Warped Compactifications and Ads/CFT

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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. The coordinate φ takes values in R and (2π) ls for the gravitational coupling and parametrizes the field theory Coulomb branch. the D8 brane charge, respectively. Seiberg argued [8] that the theory at the ori- The Einstein equations derived from (3.1) gin of the Coulomb branch obtained in the limit read (in Einstein frame metric) g = ∞ with N < 8 massless hypermultiplets is −Φ/2 2 cl f 1 2 e 2 m 5Φ/2 2Rij = gij (R − |∂Φ| − |F6| − e ) a non trivial fixed point. The restriction Nf < 8 2 2 · 6! 2 can be seen in the supergravity description as e−Φ/2 + ∂ Φ∂ Φ+ F F m1...m5 , a requirement for the harmonic function H8 in i j 5! im1...m5 j equation (2.2) to be positive when c =0and 2 −Φ/2 ∇j −5m 5Φ/2 e | |2 z = 0. At the fixed point the global symme- 0= ∂jΦ e + F6 , i 4 4 · 6! × −Φ 2 try is enhanced to SU(2) ENf +1. The Higgs ∇i / 0= e Fim1...m5 . (3.3) branch is expected to become the moduli space of ENf +1 one-instanton. Except for these formulas we will be using the The generalization to N = Q4 D4 branes string frame only. is straightforward [9]. The gauge group is now For identifying the solutions of D4 branes lo- Sp(Q4) and the global symmetry is as before. calized on D8 branes it is more convenient to The Higgs branch is now the moduli space of start with the conformally flat form of the D8

SO(2Nf ) Q4-instantons. At the fixed point the brane supergravity solution. It takes the form global symmetry is enhanced as before and the [13] Higgs branch is expected to become the mod- 2 2 2 2 ds =Ω(z)(−dt + ...+dx4 uli space of ENf +1 Q4-instanton. Our interest 2 2 2 2 + dr˜ +˜r dΩ3 +dz ) , (3.4) in the next section will be in finding dual string − 5 − 1 3 6 3 6 descriptions of these fixed points. eΦ = C Cmz Ω(z)= Cmz . 2 2 3. The Dual String Description In these coordinates the harmonic function of Q4 localized D4 branes in the near horizon limit de- The low energy description of our system is given rived from (3.3) reads by Type I’ supergravity [13]. The region between Q4 H4 = 10 3 . (3.5) / 2 2 5/3 two D8 branes is, as discussed in [14], described ls (˜r + z )

3 TMR9: Quantum aspects of gauge theories, supersymmetry and uniﬁcation Yaron Oz

This localized D4-D8 brane system solution, in a The ten dimensional space described by (3.9) 4 different coordinate system, has been constructed is a fibration of AdS6 over S .Itisthemost in [16]. One way to determine the harmonic func- general form of a metric that has the isometry tion (3.5) of the localized D4 branes is to solve of an AdS6 space [6]. The space has a bound- the Laplace equation in the background of the ary at α = 0 which corresponds to the location D8 branes. of the O8 plane (z = 0). The boundary is of 3 It is useful to make a change of coordinates the form AdS6 × S . In addition to the SO(2, 5) z = r sin α, r˜ = r cos α, 0 ≤ α ≤ π/2. We get AdS6 isometries, the ten dimensional space has also SO(4) isometries associated with the spher- −1 2 2 2 2 2 4 ds =Ω(H4 (−dt + ...dx4) ical part of the metric (3.9). In general S has 1 2 2 2 2 the SO(5) isometry group. However, this is re- + H4 (dr + r dΩ4)) , − 5 duced due to the warped product structure. As 6 1 Φ 3 − 4 e = C Cmr sin α H4 , is easily seen from the form of the spherical part 2 (3.7), only transformations excluding the α coor- − 1 3 6 dinate are isometries of (3.9). We are left with Ω= Cmr sin α , (3.6) 2 an SO(4) ∼ SU(2) × SU(2) isometry group. 1 −1 F01234 = ∂ H , The two different viewpoints of the D4-D8 r C r 4 brane system, the near horizon geometry of the where C is an arbitrary parameter of the solution brane system on one hand, and the low energy [13] and dynamics on the D4 branes worldvolume on the

2 2 2 2 other hand suggest a duality relation. Namely, dΩ4 = dα +(cosα) dΩ3 . (3.7) Type I’ string theory compactified on the background (3.9), (3.10) with a 4-form flux of Q4 The background (3.6) is a solution of the massive units on S4 is dual to an N =2supersymmet- Type IIA supergravity equations (3.3). ric five dimensional fixed point. The fixed point The metric (3.6) of the D4-D8 system can be is obtained in the limit of infinite coupling of simplified to Sp(Q4) gauge theory with Nf hypermultiplets in − 1 3 the fundamental representation and one hyper- 2 3 − 1 4 2 ds = Cmsin α (Q 2 r 3 dx 2 4 k multiplet in the antisymmetric representation, where 2 m ∼ (8 − N ) as in (3.2). The SO(2, 5) sym- 1 dr 1 f + Q 2 + Q 2 dΩ2) (3.8) metry of the compactification corresponds to the 4 r2 4 4 conformal symmetry of the field theory. The 2 2 2 where dxk ≡−dt + ...+dx4. Define now the SU(2) × SU(2) symmetry of the compactifica- 2 5 3 energy coordinate U by r = ls U . That this is tion corresponds SU(2)R R-symmetry and to the the energy coordinate can be seen by calculating SU(2) global symmetry associated with the mass- the energy of a fundamental string stretched in less hypermultiplet in the antisymmetric repre- the r direction or by using the DBI action as sentation. in the previous section. In the field theory limit, At the boundary α = 0 the dilaton (3.10) ls → 0 with the energy U fixed, we get the metric 4 blows up and Type I’ is strongly coupled. In the in the form of a warped product [5] of AdS6 × S weakly coupled dual heterotic string description 2 2 3 −1 −1 2 2 this is seen as an enhancement of the gauge sym- − 3 2 ds = ls ( C(8 Nf )sinα) (Q4 U dxk 4π metry to ENf +1. One can see this enhancement 1 2 1 2 9dU 2 2 of the gauge symmetry in the Type I’ description + Q4 + Q4 dΩ4) , (3.9) 4U 2 by analysing the D0 brane dynamics near the ori- and the dilaton is given by entifold plane [17, 18, 19]. This means that we have ENf +1 vector fields that propagate on the −5 3 − 1 3 6 AdS6 × S boundary, as in the Horava-Witten eΦ = Q 4 C C(8 − N )sinα . (3.10) 4 4π f picture [20]. The scalar curvature of the back-

4 TMR9: Quantum aspects of gauge theories, supersymmetry and uniﬁcation Yaron Oz ground (3.9), (3.10) ﬁxed point theory can be derived from the spec-

1 trum of Kaluza-Klein excitations of massive Type 2 1 − − 5 R ∼ − 3 2 3 ls (C(8 Nf )) Q4 (sin α) , (3.11) IIA supergravity on the background (3.9). We blows up at the boundary as well. In the dual will not carry out the detailed analysis here but heterotic description the dilaton is small but the make a few comments. The operators fall into representations of the F (4) supergroup. As in curvature is large, too. For large Q4 there is a re- 3 − 10 the case of the six dimensional (0, 1) fixed point gion, sin α Q4 , where both curvature (3.11) with E8 global symmetry [21] we expect E +1 and dilaton (3.10) are small and thus we can trust Nf neutral operators to match the Kaluza-Klein re- supergravity. duction of fields in the bulk geometry, and ENf +1 The AdS6 supergroup is F (4). Its bosonic charged operators to match the Kaluza-Klein re- subgroup is SO(2, 5)×SU(2). Romans constructed duction of fields living on the boundary. Among an N = 4 six dimensional gauged supergravity the E +1 neutral operators we expect to have with gauge group SU(2) that realizes F (4) [11]. Nf dimension 3k/2 operators of the type Trφk where It was conjectured in [10] that it is related to a φ is a complex scalar in the hypermultiplet, which compactification of the ten dimensional massive parametrize the Higgs branch of the theory. Like Type IIA supergravity. Indeed, we find that the in [21] we do not expect all these operators to be ten dimensional background space is the warped 4 in short multiplets. We expect that those in long product of AdS6 and S (3.9) (with Nf =0). multiplets will generically receive 1/Q4 correc- The reduction to six dimensions can be done in tions to their anomalous dimensions. Unlike the two steps. First we can integrate over the coor- hypermultiplet, the vector multiplet in five di- dinate α. This yields a nine dimensional space 3 mensions is not a representation of the supercon- of the form AdS6 × S . We can then reduce on formal group F (4). Therefore, we do not expect S3 to six dimensions, while gauging its isometry Kaluza-Klein excitations corresponding to neu- group. Roman’s construction is based on gaug- tral operators of the type Trϕk where ϕ is a real ing an SU(2) subgroup of the SO(4) isometry scalar in the vector multiplet, which parametrizes group. the Coulomb branch of the theory. Among the The massive Type IIA supergravity action in E +1 charged operators we expect to have the the string frame goes like Nf Z dimension four ENf +1 global symmetry currents √ −8 −2Φ 5/2 − R∼ that couple to the massless ENf +1 gauge fields ls ge Q4 , (3.12) on the boundary. suggesting that the number of degrees of freedom 5/2 goes like Q4 in the regime where it is an appli- 4. The M5-M5’ Brane System cable description. Terms in the Type I’ action coming from the D8 brane DBI action turn out We denote the eleven dimensional space-time co- to be of the same order. Viewed from M theory ordinates by (x||,~x, ~y,~z), where x|| parametrize point of view, we expect the corrections to the the (0, 1, 2, 3) coordinates, ~x =(x1,x2)the(4,5) 3 ∼ 3 Φ ∼ coordinates, ~y =(y1,y2)the(6,7) coordinates supergravity action to go like lp lse 1/Q4, 1 2 3 where lp is the eleven-dimensional Planck length. and ~z =(z ,z ,z )the(8,9,10) coordinates. This seems to suggest that the field theory has Consider two sets of fivebranes in M the- a1/Q4 expansion at large Q4. For example, the ory: N1 coinciding M5 branes and N2 coinciding one-loop correction of the form M5’ branes. Their worldvolume coordinates are Z 0 √ M5:(x||,~y)andM5 :(x||,~x). Such a configu- −8 − 6R4 ∼ 1/2 ls gls Q4 (3.13) raion preserves eight supercharges. The eleven- dimensional supergravity background takes the 2 is suppressed by Q4 compared to the tree-level form [22, 23, 24] action. 2 2/3 −1 2 According to the AdS/CFT correspondence ds11 =(H1H2) [(H1H2) dx|| −1 2 −1 2 2 the spectrum of chiral primary operators of the + H2 d~x + H1 d~y + d~z ] , (4.1)

5 TMR9: Quantum aspects of gauge theories, supersymmetry and uniﬁcation Yaron Oz

with the 4-form field strength F N2 2 2 2 + 2 (dy + y dψ ) . (4.5) 1 2 1 2 sin α F =3(∗dH1 ∧ dy ∧ dy + ∗dH2 ∧ dx ∧ dx (4.2)) , The metric (4.4) has the AdS5 isometry group where ∗ defines the dual form in the three di- [6]. Therefore, in the spirit of [1], M theory on mensional space (z1,z2,z3). When the M5 and the background (4.4), (4.5) should be dual to a M5’ branes are only localized along the overall four dimensional N =2SCFT. transverse directions ~z the harmonic functions Note that the curvature of the metric di- are H1 =1+lpN1/2|~z|,H2 =1+lpN2/2|~z|,where verges for small α as l is the eleven dimensional Planck length. This p 1 case has been discussed in [25]. The near hori- R∼ . (4.6) 2 2/3 8/3 zon geometry in this case does not have the AdS lpN1 sin α isometry group and thus cannot describe a dual Away from α = 0, eleven-dimensional supergrav- SCFT. Moreover, there does not seem to be in ity can be trusted for large N1. The singularity this case a theory on the intersection which de- at α = 0 is interpreted as a signal that some de- couples from the bulk physics. grees of freedom have been effectively integrated Consider the semi-localized case when the and are needed in order to resolve the singular- M5 branes are completely localized while the M5’ ity. These presumably correspond to membranes branes are only localized along the overall trans- that end on the M5’ branes. verse directions. When the branes are at the ori- The above M5-M5’ brane system can be un- gin of (~x, ~z) space the harmonic functions in the derstood as up lifting to eleven dimensions of near core limit of the M5’ branes take the form an elliptic brane system of Type IIA. It con- [26, 27] sists of N2 NS5-branes with worldvolume coordi- 4 nates (0, 1, 2, 3, 4, 5) periodically arranged in the 4πlpN1N2 lpN2 H1 =1+ 2 2,H2= . 6-direction and N1 D4-branes with worldvolume (|~x| +2lpN2|~z|) 2|~z| (4.3) coordinates (0, 1, 2, 3, 6) stretched between them The numerical factors in (4.3) are determined by as in figure 1. When we lift this brane configu- the requirment that the integral of F yields the ration to eleven dimensions, we delocalize in the appropriate charges. This solution can also be eleven coordinate (which in our notation is 7) obtained from the localized D2-D6 brane solution and since we have delocalization in coordinate 6 of [28] by a chain of dualities. as well, we end up with the semi-localized M5- It is useful to make a change of coordinates M5’ brane system. 2 2 lpz =(r sin α )/2N2,x = rcos α, 0 ≤ α ≤ π/2. The four dimensional theory at low energies N2 In the near-horizon limit we want to keep the on the D4-branes worldvolume is an SU(N1) r energy U = 2 fixed. This implies, in particular, gauge theory with matter in the bi-fundamentals. lp that membranes stretched between the M5 snd The metric (4.4) can be viewed as providing the M5’ branes in the ~z direction have finite tension eleven-dimensional supergravity description of an 4 | | 3 M5 brane with worldvolume R × Σ [29], where ~z /lp. Since we are smearing over the ~y direc- Σ is the Seiberg-Witten holomorphic curve (Rie- tions we should keep ~y/lp fixed. It is useful to mann surface) associated with the four dimen- make a change of variables ~y/lp → ~y. The near horizon metric is of the form of a warped product sional SCFT at the origin of the moduli space of of AdS5 and a six dimensional manifold M6 vacua [30]. In this brane set-up the R-symmetry × 2 group SU(2)R U(1)R is realized as the rota- 2 2 −1 3 2 3 U 2 / / 8910 × 45 ds11 = lp(4πN1) (sin α)( dx|| tion group SU(2) U(1) . The dimension- N2 less gauge coupling of each SU(N1) part of the 4πN1 2 2 2 + dU + dM ) , (4.4) gauge group is g ∼gsN2. U2 6 YM The ten dimensional metric describing the with elliptic Type IIA brane configuration is 2 2 2 2 2 sin α 2 2 −1 2 2 1 2 2 −1 2 2 dM =4πN1(dα +cos αdθ + dΩ ) / / / 6 4 2 ds10 = H4 dx|| + H4 d~x + H4 HNSdy

6 TMR9: Quantum aspects of gauge theories, supersymmetry and uniﬁcation Yaron Oz

1 localized system has been studied via a matrix description in [31]. In the following we make a NS5 few comments on this branes conﬁguration. It N 2 N 1 D4 2 can be viewed as a single M5 brane with worldvolume R4 × Σ, where Σ is a singular Riemann . surface deﬁned by the complex equation . . xN2 yN1 =0, (4.11)

. where x = x1 + ix2,y = y1 +iy2. Consider first the case N1 = N2 = 1. The complex equation xy = 0 can be written by a change of variables as 3 2 2 w1 + w2 = 0. This equation describes the singular Riemann surface at the point in the Seiberg- Witten solution [32] where a monopole becomes massless. The massless spectrum consists of a Figure 1: The Type IIA elliptic brane system corre- U(1) vector field and one hypermultiplet and the sponding to the semi-localized M5-M5’ brane config- theory at this point is free in the IR. Indeed, the uration. It consists of N2 NS5-branes with worldvol- chiral ring associated with the singular variety ume coordinates (0, 1, 2, 3, 4, 5) periodically arranged xy = 0 consists only of the identity operator. in the 6-direction and N1 D4-branes with world- Since the field theory is free in the IR we ex- volume coordinates (0, 1, 2, 3, 6) stretched between pect the dual string description to be strongly them. coupled. That means that the dual background is highly curved and the classical supergravity 1/2 2 + H4 HNSd~z , (4.7) analysis cannot be trusted. This theory has a dual string formulation [33]. The more general where 4 N1,N2 case is harder to analyse. The reason be- 4πg l N1N2 l N2 s s s ing that the singular locus of the variety (4.11) is H4 =1+ | |2 | |2,HNS = | | . [ ~x +2lsN2 ~z] 2 ~z not isolated and the notion of a chiral ring which (4.8) is used for an isolated singularity is not appropri- Its near-horizon limit is 2 2 ate. We do expect a non trivial conformal field 2 2 U 2 2 dU M2 theory in this case, but it has not been identified ds10 = ls 2 dx|| + R 2 + d 5 , (4.9) R U yet. where 2 2 2 2 2 2 sin α 2 5. BPS String Solitons dM = R (dα +cos αdθ + dΩ ) 5 4 2 N 2 The centrally extended N = 2 supersymmetry + 2 dy2 , (4.10) R2 sin2 α algebra in four dimensions takes the form [34] 2 1/2 and R =(4πgsN1N2) . The curvature of the { A ¯ } µ A µ A Qα , QαB˙ = σ ˙ PµδB + σ ˙ ZµB,, R∼ 1 Φ αα αα metric is l2R2 and it has a dilaton e = A B [AB] µν (AB) s {Q ,Q } = εαβZ + σ Z , (5.1) gsN2 −1 α β αβ µν R sin α . This supergravity solution can be A trusted when both the curvature and the dilaton where A, B =1,2andZµA =0.TheSU(2)R are small. Away from α = 0 we need large N2 part of the R-symmetry group acts on the in- 1/3 and N2 N1 . When the latter condition is dices A, B. In addition to the particle charge [AB] A not satisfied the dilaton is large and we should Z , there are in (5.1) the string charges ZµB consider the eleven-dimensional description. in the adjoint representation of SU(2)R and the (AB) The supergravity solution with both the M5 membrane charges Zµν in the 2-fold symmetric branes and the M5’ branes being completely lo- representation of SU(2)R.Thus,N=2super- calized has not been constructed yet. This fully symmetric gauge theories in four dimensions can

7 TMR9: Quantum aspects of gauge theories, supersymmetry and unification Yaron Oz have BPS strings and BPS domain walls in ad- becomes tensionless. dition to the well studied BPS particles. The BPS particles and BPS strings are real- H ized by stretching M2 brane between the M5 and M5’ branes. When the membrane worldvolume BPS strings coordinates are (0,x,y)wherexis one of the ~x components and y is one of the ~y components we get a BPS particle of the four dimensional theory. It is charged under U(1)R that acts on ~x and is a singlet under SU(2)R that acts on ~z. The BPS particles exist on the Coulomb branch SCFT of the gauge theory on which U(1)R acts, as in figure 3, and their mass is given by the Seiberg- Witten solution. C BPS particles When the membrane worldvolume coordinates are (0, 1,z)wherezis one of the ~z components Figure 2: There are BPS particles on the Coulomb we get a BPS string of the four dimensional the- branch and BPS strings on the Higgs branch. The ory. It is not charged under U(1)R and it trans- SCFT is at the intersection of these two branches. forms in the adjoint SU(2)R since ~z transforms in this representation. The BPS strings exist on We can use the AdS/CFT correspondence in the Higgs branch of the gauge theory on which order to compute the self energy of a string and SU(2)R acts, as in figure 3. The tension of the a potential between two such strings of opposite |~z| BPS string is given by 3 where |~z| is the dis- lp orientation [36]. This is obtained, in the super- tance between the M5 and M5’ branes and it is gravity approximation, by a minimization of the finite in the field theory limit. It is easy to see M2 brane action Z that we do not have in this set-up BPS domain p 1 µ ν walls since stretching a membrane with worldvol- S = 2 3 dτdωdσ detGµν ∂ax ∂bX (2π) lp ume coordinates (0, 1, 2) breaks the supersymme- (5.2) try completely. One expects BPS domain wall in the background (4.4), (4.5). Consider a static configurations when the moduli space of vacua configuration x0 = τ, x1 = ω and xi = xi(σ). has disconnected components. Then, the energy per unit length of the string is In the Type IIA picture the BPS string is given by Z constructed by stretching a D2 brane between p U sin α 2 2 2 E = 2 dU + U dα . (5.3) the D4 branes and NS5-brane. The seperation 4π N2 between these two types of branes in the ~z direction has two interpretations depending on whether Thus, the string self-energy the four dimensional gauge group has a U(1) part 2 E U ∼E N1 E = (α) (α) 2 . (5.4) or not [35]. If is does then the separation is in- N2 (δx||) terpreted as the SU(2)R triplet FI parameters ζ~. The function E parametrizes the dependence of The BPS string tension is proportional to |ζ~|.If the self-energy on the coordinate α. From the the gauge group does not have a U(1) part the field theory point of view, this parametrizes a seperation between these two types of branes in dependence on the moduli that parametrize the the ~z direction is interpreted as giving a vev to space of vacua. δx|| is a cut-off. In (5.4) we used an SU(2)R triplet component of the meson QQ˜ the holographic relation between a distance δx|| which transforms under SU(2)R as 2×2 = 3⊕1. in field theory and the coordinate U which in our The BPS string tension is proportional to this case reads vacuum expectation value. At the origin of the 1/2 (N2N1) δx|| ∼ . (5.5) moduli space, where we have SCFT, the string U

8 TMR9: Quantum aspects of gauge theories, supersymmetry and uniﬁcation Yaron Oz

This corresponds to the familiar relation [37] which again is in agreement with the ﬁeld theory

2 1/2 expectation. (gYMN1) δx|| ∼ . (5.6) The Type IIA background (4.10) is T-dual to U × 5 Type IIB on AdS5 S /ZN2 [30]. The latter is Similarly, the potential energy per units length N the dual description of the ZN2 orbifold of =4 between two strings of opposite orientation seper- theory [39]. For instance the number of degrees ated by distance L is of freedom can be understood as c(N =4)/N2 ∼ 2 N1 N1 N2. We can identify the spectrum of chiral E ∼− . (5.7) L2 primary operators with the supergravity Kaluza- Klein excitations as analysed in [40, 41]. here L is distance between two strings. The results (5.4),(5.7) do not depend on N2 which means that they do not depend on the gauge coupling 2 ∼ of the four dimensional ﬁeld theory gYM N2. Acknowledgments This is not unexpected. The gauge coupling can be viewed as a vacuum expectation value of a We would like to thank M. Alishahiha and A. scalar in the vector multiplet and does not ap- Brandhuber for collaboration on the work pre- pear in the hypermultiplet metric. Since the BPS sented in this talk. strings exist on the Higgs branch it is natural that their potential and self-energy do not depend on the gauge coupling. However, (5.4) and (5.7) are only the large N1 results and will presumably have 1/N1 corrections. As noted above, the theory on the intersection of two sets of M5 branes containes BPS particles, which arise from M2 branes stretched in the directions (0,x,y). Minimization of the M2 brane action in this case yields the potential between these two such objects of opposite charge separated by distance L

1/2 (N1N2) V ∼ . (5.8) L Again, this is expected since for the four dimen- ( 2 )1/2 ∼ gYMN1 sional ﬁeld theory it reads as V L . The 11-dimentional supergravity action goes like Z √ −9 − R∼ 2 lp g N1N2 (5.9) suggesting that the number of degrees of freedom 2 goes like N1 N2. This is also deduced from the two-point function of the stress energy tensor. N2 This is, of course, expected for SU(N1) gauge theory. It is curious to note that there is similar growth (N 3) of the entropy for a system of N parallel M5 branes [38]. TheR one√ loop correction has the following −9 − 6R4 ∼ form lp glp N2 which is suppressed 2 by N1 compared to the tree level action suggesting that the ﬁeld theory has a 1/N1 expansion,

9 TMR9: Quantum aspects of gauge theories, supersymmetry and uniﬁcation Yaron Oz

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