type I’ brane crossed. Thus, 1 theory compactified on the interval S =Z2 with X16 X16 two orientifolds (O8 planes) located at the fixed z |z − zi| |z + zi| H8(z)=c+16 − − ; l l l points. The D5 branes become D4 branes and s i=1 s i=1 s there are 16 D8 branes located at points on the (2.2) interval. They cancel the -16 units of D8 brane where the zi denote the locations of the 16 D8 charge carried by the two O8 planes. The loca- branes. tions of the D8 branes correspond to masses for Denote the D4 brane worldvolume coordi- the hypermultiplets in the fundamental represen- nates by t; x1 :::x4. The D4 brane is located at tation arising from the open strings between the some point in x5 :::x8 and z. We can consider it D4 branes and the D8 branes. The hypermulti- to be a probe of the D8 branes background. The 2 TMR9: Quantum aspects of gauge theories, supersymmetry and unification Yaron Oz gauge coupling g of the D4 brane worldvolume by massive Type IIA supergravity [15]. The con- theory and the harmonic function H8 are related figurations that we will study have Nf D8 branes as can be seen by expanding the DBI action of a located at one O8 plane and 16 − Nf D8 branes D4 brane in the background (2.2). We get at the other O8 plane. Therefore we are always between D8 branes and never encounter the situ- l g2 = s ; (2.3) ation where we cross D8 branes, and the massive H8 Type IIA supergravity description is sufficient. where g2 = c corresponds to the classical gauge The bosonic part of the massive Type IIA ac- cl ls tion (in string frame) including a six-form gauge coupling. In the field theory limit we take ls → 0 keeping the gauge coupling g fixed, thus, field strength which is the dual of the RR four- form field strength is z Z 2 ⇒ → √ g =fixed; φ= 2 =fixed;ls 0:(2.4) 1 10 −2Φ µ ls S = 2 d x −g(e (R +4@µΦ@ Φ) 2κ10 In this limit we have − 1 | |2 − 1 2 · F6 m ) (3.1) X16 X16 2 6! 2 1 1 − | − |− | | where the mass parameter is given by 2 = 2 +16φ φ mi φ+mi ; (2.5) g gcl =1 =1 √ i i 8 − Nf m = 2(8 − Nf )µ8κ10 = : (3.2) zi 2πls where the masses mi = l2 . Note that in the s 7=2 4 field theory limit we are studying the region near We use the conventions κ10 =8π ls and µ8 = + −9=2 −5 z = 0.
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