Comments on the N=1 SU(M+p)xSU(p) quiver gauge theory with flavor
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Comments on the N = 1 SU ( M + p ) × SU ( p )quiver gauge theory with flavor Francesco Benini a and Anatoly Dymarsky b a Department of Physics, Princeton University,Princeton, NJ 08544, USA b School of Natural Sciences, Institute for Advanced Study,Princeton, NJ, 08540
Abstract
We study supersymmetric vacua of the N = 1 cascading SU ( M + p ) × SU ( p ) gaugetheory with flavor – the theory on p D3-branes and M wrapped D5-branes at the tipof the conifold, and N f flavor D7-branes wrapping a holomorphic four-cycle insidethe conifold. The Coulomb branch of the moduli space is inherited from the puregauge theory without flavor and was thoroughly studied in the past. Besides, there isa Higgs branch where some D3 and/or D5-branes dissolve in the D7-branes formingthe worldvolume gauge instantons. We study the Higgs branch both from the fieldtheory and the bulk point of view. On the classical level the moduli space is closelyrelated to the one of the N = 2 C / Z orbifold theory, in particular certain vacua ofthe N = 1 theory are related to noncommutative instantons on the resolved C / Z .On the quantum level the Higgs branch acquires corrections due to renormalizationof the K¨ahler potential and non-perturbative effects in field theory. In the bulkthis is encoded in the classical D7-brane geometry. We compute the VEVs of theprotected operators and the field theory RG flow and find an agreement with theparallel computations in the bulk. ontents SU ( M + N ) × SU ( N ) theory 5 Z orbifold of N = 4 SYM with hypermultiplets . . . . . . . . . 103.2 Geometry of the D7-brane embedding . . . . . . . . . . . . . . . . . . . . . 143.3 Gauge field on the D7-brane and AdS/CFT dictionary . . . . . . . . . . . . 163.4 SUSY in the bulk and SUSY QM . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Asymptotics of the worldvolume gauge field in KS . . . . . . . . . . . . . . 223.6 SO (3) invariant flux on the D7-brane . . . . . . . . . . . . . . . . . . . . . . 233.7 Singular conifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.8 Resolved conifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.9 Deformed conifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.10 BGMPZ solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 N + N fL < N . . . . . . . . . . . . . . . . . . . 455.4 Quantum moduli space: 2 N + N fL ≥ N . . . . . . . . . . . . . . . . . . . 48 vs field theory 52 N f > A.1 Classical Higgsed mesonic directions with resolution . . . . . . . . . . . . . 61A.2 Quantum deformed Higgsed mesonic directions . . . . . . . . . . . . . . . . 63A.3 Quantum deformed Higgsed directions with resolution . . . . . . . . . . . . 66
B Page charges 67
B.1 Shift of Page charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
C Backreacted solution with massive flavors and worldvolume flux 70 Introduction
The low energy theory on D3-branes at a conifold singularity, studied by Klebanovand Witten (KW) in [1], has attracted significant attention during the last decade.This is the N = 1 SU ( N ) × SU ( N ) gauge theory with bifundamental fields and asuperpotential. Although the theory is strongly coupled, it has a simple gravity dualin the sense of the AdS/CFT correspondence [2, 3, 4]. Thanks to the simplicity of theoriginal setup and a multitude of possible variations, the KW theory has become astandard arena to study different theoretical and phenomenological phenomena. Letus mention here some of the main features that can be easily engineered within theconifold. The basic KW theory [1] is conformal. After the gauge group is modified to SU ( N + M ) × SU ( N ), the resulting Klebanov-Strassler (KS) theory exhibits a richdynamics [5, 6, 7, 8]. There is a logarithmic RG flow, which is UV complete withouta UV fixed point, and the theory has a chiral anomaly [9]. The flow takes placethrough a “cascade” [8] of Seiberg dualities [10] with many effective descriptions atdifferent scales. At low energy there is spontaneous chiral symmetry breaking andconfinement. The theory asymptotes to pure SU ( M ) Super-Yang-Mills (SYM) in theIR [8]. The theory is conjectured to have a meta-stable vacuum that dynamicallybreaks supersymmetry at an exponentially low scale [11], which might be relevant forphenomenological models [12]. Moreover this theory is a natural setup for models ofcosmological inflation [13, 14]. Adding “flavors”, i.e. fields in the (anti)fundamentalrepresentation, modifies the theory such that it asymptotes in the IR to Super-QCD(SQCD) with quartic superpotential, and the moduli space develops a Higgs branch.One can study the Veneziano limit of this theory [15, 16] which exhibits confinementand screening of charges [17] in the IR, and a “duality wall” in the UV [16]. Thegravity dual setup admits modes with localized wave-functions [18] which might berelevant for the Randall-Sundrum scenario [19, 20]. This list can go on and on.In this paper we focus on the flavored theory. In particular we study supersym-metric vacua of the cascading SU ( N + M ) × SU ( N ) theory with flavor – the theoryon N D3-branes, M fractional D3-branes (wrapped D5s) and N f flavor D7-branesinside the conifold, building on a similar analysis of the unflavored case [21]. To makeuse of the known holographic dual to the pure gauge theory, we keep the number offlavors N f much smaller (though possibly large) than the number of colors N + M throughout the paper. We focus on the vacua that are directly related to the presenceof flavor fields – the Higgs branch of the moduli space – which we study using con-2entional field theory tools as well as the dual holographic description. The N = 1supersymmetry is not enough to protect the Higgs branch from quantum corrections.Although the general structure of the classical moduli space stays intact, particularproperties of vacua, such as VEVs of various observables, get quantum corrections.The main goal of this paper is to perform a thorough analysis of Higgs vacua includingquantum effects on both sides of the duality and demonstrate how non-perturbativeeffects in field theory are manifest through the classical geometry in the bulk.The general structure of the moduli space is clear from the bulk point of view. Ifthe background has mobile D3-branes, there is a Coulomb branch associated with theirmotion on the conifold. When some D3-branes reach the D7s, they can dissolve intoworldvolume non-Abelian gauge “instantons” [22] with moduli that correspond to theHiggs branch. Besides there could be a (pseudo)-K¨ahler deformation of the conifoldmetric dual to the baryonic branch of the moduli space. There are also disconnectedbranches. For instance we can create extra D5 and D3-charge by putting worldvolumeflux at the tip of the D7s (to preserve the total charge one would need to adjust theflux at the conifold tip). Also, the mobile D3-branes can turn into Ramond-Ramond(RR) 5-form flux at the price of “shortening the throat”. We will find that all suchconfigurations have counterpart vacua in field theory.Our main interest is in the vacua associated with flavors, i.e. the D7-branes inthe bulk. The D7-branes we consider wrap a holomorphic four-cycle Σ which has thesame topology and complex structure as the Eguchi-Hanson space, albeit with a non-conventional non-Ricci-flat metric. As outlined above, the Higgs branch(es) are dualto non-trivial supersymmetric worldvolume gauge configurations on the D7s. In manycases we get conventional instantons, i.e. anti-self-dual gauge field configurations.These instantons on Σ bear close resemblance with the conventional instantons on C / Z as the two spaces coincide as complex manifolds, and so should coincide theHiggs branches of the two theories in most cases. The relation can be seen in fieldtheory: any Higgs branch solution to the N = 2 F- and D-term equations is alsoa solution to the N = 1 classical vacuum equations. However in certain cases the N = 1 supersymmetric gauge configurations satisfy non-linear equations [23]. Such“instantons” are not anti-self-dual and a priori we can not say much about theirmoduli space. Using the dual field theory we show that these “instantons” are relatedto the noncommutative instantons on C / Z .Although to explicitly find non-Abelian instantons in the N = 1 case is a difficult3ask, we find the explicit solutions for the Abelian U (1) instantons together with thecorresponding classical and quantum vacua in the dual field theory – in particular wesolve the ADHM equations of the N = 2 C / Z orbifold theory. Thus we completelycover the case of the field theories with N f = 1. This allows us to compute thequantum corrections in field theory and compare the results with gravity. For generic N f the non-Abelian instantons will emerge but we do not expect this to introduceany qualitatively new feature.The RG flow of the theory, except in very special situations, is controlled by acascade of Seiberg dualities [24, 16], in a similar but more articulated way than inthe unflavored KS case [8, 25]. A new feature is that, as we change the effectivedescription at each step of the cascade, we also get a non-trivial map between thevarious Higgs branches of the moduli space. The Seiberg duality on gravity side ismanifest through the large gauge transformation of the B-field which nicely reproducesthe map between the vacua.Finally we consider the fully backreacted supergravity solutions for smeared (pos-sibly massive and with worldvolume flux) D7-branes on the conifold [15, 16, 17, 31].We exploit such solutions to study the RG flow and show that gravity exactly repro-duces the field theory NSVZ β -functions [32] in all vacua.The paper is organized as follows. In section 2 we review the conformal KW andthe cascading KS theories without flavor, gaining enough familiarity to be ready toadd probe flavor D7-branes in section 3. First we digress to consider the N = 2 C / Z orbifold theory, which gives us basic intuition about the moduli space of instantons.Then we move to study the D7-branes inside the conifold, and find the solutions to thelinearized perturbations of the worldvolume gauge fields thus building the AdS/CFTdictionary in the sense of [3, 4]. Finally we explicitly construct the U (1) instantonsfor D7-branes in all SUSY vacua of the KW/KS theories and calculate the VEVs ofthe protected operators from the flavor sector. In section 4 we go beyond the probeapproximation and compute the backreaction of the flavor branes on the geometryin the Veneziano large N limit, with N f /N small but fixed. To solve the equationswe place the D7s in a way that preserves the isometries of the conifold. We read offthe RG flow, corrected by the flavors, to compare with the dual gauge theory laterin section 6. This ends the gravity analysis. In section 5 we study the moduli space Also the N = 2 C / Z orbifold theory admits cascading RG flows [26, 27, 28, 29, 30] althoughthe physics is different than in the N = 1 case. N = 1 field theory techniques. We first consider the action of Seiberg duality,then we perform a classical analysis of the moduli space and eventually we includequantum effects. Finally section 6 is devoted to the comparison between gravity andfield theory results. We draw our conclusions in section 7. Various computations areexiled to appendices. SU ( M + N ) × SU ( N ) theory This section is a review of the unflavored conifold theories and might be skipped bya reader familiar with the subject. A thorough discussion of these theories can befound in [33, 34, 21].
Following [1] we start by placing a stack of N D3-branes at the tip of the conicalsingularity X i =1 z i = 0 . (2.1)The resulting field theory on the D3-branes is an N = 1 superconformal quiver gaugetheory with gauge group SU ( N ) × SU ( N ) and global symmetry SU (2) A × SU (2) B × U (1) R × U (1) baryon . Besides the vector multiplets there are bifundamental fields A α , B ˙ α in the ( N , N ) and ( N , N ) representations with charges ( , , ,
1) and ( , , , − W KW = 12 h ǫ αβ ǫ ˙ α ˙ β Tr A α B ˙ α A β B ˙ β . (2.2)At the conformal point the theory is always strongly coupled, and the conformalmanifold is described by h ( g , g ) [25].The moduli space can be found from the F- and D-flatness conditions. The formerimplies the matrix equation ǫ αβ ǫ ˙ α ˙ β A α B ˙ α A β B ˙ β = 0 . (2.3)It is convenient to introduce the new variables w ˙ αα = w w w w ! = √ h B ˙ α A α , (2.4)5here the prefactor has been introduced for convenience, and rewrite the F-flatnesscondition in the form det w ˙ αα = 0. This coincides with the conifold equation (2.1).Assuming the matrices A, B are diagonal, the F-flatness condition describes the mo-tion of N D3-branes on the singular conifold. The D-flatness condition is A A † + A A † − B † B − B † B = U A † A + A † A − B B † − B B † = U , (2.5)where both identity matrices are N × N and U is a constant. For U = 0 a genericsolution – up to gauge equivalencies – describes N points on the singular conifold; for U = 0 the solution describes N points on the resolved conifold. The resolved conifoldis the singular conifold with S blown up at the tip. Instead of det w ˙ αα = 0 the spaceis described by the equation ( w ˙ αα ) ν ν ! = 0 (2.6)with ( ν , ν ) ∈ CP . For w i = 0, the space is bi-holomorphic to (2.1), while at w i = 0we have a non-trivial CP . The resolved conifold has the same complex structure asthe singular conifold but different metric [35].The dual geometry in ten dimensions is a warped product of Minkowski space andthe Ricci-flat conifold ds = h − / dx + h / d ˜ s . (2.7)In fact there is a one-parameter family of Ricci-flat metrics on (2.1). The simplestone is the cone over the T , (a homogeneous Sasaki-Einstein space) d ˜ s = dr + r ds T , . (2.8) T , can be defined as a quotient SU (2) × SU (2) U (1) which makes the global SU (2) A × SU (2) B symmetry manifest (it is also invariant under U (1) R ). The remaining global U (1) baryon of the field theory is not geometrical. Topologically T , ∼ = S × S , and one can definethe generators of H ( T , , Z ) and H ( T , , Z ) Z S ω = 4 π , Z S ω = 8 π . (2.9)Metrically T , can be represented as a U (1) fibration over S × S .The geometry (2.7)-(2.8) is the singular conifold. It is invariant under the Z symmetry that flips the sign of z . This symmetry exchanges the two S in the base6f T , . On the field theory side this symmetry exchanges A α ↔ B † ˙ α accompanied bya charge conjugation. This symmetry flips the sign of U and is spontaneously brokenif U = 0. Hence the singular conifold corresponds to the vacuum with U = 0. Thevacua with U = 0 correspond to the resolved conifold geometry [36].The supergravity background is of the GKP type [37] and the warp factor dependsonly on the location of the D3-branes on the conifold: − ˜ ∇ h = (4 π α ′ ) N X i ∈ D3-branes δ (6) ( x − x i ) , (2.10)where tilde corresponds to the unwarped metric d ˜ s . The AdS × T , solution corre-sponds to h = L r , i.e. all D3-branes located at the singularity r = 0. As evident fromthe field theory the D3-branes can move anywhere on the conifold. The correspondingbackground is given by (2.7)-(2.10) and the RR form C = h − dx ∧ · · · ∧ dx .In the case with U = 0 in (2.5) the D3-branes can still freely move, and the warpfactor is determined by (2.10). If all D3-branes are smeared on the S at the tip, SU (2) A × SU (2) B is preserved but the solution is singular [38]; if the D3-branes arelocalized, the solution is regular but the global symmetries are broken [39].In the dual geometry the gauge couplings are controlled by the value of the dilaton e φ and the flux of the B -field through S g + 1 g = 14 πe φ , g − g = 12 πe φ h b −
12 (mod 1) i , (2.11)where we defined b = π α ′ R S B . The background with vanishing B -field correspondsto g = ∞ and b = 1 / g = g . The conformal SU ( N ) × SU ( N ) KW theory can be generalized to SU ( N + M ) × SU ( N ) gauge group. The theory is no longer conformal but instead experiences acascade of Seiberg dualities, each decreasing the rank of the gauge groups by M .Each description gives rise to a branch of perturbative vacua given by the deformedconifold equation X i =1 z i = det w ˙ αα = ǫ , (2.12)7here w ˙ αα is defined as in (2.4) and the constant ǫ is related to the scales Λ , ofthe gauge sector. The eigenvalues of w ˙ αα parametrize the locations of D3-branes onthe deformed conifold. The chiral U (1) R symmetry is broken to Z M by the anomaly,and further spontaneously broken to Z by a gaugino condensate that gives rise to M vacua. The remaining Z stays unbroken. The whole moduli space is the collectionof the mesonic branches [21]Moduli space = ⊕ kl =0 ⊕ Mr =1 Sym N − lM ( C r,l ) , (2.13)where k = [ N/M ] − is the number of steps in the cascade, r labels the values of thegaugino condensate and C r,l is the deformed conifold with the deformation parameter ǫ r,l = ǫ e πi rM I lM [21, 40]. The RG-invariant parameter I of the field theory is dualto the string coupling constant I = e πiτ . In the regime g s M ≫ I lM = 1 at the leading order in g s M . Since all branches with different r areequivalent, in what follows we drop the index of the deformation parameter r, l andassume real ǫ .In the special case N = kM , the IR gauge group reduces to SU (2 M ) × SU ( M )and this requires a special treatment. The strongly coupled SU (2 M ) group has asmany colors as flavors, and its moduli space is described by mesons M ˙ αα = B ˙ α A α and baryons A = 1( M !) ǫ i ··· i M ǫ j ··· j M ǫ k ··· k M ( A ) i j . . . ( A ) i M j M ( A ) i M +1 k . . . ( A ) i M k M B = 1( M !) ǫ i ··· i M ǫ j ··· j M ǫ k ··· k M ( B ) j i . . . ( B ) j M i M ( B ) k i M +1 . . . ( B ) k M i M , (2.14)which are singlets of SU ( M ) × SU (2) A × SU (2) B , subject to the quantum constraintdet M ˙ αα − AB = Λ M . The constraint can be enforced by a Lagrange multiplier X and the superpotential W eff = W KW + X (det M ˙ αα − AB − Λ M ) . (2.15)There are two distinct branches resulting from (2.15). If X = 0, F-flatness requires A = B = 0 and w ˙ αα must satisfy det w ˙ αα = ǫ . This is one of the mesonic branchesdiscussed before. If X = 0, the F-flatness condition requires M ˙ αα = 0 and hence AB = − Λ M . This is the baryonic branch. It has one complex dimension and can We define [ x ] − as the largest integer less than or equal to x .
8e parametrized by the VEV of the baryons. Therefore in (2.13) the possible factor
Sym ( C r,k ) is assumed to be the baryonic branch C .The gravity dual of the SU ( M + N ) × SU ( N ) theory is the Klebanov-Strasslersolution [8], possibly generalized by extra mobile D3-branes. It is of the GKP typewith metric (2.7) and RR five-form, where d ˜ s is the Ricci-flat metric on the deformedconifold. Besides, the solution also has RR and NSNS three-forms. The solution isengineered by placing M fractional D3-branes and N regular D3-branes at the conifoldsingularity and is characterized by 14 π α ′ Z S F = M (2.16)while F is running. The pure KS solution has no mobile D3-branes and it is invariantunder the Z symmetry. Hence it corresponds to the point A = B of the baryonicbranch [27]. The rest of the baryonic branch is given by the BGMPZ solutions [41].They have metric ds = e A dx + ds , where e − A ds is some pseudo -K¨ahler metricon the deformed conifold, running dilaton and the three-form flux is not imaginary-self-dual. The VEV of the baryons A , B is related to the D-term parameter U . Belowthe scale of baryon VEV the gauge symmetry is broken to SU ( M ). That is why forlarge U the geometry near the tip approaches the MN solution [42, 43] dual to the SU ( M ) SYM [41, 21, 44].To describe the solutions dual to the mesonic branch we need to introduce mobileD3-branes on the conifold. As in the KW case, the extra p D3s only affect warpfactor and 5-form flux, through the same equation (2.10) where now h tot = h KS + h . While the original solution is dual to SU (cid:0) ( k + 1) M (cid:1) × SU ( kM ), the new one is dualto SU (cid:0) ( k + 1) M + p (cid:1) × SU ( kM + p ). Unless p = 0 (mod M ), the two theories aredifferent. The new theory does not have a baryonic branch. If we put D3-branes onthe BGMPZ solution SUSY is broken and the baryonic branch is lifted by a potentialthat returns the system to the vacuum described by the KS solution with mobileD3-branes [21]. If p = 0 (mod M ), the new solution describes one of the mesonicbranches of the original SU (cid:0) ( k + 1) M (cid:1) × SU ( kM ) theory.In conclusion let us mention here that besides the regular Klebanov-Strassler grav-ity background discussed above there is an “approximate” version of this backgroundfound by Klebanov and Tseytlin (KT) [7]. This background approaches KS in theUV but is singular in the IR. Although it does not correctly describe physics at low Such background can be solved explicitly [45, 46].
In this section we add probe D7-branes to the conifold backgrounds. In particular weexplicitly construct the Abelian U (1) instantons which are dual to the Higgs vacuain the field theory with N f = 1. Z orbifold of N = 4 SYM with hypers
Before adding D7-branes to the conifold theory, let us consider a simpler but closelyrelated example of the N = 2 Z orbifold of N = 4 SYM with flavors. We start withthe N = 4 SU ( N ) SYM theory which lives on N D3-branes. Then we add a smallnumber N f ≪ N of D7-branes [47] that span the Minkowski space R , and wrapthe holomorphic non-compact cycle Σ = C ⊂ C : they add N f hypermultiplets inthe fundamental representation, and break SUSY to N = 2. Choosing coordinates z , z , z on C , the embedding Σ = { z = m } introduces hypermultiplets of mass m . The D3-branes are free to move on C , and their positions parametrize theCoulomb branch. When k D3-branes reach the D7s, they can dissolve into them if N f > k non-Abelian U ( N f ) instantons. This corresponds to Higgsing SU ( N ) → SU ( N − k ). The field theory analysis of the moduli space relies on the F-and D-term equations[Φ , Φ ] + Q ˜ Q = 0 , [Φ , Φ † ] + [Φ , Φ † ] + QQ † − ˜ Q † ˜ Q = 0 (3.1)together with Φ = m for the k × k block of the N × N matrices Φ i . These equa-tions exactly coincide with the ADHM description of the moduli space of k U ( N f )instantons – the worldvolume gauge instantons on the D7s [48, 22]. The equivalencebetween field vacuum equations and the ADHM construction (see [49] for a pedagog-ical review) is at the core of the holographic description of the Higgs branch. Besidesthe moduli space itself, it can be extended to various observables in field theory: themoduli space metric, chiral operators, etc [50, 51, 52, 53].10hen we take a Z orbifold of Σ = C . The resulting geometry is singular, butit can be smoothened out. One can parametrize C / Z by two complex variables( w , w ) subject to identification ( w , w ) ∼ ( − w , − w ). Alternatively one can intro-duce invariant coordinates z , = ( w ± w ) / z = iw w , subject to the constraint P i =1 z i = 0. The singular orbifold admits a simultaneous deformation of the complexstructure X i =1 z i = ǫ (3.2)and a resolution: both replace the singularity by a finite size S . This is the smoothEguchi-Hanson space. Deformation and resolution are measured by the self-dualforms ω (2 , and J (1 , : Z S ω (2 , ≡ ξ C = ǫ , Z S J (1 , = ξ R . (3.3)The resolution and deformation parameters ξ R , ξ C transform as a triplet under SU (2) R that rotates the complex structures on the hyper-K¨ahler Eguchi-Hanson space.Since the deformed/resolved orbifold has an exceptional 2-cycle S , it admits U (1)instantons. Hence the orbifold theory has a Higgs branch even for N f = 1. In generalthe U ( N f ) instantons on C / Z are characterized by the first and second Chern classesch = 12 π Z S Tr F , ch = 18 π Z C / Z Tr F ∧ F (3.4)and the conjugacy class of the monodromy matrix ˆ ρ : Z → U ( N f ). The latter isdefined as follows. One considers a radial section S / Z of the orbifold at infinity,where F = 0, and computes the holonomy ˆ ρ = Pexp i H ∂ Γ A ∈ U ( N f ) along thegenerator ∂ Γ of π ( S / Z ) = Z . Such a matrix must satisfy ˆ ρ = 1I, and its conjugacyclass is a gauge-invariant observable.The gauge instantons on the D7s’ worldvolume are D3-branes on their Higgsbranch, dissolved in the D7s. The corresponding moduli space was analyzed from theD-brane point of view in [54], showing that it agrees with the ADHM constructionput forward by Kronheimer and Nakajima [55] (see also [56, 57] for a review). As inthe case of pure SU ( N ) SYM with flavor, we can reproduce the ADHM quiver andequations by analyzing the vacuum equations of the field theory. The Z orbifoldgives an SU ( N ) × SU ( N ) quiver theory with N fL left and N fR right flavors, N f = N fL + N fR . Invariance under the Z orbifold action dictates that only the non-diagonal11 × N blocks of Φ , are non-vanishing, while Φ is block diagonal:Φ α = A α ǫ αβ B β ! , Φ = φ − ˜ φ ! . (3.5)The fields with index α = 1 , SU (2) symmetry, while SU (2) R acts on ( A, B † ) as a doublet. The resulting superpotential is W = φ ( A α B α − Q L ˜ Q L ) + ˜ φ ( B α A α + Q R ˜ Q R ) , (3.6)where sum over α is implicit.The N D3-branes can freely move on C / Z × C , realizing the Coulomb branch;when the D3s reach the D7s they can dissolve turning into instantons, and Higgsingpart of the gauge symmetry. Let us denote with k , the ranks of the broken symmetry.They might be different, corresponding to the presence of D5-branes dissolved in theD7s and wrapping the 2-cycle of C / Z . The Higgsed directions of Φ , are the k × k and k × k blocks where φ , ˜ φ are equal to m multiplied by the k × k and k × k identity matrices to allow non-trivial values of flavor fields. Eliminating Φ , the F-and D-term equations describing the Higgs branch effectively represent the quiver infigure 1, which we will concisely denote as N fL × k × k × N fR .To present the F- and D-term equations in a concise form we define the combina-tions C α = B α A α † ! , P L = − Q L ˜ Q † L ! , P R = Q R ˜ Q † R ! , (3.7)and use the Pauli matrices Γ µ = (cid:0) (cid:1) , (cid:0) − ii (cid:1) , (cid:0) − (cid:1) to represent the F-term andD-term equations in a SU (2) R covariant form C † α Γ µ C α + P L Γ µ P † L = − ξ Lµ , C α Γ ∗ µ C † α + P R Γ ∗ µ P † R = − ξ Rµ . (3.8)Here † acts on gauge indices, while transposition of SU (2) R indices is implicit. Ingeneral ξ L , ξ R should be understood as some parameters of the solution. In the caseof the U ( N ) × U ( N ) orbifold theory these are the Fayet-Iliopoulos (FI) terms. If k = k = N i.e. there is no remaining unbroken gauge group ξ L and ξ R can beturned on independently. Otherwise ξ L = ξ R = ξ as follows from the components of(3.8) with trivial Q . In the bulk the triplet ξ µ controls the resolution/deformation of C / Z as seen from (3.3). 12 fL N fR k k A α B α ˜ Q R Q R ˜ Q L Q L Figure 1: ADHM quiver describing instantons on the Eguchi-Hanson space C / Z .We will concisely refer to this quiver as N fL × k × k × N fR .In components the equations (3.8) are A α B α − Q L ˜ Q L = ξ C , A α A α † − B † α B α + Q L Q † L − ˜ Q † L ˜ Q L = ξ R ,B α A α + Q R ˜ Q R = ξ C , B α B † α − A α † A α + Q R Q † R − ˜ Q † R ˜ Q R = − ξ R , (3.9)where we defined ξ R ≡ ξ , ξ C = − ( ξ − iξ ) / N fL , N fR , k , k in field theory and the propertiesof the instanton in the bulk is as follows. The conjugacy class of the monodromymatrix ˆ ρ defines splitting of N f into N fL,R . Since ˆ ρ = 1I, its eigenvalues are ± N fL,R = Tr(1I ∓ ˆ ρ ) /
2. The ranks k , k are related to the Chern classes (3.4) as follows[54]: ch = 2( k − k ) − N fL , ch = k + N fL . (3.10)Finally, the dimension of the moduli space of instantons with given ch , and ˆ ρ , i.e. k , k , N fL,R , is equal todim M = 4 (cid:0) N fL k + N fR k − ( k − k ) (cid:1) . (3.11)The quiver in figure 1 as well as its space of vacuums are invariant under a Z flipthat exchanges the left and right groups: k ↔ k and N fL ↔ N fR . Although suchsymmetry is trivial in the field theory, it acts non-trivially on the space of instantons.It multiplies ˆ ρ by − → − N f − ch , ch → N f k + k ) − ch . (3.12)The relation between geometric properties of instantons and ranks k , k , N fL,R of the quiver provides a simple holographic picture. The splitting of flavors intoleft and right is determined by the worldvolume gauge field on the D7s and the Our description differs from the one in [54] by the sign in the definition of ch and a Z flip ofthe quiver. ρ . Above the scale m the field theory has gaugegroup SU ( N ) × SU ( N ) with N f hypermultiplets while below m the gauge theory ispure SU ( N − k ) × SU ( N − k ). The resulting low-energy gauge theory is describedholographically by min ( N − k , N − k ) D3-branes and | k − k | D5-branes wrappinghomologically non-trivial S of C / Z .To get some intuition about how k , k , N fL,R are related to the instanton charges,let us consider a simple Abelian instanton of charge n , i.e. a U (1) gauge field withch = n . The charge n is integer while one finds ch = n and ˆ ρ = ( − n [54]. Wehave two distinctive cases. When n = 2 r − × r × r ( r − × , (3.13)while when n = 2 r is even the quiver is0 × r ( r + 1) × r × . (3.14)The explicit matrices that solve the quiver equations in these cases can be found insection 5.2.After we developed some intuition in the N = 2 case we return our attention tothe conifold geometry in the next section. Throughout this paper we consider D7-branes along the so-called Kuperstein embed-ding [58] – a holomorphic non-compact 4-cycle Σ defined by z = µ √ . (3.15)If we start with the C / Z × C N = 2 case and the usual D7-brane discussed in theprevious subsection and introduce the massive deformation of the orbifold theory thatleads via the RG flow to the conifold theory the original D7-brane result in a D7-braneembedded along (3.15) [24]. This provides us with the field content and superpotentialof the flavor sector. A stack of k D7-branes introduces k flavors of hypermutiplets˜ Q, Q in the fundamental of one of the gauge groups, with superpotential of the form W flavor ∼ ˜ Q ( A B + A B − µ ) Q + ˜ QQ ˜ QQ . (3.16)14e will be more precise in sections 5 and 6, clarifying also under which gauge groupthe quarks are charged.The embedding (3.15) preserves the anti-diagonal SU (2) AB of the global SU (2) A × SU (2) B symmetry. In what follows we assume that the deformation parameter ǫ andmass µ are non-zero. The corresponding limits of singular conifold ǫ = 0 or zero massembedding z = 0 are straightforward.The 4-cycle Σ has the complex structure of C / Z with deformation parameter ǫ − z
24 3 X i =1 z i = ǫ − z . (3.17)As we discussed in section 3.1, at infinity this space approaches a cone over S / Z ,and given a flat bundle on it one can construct a monodromy matrix ˆ ρ = Pexp i H ∂ Γ A whose conjugacy class is a gauge-invariant. We can also think of S / Z as a Hopf S fibration over S , with S shrinking at the tip of Σ while S staying finite. Hence Σcan support Abelian flux.We can parametrize Σ by the radial coordinate of the conifold X i =1 | z i | ≡ r ≡ ǫ cosh t , (3.18)which takes value in the range r ≥ | z | + | ǫ − z | , together with some angularcoordinates on S / Z . In practice it is convenient to use the one-forms g , dz i of thefull conifold geometry pulled-back on Σ. In terms of the usual conifold coordinates,let us define g = dψ − X i =1 , cos θ i dϕ i , Vol i = sin θ i dθ i ∧ dϕ i . (3.19)Expressing z through z , z one finds the following useful relations [23] − dg ∧ dg (cid:12)(cid:12)(cid:12) Σ = f ( t ) dt ∧ g ∧ dg (cid:12)(cid:12)(cid:12) Σ = 4 | z cosh t − ¯ z | ǫ sinh t | z | dz ∧ d ¯ z ∧ dz ∧ d ¯ z (cid:12)(cid:12)(cid:12) Σ (3.20)where the function f ( t ) = − | z cosh t − ¯ z | sinh t ( ǫ sinh t − | z | cosh t + z + ¯ z ) (3.21)is defined through a ′ / a = − f , and a ( t ) is a “volume” of Σ at the given radius t Z S / Z at t g ∧ dg = 32 π a ( t ) = 32 π ǫ sinh t − | z | cosh t + z + ¯ z ǫ sinh t . (3.22)15et us remark that on the deformed conifold the 2-form dg is singular at the tip, asone can check by computing the norm | dg | using the inverse metric: the magnitudediverges as 1 /t (whilst g is regular). In the case of the massless embedding z = 0,the pull-back of dg is likewise singular at the tip. Therefore when expressing a gaugefield on Σ, we should be careful to ensure that the coefficient in front of dg vanishes.Let us stress that the geometry of Σ on the deformed conifold is regular when z = 0,and all physical quantities should be continuous in this limit.A similar subtlety arises in the resolved conifold case. Since at the tip one of the2-spheres in the base S × S of T , (as U (1) fibration) vanishes, the correspondingvolume form – say Vol – diverges. This again can be checked by computing | Vol | .As a result, both 2-forms dg = Vol + Vol , ω = 12 (Vol − Vol ) (3.23)are divergent at the tip of the resolved conifold. An easy way to avoid the difficultyis to combine dg and ω at the tip into the volume form Vol = dg + ω which iswell-defined. We also remark that the limit z → C / Z singularity, however theresolution of the conifold induces a resolution of Σ, and in fact the blown-up 2-sphereof the conifold coincides with the blown-up 2-sphere of Σ. In this section we solve the linearized equations for SO (3) invariant fluctuations ofthe worldvolume fields on Σ and identify them with field theory operators accordingto the AdS/CFT correspondence. We will be mainly concerned with the UV (large r ) behavior of the bulk fields, therefore we will work in the singular conifold limit ǫ = 0. We can introduce a set of real coordinates r, X I , Y I (with I = 1 , · · · ,
4) on theconifold z I = r / ( X I + iY I ) (3.24)where X I , Y I are subject to the constraints X = Y = 12 , X · Y = 0 . (3.25)The base T , of the singular conifold is represented as the product of two 3-sphereswith an orthogonality condition and metric ds T , = 23 ( dX + dY ) −
29 (
XdY − Y dX ) . (3.26)16n order to introduce local coordinates on Σ and calculate the induced metric werepresent the conifold as a foliation of the Kuperstein embeddings parametrized by X , Y , the radial coordinate r and three angular coordinates t i . First we fix z = µ ( r ) √ r / ( X + iY ) . (3.27)We can think of this equation with X , Y ( r ) as a parametrization of a generic SO (3)-invariant embedding Σ. Then we arbitrarily choose X (0) I , Y (0) I , for I = 1 , , t i as the “Euler angles” of the SO (3) rotation which transforms the point X (0) + iY (0) into some other point on S / Z . We can use the conventional 3 × T i of so (3) embedded into the upper left corner of the 4 × z I as follows X + iY = e t i T i ( X (0) + iY (0) ) . (3.28)Clearly this transformation leaves z invariant. Then the tangent vector is d ( X + iY ) = dt i T i ( X + iY ) + ∂ ( X + iY ) ∂X dX + ∂ ( X + iY ) ∂Y dY . (3.29)The one-forms dt i are the left-invariant one-forms e i on S ∼ = SU (2) calculated at theorigin e i = dt i + ǫ ijk t j dt k + O ( t ) . (3.30)To obtain the expression valid everywhere on S / Z we can simply substitute dt i by e i . Now, if we substitute (3.29) into the conifold metric ds = dr + r ds T , with(3.26), we obtain the metric in terms of ( r, X , Y , t i ). If we instead interpret X , Y as radial functions defined by the generic SO (3)-invariant embedding µ ( r ), we obtainthe induced metric on Σ. In the special case µ = const the unwarped metric on Σ is ds = 4 r − | µ | r − | µ | ) dr + r − | µ | r e + r e + 4 r − | µ | r e . (3.31)Our next step is the quadratic action for the fluctuation of the worldvolume gaugefield given by Z d x dr h √ g g AB g A ′ B ′ F AA ′ F BB ′ − h − Pf F i . We chose the parametrization X (0)1 = p / − X , X (0)2 = 0 , X (0)3 = 0 and Y (0)1 = − X Y / p / − X , Y (0)2 = p / − X − Y / p / − X ) , Y (0)3 = 0. A, B run through the Minkowski and internal r, e i directions. Theinduced metric g AB is a warped product of the flat Minkowski metric and the metric(3.31) on Σ. The Pfaffian Pf F is calculated with the 4 × F AB with all indexestaken along the internal directions r, e i .We are focusing on the lowest SU (2)-invariant modes in the KK expansion. Thecorresponding woldvolume gauge field can always be brought to the form A = A µ ( r, x µ ) dx µ + A i ( r, x µ ) e i , (3.32)with vanishing component along dr . To fix the residual gauge symmetry we requirethe Minkowski vector A µ to be transverse, ∂ µ A µ = 0, therefore A splits into a trans-verse space-time vector and three space-time scalars. The effective Lagrangian (to beintegrated over space-time and radius from r = | µ | to infinity) for A µ is L A µ = 4( r − | µ | ) | ∂ r A µ | + h (4 r − | µ | ) | ∂ ν A µ | . (3.33)The Lagrangian for the scalars A i is L A i = 12 h (cid:0) ρ i A i + ρ − i A ′ i (cid:1) + (4 r − | µ | )2( r − | µ | ) ρ − i | ∂ µ A i | ρ = 32 r , ρ = 3 r r − | µ | ) , ρ = (4 r − | µ | )2 r ( r − | µ | ) . (3.34)The linear in derivative term h − A i ∂ r A i in L A i comes from the CS term in the action.Eventually the Lagrangian for the perturbation δµ of the geometrical profile, i.e. µ = const + δµ , is L δµ = 4 r ( r − | µ | )(4 r − | µ | ) | ∂ r δµ | + hr | ∂ µ δµ | . (3.35)We can now analyze the resulting equations for the fluctuations A µ , A i , δµ andidentify the dual field theory operators. As a by-product we will also find the massspectra for the corresponding mesons in the KW case. Using the explicit form ofthe KW warp factor h = L r we find the asymptotic static (space-time independent)solutions A = c r − / + c r − / , A µ = c + c r − + O ( r − ) ,A = ( r − | µ | ) − / ( c + c r − ) , δµ = c + c r − + O ( r − ) ,A = ( r − | µ | ) − / r − / h c + c (cid:16) log r + | µ | r (cid:17)i . (3.36)18he asymptotic behavior in AdS of a canonically normalized field φ ( r ) dual to anoperator of dimension ∆ is φ ( r ) ∼ c source r ∆ − s + c VEV r − ∆+ s (3.37)where s = 0 for a scalar and s = 1 for a vector. This reveals that the vector A µ is dualto an operator of dimension 3 (the conserved current J µ of the flavor U (1) symmetry)and the scalar A is dual to an operator of dimension 2 (the bottom component | Q | − | ˜ Q | of the U (1) current multiplet). These operators, as we show in section3.4, are manifestly related in the bulk by a SUSY transformation. We used Q, ˜ Q forthe bottom component of the corresponding chiral superfields.The real and imaginary parts of δµ are degenerate since µ is dual to a complexchiral superfield in field theory. It follows from (3.36) that δµ corresponds to operatorsof dimension either 3 / /
2. To distinguish between the two [36] we notice that,because of the superpotential (3.16), µ couples to the operator R d θ ˜ QQ of dimension5 /
2, and by AdS/CFT this is the operator dual to δµ . The fluctuations A , A (whichafter an appropriate change of variables satisfy the same equation) combine into acomplex scalar dual to the bottom component ˜ QQ – an operator of dimension 3 / A , ∼ r − / is dual to the VEV of ˜ QQ , while the sub-leading A , ∼ r − / to the source of ˜ QQ in the Lagrangian. Was one interested incalculating the mass spectrum of the corresponding meson excitations, such bound-ary conditions would lead to a complication because one would have to ensure thatthe subleading asymptotic vanishes. It is more convenient to calculate the spectrumof the superpartner δµ , since four-dimensional SUSY guarantees the degeneracy ofmasses within the multiplet. In the bulk this follows from the SUSY Quantum Me-chanics transformation that relates the equations for δµ and A , and also for A µ and A . To see how the supersymmetric quantum mechanics works, let us consider a familyof one-dimensional effective actions of the form There might be logarithms in (3.37) as in A from (3.36) if the two series expansions overlap. We thank D. Melnikov for his input on the following subsection. = Z dr (cid:0) F ψ ′ − Hψ − m Gψ (cid:1) , (3.38)where F, H, G are functions of r and ′ denotes derivative with respect to r . To bringthe corresponding EOM to the canonical form we perform the change of variables ψ = φ √ F resulting in φ ′′ − V φ = − m GF φ , V = F ′′ F − F ′ F − HF . (3.39)In fact the potential V can be expressed as V = W ′ + W − HF (3.40)with the function W given by W = 12 (log F ) ′ + (cid:20) F (cid:16) const + Z r F − (cid:17)(cid:21) − . (3.41)In all cases below the “const” in the formula above will be infinite and W = (log F ) ′ .If H = 0, then V is entirely captured by the superpotential W . In this case theequation (3.39) can be written in a form that makes the SUSY QM explicit Q Q φ = − m φ , (3.42)with Q = α (cid:18) ddr + W − (log α ) ′ (cid:19) , Q = α (cid:18) ddr − W (cid:19) , α = FG . (3.43)Clearly equation (3.42) has a superpartner which shares the same mass spectrum (upto a possible zero-mode m = 0) Q Q φ = − m φ . (3.44)This equation can be written in the canonical form (3.39) using new functions ˜ F , ˜ H, ˜ G .In this case ˜ F / ˜ G = F/G and the new potential is˜ V = ˜ F ′′ F − ˜ F ′ F − ˜ H ˜ F = ( − W ′ + W ) + α ′′ − W α ′ α . (3.45)Let us apply this to the equations for A µ , A i , δµ . We start with the equations for A , and cast them in the form (3.39). It turns out that for both modes the potential V vanishes and the equations coincide φ ′′ = − m L (4 r − | µ | )4( r − | µ | ) r φ . (3.46)20he equation governing the complex scalar δµ brought to the canonical form is: φ ′′ = − r (16 r + r | µ | − | µ | ) | µ | r − | µ | ) (4 r − | µ | ) φ − m L (4 r − | µ | )4( r − | µ | ) r φ . (3.47)This equation is the SUSY QM partner of (3.46): if we compute the effective potentialfor the superpartner of (3.47) using (3.45) and F = 4( r − | µ | ) r (4 r − | µ | ) , H = 0 , α = FG = 4( r − | µ | ) r (4 r − | µ | ) , (3.48)we find that ˜ V vanishes and we arrive at (3.46).The equation for A µ written in a canonical form is φ ′′ = 3 r ( r − | µ | )4( r − | µ | ) φ − m L (4 r − | µ | )4( r − | µ | ) r φ . (3.49)Using H = 0 and F = 4( r − | µ | ) we can calculate the potential for the superpartnerequation arriving at φ ′′ = − r − r | µ | − | µ | ) r (4 r − | µ | ) φ − m L (4 r − | µ | )4( r − | µ | ) r φ , (3.50)which is the equation for A written in a canonical form.Notice that SUSY QM relates the equations for ( A µ , A ) and ( A , , δµ ) for anywarp factor h , because supersymmetry is unbroken for any distribution of D3-braneson the conifold.Now let us briefly address the question of computing the four-dimensional spec-trum of A , . The leading asymptotic behavior of φ following from (3.46) is φ = c + c r . To calculate the spectrum numerically, say by shooting, one needs to im-pose the exotic boundary condition that φ does not have a constant part at infinitywhile may have the linearly divergent term. In practice this is difficult to control.Instead of dealing with (3.46) one can calculate the spectrum of equation (3.47). Ithas the same asymptotic behavior but the conventional boundary condition, i.e. φ may go to a constant at infinity but should not diverge. This gives the followingspectrum for m (in units of | µ | L − ): 3 .
6, 19 . A fluctuations. The asymptoticbehavior of the wave-function is φ = r / ( c + c log r ). The subleading term is onlylogarithmically suppressed and to impose the boundary condition of vanishing c inpractice may require a very large cutoff. It is better instead to deal with the equation213.49) which results in the asymptotic behavior φ = c r − / + c r / . The boundarycondition is simply that φ vanishes at infinity yielding the spectrum of masses m (inunits of | µ | L − ): 6 .
6, 24 . Knowing the asymptotic behavior in the KW case is usually good enough to deal withthe KS and BGMPZ solutions as well, because these solutions – up to logarithmiccorrections – approach the KW background at large radius. The corrections are notimportant when the leading and subleading asymptotics have two different powers of r . This is not the case for A . Therefore we repeat the analysis of the UV behaviorfor this mode in the case of the deformed conifold. The leading UV behavior is notsensitive to the value of µ and therefore we put it to zero, significantly simplifyingthe calculation. The cycle Σ can be parametrized by the radial coordinate t and theangles θ = θ ≡ θ , φ = φ ≡ φ , ψ . Using the relation between e and g (for thesingular conifold case) g = r r r − | µ | r e , (3.51)the gauge field A = A ( r, x µ ) e can be written as A = ξ ( t, x µ ) g . The Lagrangian forthe static (Minkowski-independent) ξ ( t ) is L = Z ∞ dt h ( t ) (cid:0) ρ ξ ξ + ρ − ξ ξ ′ (cid:1) , ρ ξ ( t ) = r
23 sinh t p sinh( t ) cosh( t ) − t . (3.52)The resulting equation (with the restored x µ -dependence) is a superpartner, in thesense of the SUSY QM discussed before, of the equation for the vector mode discussedin [59], and such relation holds for any warp factor h ( t ).The EOM for ξ has two solutions. The subleading solution that corresponds tothe VEV of the operator | Q | − | ˜ Q | represents the anti-self-dual flux on the D7 anddoes not break supersymmetry ξ ( t ) = c (cid:0) cosh( t ) sinh( t ) − t (cid:1) − / . (3.53)The general asymptotic behavior at infinity in the KS case is ξ ( t ) = (cid:0) c + c (4 t − (cid:1) e − t/ + O ( e − t/ ) . (3.54)22he leading solution has an extra t ∼ log r compared with (3.36), as can be under-stood in the KT limit from (3.34) using the warp factor h ∼ L r log r .Something interesting occurs when we turn on the baryonic branch parameter U . The corresponding background is the BGMPZ solution that approaches the KSsolution at infinity, but this does not guarantee that the asymptotic of the fields on theD7 are the same. The BGMPZ solution approaches the KS solution slowly enough tocreate a non-trivial source at large r for some fluctuations of the worldvolume fields.This happens to A , and not to A , . When U = 0 the B -field acquires an extra term B BGMPZ = B KS + χ ′ dg + O ( U ) , χ ′ → U t − e − t/ . (3.55)The new term has exactly the structure to couple to ξ as both fluctuations correspondto the operators of dimension 2 – the bottom components of the U (1) baryon and U (1) flavor currents. Therefore χ causes a non-homogeneous term in the linearizedequation for ξ , and the asymptotic behavior takes the form ξ = (cid:18) c + c (4 t − + 3 U
16 (2 t − (cid:19) e − t/ + O ( e − t/ ) . (3.56)This is the bulk manifestation of the mixing between U (1) baryon and U (1) flavor . SO (3) invariant flux on the D7-brane In this section we will find a general expression for the real SO (3)-invariant closed(1 ,
1) two-form F , = dA on Σ : { z = const } , which combines with the pull-back of B to form the gauge-invariant flux F = P [ B ] + 2 πα ′ F on the D7. Supersymmetryrequires F to be of (1 , F to be (1 ,
1) as well.There are four (1 , SO (3)-invariant 2-forms on Σ that can be combined witharbitrary r -dependent real coefficients ζ , ζ , λ , λ F , = F I + F II , F I = i (cid:0) ζ dz i ∧ d ¯ z i + ζ ¯ z i dz i ∧ z j d ¯ z j (cid:1) , (3.57) F II = iǫ ijk (¯ λz i − ¯ z i λ ) dz j ∧ d ¯ z k , (3.58)where we introduced a complex λ = λ + iλ . The constraint dF , = 0 boils down tothe two independent equations dF I = dF II = 0. The first can be rewritten in termsof the 1-forms dt , g and dg using¯ z i dz i = r (cid:16) drr + ig (cid:17) . (3.59)23he only possible closed combination is exact F I = d A I , A I = ξ ( t ) g . (3.60)The second constraint implies (now ′ stands for derivative with respect to r )2 λ + (cid:2) λ ′ ( r − | z | ) + ¯ λ ′ ( z − ǫ ) (cid:3) = 0 . (3.61)The general solution is λ = − n p z − ǫ (cid:0) r − | z | + | z − ǫ | (cid:1) + i m p z − ǫ (cid:0) r − | z | − | z − ǫ | (cid:1) (3.62)with n, m real coefficients.Locally we can express F II as F II = d A II , in terms of an SO (3)-invariant potential A II . The most general ansatz is A II = σ ǫ ijk z i ¯ z j dz k + c.c. (3.63)The constraint that d A II be of (1 , σ (cid:0) ( r − | z | ) − | z − ǫ | (cid:1) σ ′ + 2( r − | z | ) σ = 0 , (3.64)with solution σ = C ( r − | z | ) − | z − ǫ | . (3.65)To relate the complex constant C to n, m we compute d A II and cast it in the form(3.58): iλ = ¯ σ + ( r − | z | ) ¯ σ ′ + ( z − ǫ ) σ ′ . Eventually comparing with (3.62) we find C = ( m − in ) p ¯ z − ¯ ǫ .If m = 0, F II is singular at the tip of Σ and should be discarded. If m = 0, F II isregular but A II is still singular at the tip because F II is cohomologically non-trivialon S . We can parametrize the tip as z i = ix i p z − ǫ in terms of real coordinates x i with P i =1 x i = 1. Then F II = n ǫ ijk x i dx j ∧ dx k which gives Z S F II = 2 πn . (3.66)Quantization requires n to be integer. Notice that in the resolved conifold case F II isproportional to the Betti-form ω on T , (3.23) ω = − ir ǫ ijkl z i ¯ z j dz k ∧ d ¯ z l (3.67)24ulled-back on Σ: F II = n P [ ω ]. This confirms that F II is cohomologically non-trivial.We are interested in the Page D3- and D5-charge induced by the worldvolumegauge field on the D7-brane. The D3-charge is given by the integral of the current J D3 = F − B ∧ F + B ∧ B ∧ F on T , , and the contribution from the D7 isgiven by the difference between the tip of the D7 at r min and very large radius:(4 π α ′ ) N D3 = R r = ∞ J D3 − R r = r min J D3 = R M dJ D3 . Using dJ D3 = (2 πα ′ ) F ∧ F ∧ δ D72 (where δ D72 is a 2-form delta-function localized on the D7), we get N D3 = 18 π Z Σ F ∧ F . (3.68)The computation is performed in appendix B and the result is N D3 = n . (3.69)The D5-charge is given by the integral of the current J D5 = F − B ∧ F on S ⊂ T , , and the contribution from the D7 is (4 π α ′ ) N D5 = R r = ∞ J D5 − R r = r min J D5 = R S × R + dJ D5 . Using dJ D5 = (2 πα ′ ) F ∧ δ D72 we get N D5 = 12 π Z Γ=Σ ∩ ( S × R + ) F . (3.70)The computation is performed in appendix B and the result is N D5 = n . (3.71)As we saw in section 3.1, another gauge-invariant is the conjugacy class of theWilson loop ˆ ρ = Pexp i H A , computed at large radius over the non-contractiblecontour ∂ Γ = S on S / Z . Note that such class is invariant under regular gaugetransformations but can change under large gauge transformations of B . To computeˆ ρ we integrate the field strength over Γ, which for a single D7-brane coincides withthe calculation of the D5-charge (see appendix B for details)ˆ ρ = e i H A = e πiN D5 = ( − n . (3.72)Eventually we interpret n from the field theory point of view. Expanding A II at infinity and comparing the leading r − / asymptotic with (3.36) we find that n corresponds to the VEV ˜ QQ = n q z − ǫ . (3.73)25he expectation value of | Q | − | ˜ Q | depends on the 1 /r asymptotic of ξ and variesin different cases. Let us note here that identifying the asymptotic behavior withthe VEVs of the field theory operators as outlined in section 3.3 is too naive becausedifferent operators may have the same quantum numbers and mix. Generically thishappens when ˆ ρ is non-trivial, so that some extra fields are turned on at the boundaryand the AdS/CFT dictionary needs to be corrected. We will return to this problemin section 6.1.Let us now consider in more details various setups and find explicitly the corre-sponding Abelian U (1) instantons. Consider the singular conifold with a D7-brane along Σ : { z = µ/ } and an arbitrarydistribution of D3-branes. The latter only affect the warp factor which does not alterthe supersymmetry condition for the D7-brane flux: P [ J ] ∧ F = 0 , F , = 0 (3.74)with F = P [ B ] + 2 πα ′ F . The K¨ahler form on the singular conifold J = d ( kg ) , k = r , (3.75)is of the form (3.57) and is orthogonal to the flux of type (3.58), see (B.4). The B -field of the KW solution has the form B = πα ′ b ω , where ω is given in (3.67) andits pull-back is of the form (3.58), so it is automatically primitive. Therefore F II isnot constrained and the resulting differential equation for ξ can be easily solved ξ = ξ a k , (3.76)with ξ a constant. The resulting ξ is singular at r min either because a ( r min ) = 0when µ = 0 or k (0) = 0 in the massless case µ = 0. Hence we must set ξ = 0. Theonly surviving degree of freedom is the integer n that parametrizes the flux F II (3.62).Empowered by the AdS/CFT dictionary developed in section 3.3, we derive thatthe background with n units of D7 worldvolume flux is dual to a vacuum with VEVs | Q | − | ˜ Q | = 0 , ˜ QQ = nz . (3.77)26n fact this is correct only when ˆ ρ = 1 (so that there are no Wilson lines at theboundary), otherwise we should expect corrections to the AdS/CFT dictionary. Suchcorrections comes from the mixing of the operators above with other operators withthe same quantum numbers, for instance | Q | − | ˜ Q | can mix with | A | − | B | , while˜ QQ can mix with µ Next consider the resolved conifold with a D7-brane along Σ : { z = µ/ } and anarbitrary distribution of D3-branes. The K¨ahler form compared with (3.75) containsan extra term a Vol J = d (cid:0) k g (cid:1) + a dg + a ω , k = r . (3.78)The B-field is B = πα ′ b (cid:16) ω + 12 d ( f g g ) (cid:17) (3.79)where f g is some radial function. The 2-form ω is singular at the tip of the resolvedconifold, and the extra piece makes B regular provided that f g (0) = 1 (similarly to ξ (0) = n/ z = 0 case). At infinity f g → f g is a pure gauge degree of freedom and is not fixed bythe EOM. We choose it such that a trivial gauge field F = 0 preserves supersymmetry f g = a k ( r ) + a . (3.80)Since P [ ω ] can be expressed as F II with n = 2 , m = 0, we can absorb f g into ξ andimpose primitivity of F ξ + b f g = a b + n k + a / a − ξ k + a / . (3.81)If z = 0, the second term is divergent at r min due to a − and we must set ξ = 0.The first term is non-trivial and we derive | Q | − | ˜ Q | = n a , ˜ QQ = nz . (3.82)27f z = 0 we cannot use (3.81) because it was obtained by simplifying a ′ on bothsides and a ′ = 0 in this case: we need to do the analysis anew. First we set ξ (0) = n/ F , at the tip, and now R tip F , = 2 πn . Since F II is zeroaway from the tip, P [ J ] ∧ F = 0 implies ξ + b f g = ξ k ( r ) + a / . (3.83)To satisfy ξ (0) = n/ ξ = a (cid:0) b + n (cid:1) and find again (3.82) with µ = 0.As discussed at the end of section 3.2, the z = 0 limit is smooth. Moreover thesolution (3.83) is such that the coefficient in front of dr ∧ g in F at r = 0 vanishes:( ξ + b f g ) ′ (0) = 0. In the deformed conifold case, on the KS background the K¨ahler from is J = d (cid:0) k dg (cid:1) , k ( t ) = (cid:0) cosh( t ) sinh( t ) − t (cid:1) / (3.84)which has the form (3.57). The B-field is B KS = h ( t ) cosh( t ) 2 i ǫ ijkl z i ¯ z j dz k ∧ d ¯ z l ǫ sinh t cosh t (3.85)where h ( t ) is a suitable function, and B has the form (3.58). Therefore n is notconstrained and ξ ( t ) must satisfy the differential equation giving ξ ( t ) = ξ a ( t ) k ( t ) , (3.86)which coincides with (3.53) when µ = 0. This is singular at t min and hence ξ = 0 ,leaving only n as free parameter. Correspondingly we derive | Q | − | ˜ Q | = 0 , ˜ QQ = n q z − ǫ . (3.87) The BGMPZ solutions [41], based on SU (3)-structure geometries, have a more com-plicated κ -symmetry condition. The computation for the type I flux (3.57) was carriedout in [23] – here we add the type II flux (3.58). The κ -symmetry condition reads U (cid:0) J ∧ J − F ∧ F (cid:1) + e A J ∧ F (cid:12)(cid:12)(cid:12) Σ = 0 (3.88)28here U is the parameter along the baryonic branch and A is the warp factor. Thepseudo-K¨ahler form J is the sum of two terms of type I and II: e A J = U B − d (cid:2) ( λ + U χ ) g (cid:3) , B = B KS + χdg , λ = U e φ a ( t cosh t − sinh t )2( a cosh t + 1)(3.89)and so is F F = B KS + χdg + d ( ξ g ) + F II ( n ) . (3.90)Here B KS is given by (3.85) but with some different function h ( t ). The term B KS ∧ F II identically vanishes, and we get a differential equation for ξ : − a ddt (cid:20) a (cid:16) ( ξ + χ ) + 2 λU ( ξ + χ ) + (cid:16) e − φ h sinh t − λ U ( e − φ − (cid:17)(cid:17) ++ n | z − ǫ | r − | z | + | z − ǫ | ) (cid:21) = 0 . (3.91)The equation can be integrated, in terms of a constant c . At infinity λ diverges as − e t/ + U ( t −
1) + O ( e − t/ ), while h sinh t remains finite, therefore only one rootof the quadratic equation is meaningful ξ + χ = − λ − e − φ p λ − U h sinh t − e φ U a − c U , (3.92)c( t ) = c + n | z − ǫ | ( ǫ cosh t − | z | + | z − ǫ | ) . (3.93)At the minimal radius t min the functions λ , h sinh t , φ are regular but a − is singular,hence to avoid singularities we set c = − n /
16 and the large t asymptotic is ξ → U t + 20 t + 35 − n e − t/ + O ( e − t/ ) , (3.94)in agreement with (3.56). We interpret the asymptotic with n = 0 as the one corre-sponding to the vacuum with ˜ Q = Q = 0 effectively absorbing non-trivial d ( ξg ) into B . Then we derive | Q | − | ˜ Q | = − n U | ǫ | / / , ˜ QQ = n q z − ǫ . (3.95)In the special case z = 0, although a ≡ dg is singular at the tip and we must require ξ + χ to vanish at t = 0. Thisfixes c as in (3.93). 29 Backreaction of D7-branes
In order to extract full information about the dynamics of the dual field theory, inparticular its RG flow, one has to go beyond the probe approximation and construct afully backreacted gravity solution. To do so for localized D7-branes is a hard problem.One possibility is to consider smeared solutions. In the Veneziano large N c limit, with N f /N c ≪ N f of D7-branes is large and one can distributethem uniformly along the angular directions. This can be done supersymmetrically,and such a configuration has a precise field theory dual (discussed in section 6.3).One can consider both massless and massive embeddings of D7-branes into theKW, KT, and KS backgrounds [15, 16, 61, 17, 31, 62, 63]. We will consider heremassless embeddings in KT [16], and move the massive embeddings in KT with extraworldvolume flux to appendix C. We consider an SU (2) × SU (2) × U (1) invariantansatz ds = h − dx , + h h e u (cid:16) dρ + 19 g (cid:17) + e g X (cid:0) dθ i + sin θ i dϕ i (cid:1)i J = e u dρ ∧ g + e g X sin θ i dθ i ∧ dϕ i Ω = 16 e iψ + u +2 g (cid:16) dρ + i g (cid:17) ∧ (cid:0) dθ + i sin θ dϕ (cid:1) ∧ (cid:0) dθ + i sin θ dϕ (cid:1) δ smeared2 = N f π dg = N f π X sin θ i dθ i ∧ dϕ i ,F = N f π g , B = α ′ πb ( ρ ) ω , H = α ′ πb ′ ( ρ ) dρ ∧ ω , (4.1)where u, g, b, h are functions of ρ to be determined. ρ is a new radial coordinate, whichranges from −∞ in the deep IR to 0 at the UV Landau pole. Roughly ρ ∼ log rr L where r L is the radius associated to the Landau pole scale. The smeared chargedistribution 2-form δ smeared2 is essentially fixed by symmetries, and F has been chosento satisfy dF = δ smeared2 . (4.2)The ansatz also includes the SU (3)-structure of the conifold: the K¨ahler form J andthe (3 , One could consider smearing orientifold planes as well, as in [60]. φ ′ = 3 N f π e φ , u ′ = 3 − e u − g − N f π e φ , g ′ = e u − g , (4.3)while the solution with the proper boundary conditions is [15] e φ = 4 π N f − ρ ) , e u = − ρ (1 − ρ ) − / e ρ , e g = (1 − ρ ) / e ρ . (4.4)Another SUSY equation is H = e φ ∗ F , from which we get F = N f α ′ − ρ ) b ′ g ∧ ω . (4.5)Then we have d F = P [ H ], with F , = 0 and F ∧ P [ J ] = 0. In the massless case P [ B ] = 0 because P [ ω ] = 0, and the only solution is F = 0. That is because anynormalizable flux on Σ must be supported on the 2-cycle at the tip, while in themassless case, and within the KT approximation, such 2-cycle is shrunk to zero size.Had we considered the KS setup, worldvolume flux on the massless D7s would bepossible. Finally the Bianchi identity dF = H ∧ F fixes b ( ρ ) = c ( − ρ ) + c , (4.6)where c , c are integration constants. The self-dual 5-form flux F is fixed by theBianchi identity dF = H ∧ F (where we neglected gravitational corrections on theD7s), and F in turn fixes the warp factor via C = h − dx ∧ · · · ∧ dx .The integration constant c is constrained by quantization of the Page D5-charge Q D5 = 14 π α ′ Z S (cid:0) F − B ∧ F (cid:1) = − N f c . (4.7)This charge is sourced by D5-branes and the worldvolume flux on D7-branes, andhas to be quantized in terms of the minimal charge in the setup. According to table In particular the SU (3)-structure satisfies the relations dJ = 2( g ′ − e u − g ) dρ ∧ J = 0 , d Ω = (2 g ′ + u ′ − dρ ∧ Ω = − dφ ∧ Ω . In section 5 we discuss the corresponding field theory. For µ = 0, classically there are no vacuacorresponding to a non-trivial worldvolume flux. Those vacua reappear, though, in the quantumtheory which, on the gravity side, corresponds to the KS background. fL N fR N N A α B ˙ α ˜ Q R Q R ˜ Q L Q L Figure 2: Quiver of the flavored conifold theory.(6.1), it must be semi-integer. The integration constant c is free and correspondsto changing the gauge couplings. We will use this solution in section 6.3 to extractthe RG flow. The field theory dual to (fractional) D3-branes on the conifold, as reviewed in section2, is the N = 1 SU ( M + p ) × SU ( p ) quiver gauge theory [1, 5, 6, 7, 8]. The leftnode corresponds to wrapped D5-branes, the right node to D5s each with − { z = µ/ } introduces a pair of quarks Q, ˜ Q (one “flavor”) of mass √ h µ (thesuperpotential coupling h appears because of a choice of normalization). The cycleΣ contains a topologically non-trivial S and therefore there are two fractional D7-branes of minimal tension, distinguished by a monodromy ˆ ρ at infinity and by theflux at the tip. Similarly to the N = 2 Z orbifold case discussed in section 3.1, apure D7 introduces flavors coupled to the right node, while a D7 with − N = 2 orbifold C × C / Z and follow the RG flow discussed in [1, 24]. The precise map between theD-brane charges and ranks in field theory is given in section 6.Summarizing, the gauge theory is a quiver with the gauge group SU ( N = M + p ) × SU ( N = p ) (we do not necessarily restrict to N ≥ N ) and bifundamental fields A α , B ˙ α ( α, ˙ α = 1 ,
2) as in the pure conifold theory, with the addition of N fL flavorscharged under SU ( N ) and N fR flavors under SU ( N ). We set N f = N fL + N fR .The corresponding quiver in figure 2 exactly coincides with the ADHM quiver of the N = 2 Z orbifold theory shown in figure 1. To denote this quiver we adapt the same Alternatively, one could compute the charge π α ′ R S (cid:0) F − B ∧ F − πα ′ A ∧ δ D72 (cid:1) which issourced by D5-branes only, and needs to be integer. N fL × N × N × N fR . The full superpotential is (compare with (3.16)) W = h ( A B A B − A B A B ) −√ h η L ˜ Q L (cid:16) A B + A B − µ √ h (cid:17) Q L + η L Q L Q L ˜ Q L Q L − √ h η R ˜ Q R (cid:16) B A + B A − µ √ h (cid:17) Q R − η R Q R Q R ˜ Q R Q R (5.1)where trace is implicit. Various factors of h have been inserted for convenience. Thecoefficients of the quartic quark terms have specific values, which come from the N = 2 orbifold theory broken to N = 1 [1, 24]. One could consider deforming thetheory by the marginal operators Tr ˜ Q L Q L ˜ Q L Q L and Tr ˜ Q R Q R ˜ Q R Q R . These oper-ators contain two traces over color indices and therefore correspond, on the gravityside, to a change of boundary conditions for the modes dual to Tr ˜ Q L Q L , Tr ˜ Q R Q R [65]. If the superpotential is ignored, the instanton factors related to the 1-loop-exactholomorphic (RG invariant) β -functions areΛ N − N − N fL ≡ Λ b , Λ N − N − N fR ≡ Λ b . (5.2)The non-Abelian symmetries (vector-like and non-anomalous) are SU ( N fL ) × SU ( N fR ) × SU (2) AB , where SU (2) AB is the anti-diagonal subgroup of SU (2) A × SU (2) B that preserves A α B ˙ α δ α ˙ α . The Abelian symmetries can be analyzed in thebasis of table 1, and the exact symmetries are the subgroup under which no couplingor instanton factor is charged. For µ = 0 it is U (1) b × U (1) fL × U (1) fR × Z q , where Z q ⊂ ] U (1) R and q = gcd(2 N − N − N fL , N − N + N fR ). µ = 0 completelybreaks ] U (1) R . The generators of the unbroken symmetries are U (1) fL,fR = U (1) Q L,R − U (1) ˜ Q L,R , U (1) b = U (1) A − U (1) B Z q ⊂ ] U (1) R = U (1) R −
12 [ U (1) A + U (1) B + U (1) Q L + U (1) ˜ Q L + U (1) Q R + U (1) ˜ Q R ](5.3) Precisely, the superpotential (5.1) is obtained from the N = 2 theory with U ( N ) gauge groups.Starting with SU ( N ) gauge groups, one obtains other terms with different contraction of flavorindices. The difference is negligible in the large N limit. As in [66] we use holomorphic normalization for the gauge sector, g F ∧ ∗ F , and distinguishbetween holomorphic β -functions, where chiral matter fields are not renormalized, and physical β -functions, where chiral matter fields do have anomalous dimensions. (1) A U (1) B U (1) Q L U (1) ˜ Q L U (1) Q R U (1) ˜ Q R U (1) R ] U (1) R A / B / Q L / Q L / Q R / Q R / h − − − η L − − − η R − − − µ b N N N fL N fL N N − N − N fL Λ b N N N fR N fR N − N + 2 N − N fR Table 1: Basis for the global Abelian symmetries.where U (1) b is the usual baryonic symmetry of the conifold. It will also be convenientto define U (1) ≡ U (1) b + U (1) fL U (2) ≡ − U (1) b + U (1) fR (5.4)which are the “baryonic symmetries” of the two SU nodes. We can form combinationsof couplings that are invariant under flavor symmetries. They will correspond tosupergravity parameters. First of all we take L ≡ Λ b h N η N fL L , L ≡ Λ b h N η N fR R (5.5)with R-charges R [ L ] = 2 N − N − N fL and R [ L ] = − N + 2 N − N fR . In themassless case we associate the following combinations to supergravity fields: I ≡ L L ∼ e πiτ , L L ∼ exp Z S (cid:0) B + iC (cid:1) (5.6)as in [21]. In the massive case we can construct the dimensionless invariants µ N f L L and µ − N +4 N + N fL − N fR L L .By a field redefinition we can take η L,R = √ h . This will lead to a simplified formof the superpotential (5.1) which we will use in what follows.34 .1 Seiberg duality, parameters and vacua The field theory with superpotential (5.1) has a remarkable property to be self-similarunder Seiberg duality, up to a shift of ranks. The superpotential is such that left andright quarks become simultaneously massless on the mesonic branch. This must be so,as quarks come from the D3-D7 strings and what distinguishes left quarks from rightquarks is the flux on the D7s, not the embedding equation. In fact the coefficientsof quartic quark terms are precisely such that the property that left and right flavorsbecome simultaneously massless is invariant under the Seiberg duality. Besides beinga map between theories, the duality is also a map between vacua, e.g. what looks likea simple vacuum in one description may look complicated in another. We analyzehere such issues.Consider the superpotential W (5.1). We perform a Seiberg duality on the rightnode SU ( N ). The mesons are M α ˙ α = 1Λ A α B ˙ α , N α = 1Λ A α Q R , ˜ N ˙ α = 1Λ ˜ Q R B ˙ α , Φ = 1Λ ˜ Q R Q R (5.7)and the dual quarks are A α → c α , B ˙ α → d ˙ α , ˜ Q R → r , Q R → ˜ r . (5.8)The magnetic gauge group is SU (2 N + N fR − N ). The magnetic superpotentialis W , written in terms of the magnetic variables, plus the extra terms M α ˙ α d ˙ α c α + N α ˜ rc α + ˜ N ˙ α d ˙ α r + Φ˜ rr . On a branch of the moduli space where the fields M α ˙ α , N α ,˜ N ˙ α , Φ are massive they can be integrated out via their F-term equations. We thusobtain the superpotential in the dual magnetic theory W mag = 1Λ h ( c d c d − c d c d ) + 1Λ ˜ Q L (cid:0) d c + d c + Λ √ h µ (cid:1) Q L − h Q L Q L ˜ Q L Q L + 1Λ h ˜ r (cid:0) c d + c d + Λ √ h µ (cid:1) r + 12Λ h ˜ rr ˜ rr where a constant term has been dropped. We can now redraw the quiver, flipping ithorizontally and perform a further field redefinition c α = ǫ αβ √ Λ h a β ˜ r = √ Λ h ˜ q L ˜ Q L = ˜ q R d ˙ α = − ǫ ˙ α ˙ β √ Λ h b ˙ β r = √ Λ h q L Q L = q R . (5.9)35he resulting superpotential W mag = h ( a b a b − a b a b ) − h ˜ q L (cid:16) a b + a b − µ √ h (cid:17) q L + h q L q L ˜ q L q L − h ˜ q R (cid:16) b a + b a − µ √ h (cid:17) q R − h q R q R ˜ q R q R (5.10)is manifestly identical to the initial one (5.1). Notice that the flip exchanges both thegauge ranks and the number of flavors N fR and N fL .To conclude, we can map mesonic gauge-invariant operators with respect to thedualized node from the electric theory to the magnetic one: b ˙ α a α = A α B ˙ α − δ α ˙ α Q L ˜ Q L ˜ q L a α = − ǫ ˙ βα ˜ Q R B ˙ β q R = Q L ˜ q L q L = ˜ Q R Q R − µ √ h N fR b ˙ β q L = ǫ α ˙ β Q α Q R ˜ q R = ˜ Q L . (5.11)When a Seiberg duality is performed on the left node, the same formulæ hold byexchanging the electric with the magnetic theory.Let us now look at the real operators in the bottom component of current su-permultiplets. To simplify the discussion, consider SQCD n c ,n f with quarks Q, ˜ Q andbaryons B = Q n c , ˜ B = ˜ Q n c . The dual description SQCD n f − n c ,n f has quarks q, ˜ q andbaryons b = q n f − n c , ˜ b = ˜ q n f − n c . The map b = Λ n f − n c B · ǫ , and similarly for tildedquantities, implies the following map for the bottom component of the baryonic cur-rent multiplet at weak coupling:1 n c (cid:0) | Q | − | ˜ Q | (cid:1) = 1 n f − n c (cid:0) | q | − | ˜ q | (cid:1) . Now consider dividing the quarks into two groups: Q → ( Q R , P ) in number ( n fR , n f − n fR ) (and similarly for tilded quarks). This amounts to considering a subgroup ofthe global symmetry SU ( n f ) → SU ( n fR ) × SU ( n f − n fR ) × U (1) aux , and defines asplitting of the dual quarks q → ( q R , p ). From U (1) baryon and U (1) aux we can constructa symmetry U (1) fR that only gives charge ± Q R , ˜ Q R respectively. From thecharges of quarks and dual quarks we get the map1 n fR (cid:0) | Q R | − | ˜ Q R | (cid:1) = − n f − n c − n fR n fR ( n f − n c ) (cid:0) | q R | − | ˜ q R | (cid:1) + 1 n f − n c (cid:0) | p | − | ˜ p | (cid:1) . If we now translate that relation in terms of our quiver, we obtain for the bottom36omponents of the current supermultiplets of U (1) fL,R in the electric description: | ˜ Q L | − | Q L | = | ˜ q R | − | q R | (2 N − N + N fR ) (cid:0) | ˜ Q R | − | Q R | (cid:1) = (2 N − N ) (cid:0) | ˜ q L | − | q L | (cid:1) + N fR (cid:0) | a | − | b | (cid:1) . (5.12) We start our quest of understanding the moduli space with the classical analysis byfinding the space of solutions of the F-term and D-term equations, modded out bygauge equivalences. The F-term equations are0 = B A B − B A B − B Q L ˜ Q L − Q R ˜ Q R B B A B − B A B − B Q L ˜ Q L − Q R ˜ Q R B A B A − A B A − Q L ˜ Q L A − A Q R ˜ Q R A B A − A B A − Q L ˜ Q L A − A Q R ˜ Q R (cid:16) A B + A B − Q L ˜ Q L − µ √ h (cid:17) Q L = ˜ Q L (cid:16) A B + A B − Q L ˜ Q L − µ √ h (cid:17) (cid:16) B A + B A + Q R ˜ Q R − µ √ h (cid:17) Q R = ˜ Q R (cid:16) B A + B A + Q R ˜ Q R − µ √ h (cid:17) (5.13)while the D-term equations following from the canonical K¨ahler potential (at theclassical level we disregard corrections to the K¨ahler potential) are ξ N = A α A † α − B † ˙ α B ˙ α + Q L Q † L − ˜ Q † L ˜ Q L ξ N = B ˙ α B † ˙ α − A † α A α + Q R Q † R − ˜ Q † R ˜ Q R . (5.14)Here ξ , are free parameters to be determined. If the global symmetries U (1) , defined in (5.4) are gauged, then ξ , become FI terms. In general only one linearcombination of ξ , can be turned on such that the equations above are satisfiedand supersymmetry is preserved, and we will later specify which linear combinationdepending on the branch of the moduli space. In the following we use the notationsfor the (deformed) conifold introduced in section 2.2 C ǫ = (cid:8) det ˙ αα w ˙ αα = ǫ (cid:9) . (5.15)37he classical moduli space has an intricate structure that we summarize here.First, there are mesonic directions where A α , B ˙ α take VEV with ξ , = 0 and Q L,R =˜ Q L,R = 0. For suitable choices of N , N there can be a baryonic direction where A α , B ˙ α take VEV with ξ , = 0 while still Q L,R = ˜ Q L,R = 0. These two branchesare essentially the same as in the unflavored theory. Second, there are instanton-likedirections, when VEVs of Q L,R , ˜ Q L,R partially break the gauge group while preserving N − N . This time Q L,R , ˜ Q L,R have moduli and these branches are continuouslyconnected with the mesonic/baryonic directions. Finally, there are Higgsed mesonicdirections (only for µ = 0) when Q L,R , ˜ Q L,R take VEV and break the gauge group SU ( N ) × SU ( N ) to two smaller SU factors and changing the difference N − N .These vacua are disconnected from the previous ones. For both Higgsed mesonicand instanton-like directions, the low energy theory with the unbroken gauge groupusually sits in a mesonic vacuum although in certain cases the parameters ξ , can beturned on as well. Mesonic directions.
Up to gauge transformations, the mesonic vacua are A α = A (1) α . . . A ( p ) α . . . . . . B T ˙ α = B (1)˙ α . . . B ( p )˙ α . . . . . . X α | A ( a ) α | − X ˙ α | B ( a )˙ α | = 0 ∀ a (5.16)and Q L = ˜ Q L = Q R = ˜ Q R = 0. Here we assumed M = N − N > SU ( M + p ) × SU ( p ) isbroken to SU ( M ) × U (1) p − × Weyl (for
M > U (1) factors are diagonallyembedded, SU ( M ) ⊂ SU ( N ) and the Weyl group permutes the U (1)s. The moduliare characterized by the coordinates w a ˙ αα ≡ √ h B ( a )˙ α A ( a ) α (5.17)( √ h inserted for convenience) which satisfy det ˙ αα w a ˙ αα = 0. This gives a symmetricproduct (because of the Weyl group) of p copies of the singular conifoldSym p ( C ) .
38t a generic point the low energy spectrum contains the SU ( M ) gauge multiplet,3 p neutral chiral multiplets (parametrizing the moduli) and p − U (1) p − groups have N f flavors, generically of mass √ h (cid:0) µ − Tr ˙ αα w a ˙ αα (cid:1) . SU ( M ) has N fL flavors of mass √ h µ , which become massless for µ = 0, and quarticsuperpotential. If we start from the origin of the mesonic branch with gauge group SU ( N ) × SU ( N ) and give large VEV to only one mesonic component, the gaugegroup is broken to SU ( N − × SU ( N − × U (1) and U (1) gauges the symmetry U (1) − U (1) = 2 U (1) b + U (1) fL − U (1) fR of the low energy theory. At the laststep, where we are left with SU ( M ) and N fL flavors, one linear combination of the U (1) p − gauges U (1) fL .We have three baryonic symmetries – U (1) b and U (1) fL,R – and we can gaugeany linear combination. For instance if we gauge U (1) b , at low energy we get p N = 4 Abelian vector multiplets (at special points on the mesonic branch there willbe massless N = 2 flavors). We can also add a FI term ξ ≡ ξ = − ξ . If N = N = N we have SUSY vacua describing p symmetrized copies of the resolved conifold X α | A ( a ) α | − X ˙ α | B ( a )˙ α | = ξ ∀ a (5.18)with Q L,R = ˜ Q L,R = 0. To parametrize the tip we need, besides the mesons, thebaryons (2.14). If N > N , supersymmetry is broken for N f = 0 but it mightbe preserved for N f > N , N allow for a Higgsed mesonic vacuum(discussed below) whose low energy theory is SU ( ˜ N ) × SU ( ˜ N ).Let us comment here on the SU ( M ) non-perturbative dynamics at low energiesif N > N . We will distinguish between the massless and massive cases in whatfollows.We start with the massless case µ = 0. Since SU ( M ) ⊂ SU ( N ), to get theinstanton factors by scale matching we give large VEV to N components of A, B .Each time we turn on one component, the breaking pattern is SU ( N ) × SU ( N ) → SU ( N − × SU ( N − × U (1). The SQCD n c ,n f theory goes to SQCD n c − ,n f − as aresult of a VEV h AB i and a mass term h h AB i from the superpotential. In the finalexpression the value of VEV cancels out and we are left with instanton factorsΛ N − N − N fL −
11 low = Λ N − N − N fL h , Λ N − N − N fR −
12 low = Λ N − N − N fR h . Repeating N times, we are left with SU ( M ) with N fL flavors, instanton factorΛ M − N fL = Λ N − N − N fL h N (5.19)39nd a quartic superpotential W = h ˜ Q L Q L ˜ Q L Q L . The dynamically generated on-shell superpotential on the mesonic branch is W eff (vacua) = 2 N − N − N fL (cid:0) Λ N − N − N fL )1 h N + N fL (cid:1) N − N − NfL . (5.20)In the massive case µ = 0, we can discuss two different scenarios: large or smallmass √ h µ . For large mass we integrate out the flavors first and obtain the instantonfactors Λ N − N = ( √ h µ ) N fL Λ N − N − N fL and Λ N − N = ( √ h µ ) N fR Λ N − N − N fR .Then we break SU ( N ) by moving on the mesonic branch, while preserving unbroken SU ( M ) ⊂ SU ( N ) with instanton factor and on-shell superpotentialΛ M = Λ N − N − N fL h N ( √ h µ ) N fL , W eff ∼ (cid:0) Λ M (cid:1) M . (5.21)For small mass we break SU ( N ) on the mesonic branch first and obtain a massivequartic SQCD N − N ,N fL with Λ N − N ) − N fL low = Λ N − N − N fL h N . For ( √ h µ ) n c − n f ≪ Λ n c − n f low h n c the theory is essentially massless and we recover (5.20). In the oppositelimit the theory has vacua where SU ( n c ) is broken to SU ( n c − j ) and the on-shellsuperpotential is W eff ∼ (cid:0) Λ n c − n f low h j ( √ h µ ) n f − j (cid:1) nc − j ∼ (cid:0) Λ N − N − N fL h N + j ( √ h µ ) N fL − j (cid:1) N − N − j . (5.22)For j = 0 we recover the vacua in (5.21). For 1 ≤ j ≤ n c we have Higgsed mesonicvacua, more precisely j blocks with n = − Baryonic direction.
These vacua are present if N = ( k + 1) M , N = kM . Letus first define the Upper and
Lower ( k + 1) × k matrices [21] U k = √ k . . . √ k − . . . . . . √ . . . . . . L k = . . . . . . √ . . . . . . √ k − . . . √ k . (5.23) SQCD n c ,n f with quartic superpotential has an intricate structure [67]. For n f < n c , the numberof vacua is (2 n c − n f )2 n f − , all with the same dynamical scale W eff = 2 n c − n f (cid:0) Λ n c − n f ) h n f (cid:1) / (2 n c − n f ) . U T k U k + L T k L k = ( k + 1)1I k , U k U T k + L k L T k = k k +1 , U k +1 L k = L k +1 U k . (5.24)Up to a gauge transformation the classical vacua are given by A = C U k ⊗ M , A = C L k ⊗ M , B = B = 0 , Q L,R = ˜ Q L,R = 0 . (5.25)There is another set with A ↔ B T . Here C is an arbitrary complex number. Thevacua (5.25) satisfy the D-term equations with ξ = k | C | , ξ = − ( k + 1) | C | and ξ ↔ − ξ when A ↔ B T . The branches are parametrized by either the baryon A ∼ ( A A ) k ( k +1) M/ or the anti-baryon B ∼ ( B B ) k ( k +1) M/ . The origin of thebaryonic branch touches (classically) the origin of the mesonic branch.For ˜ p ≡ N mod M(= N − N ) = 0 there is no baryonic flat direction. One wayto see that is to give mesonic VEVs to ˜ p directions. This breaks the gauge group to SU (cid:0) ( k + 1) M (cid:1) × SU ( kM ) × U (1) ˜ p . Although this is very close to the theory with thebaryonic branch discussed above the low energy bifundamentals are charged under alinear combination of U (1) ˜ p . Hence the D-term equations set C = 0 and the resultingvacuum belongs to the mesonic direction. Instanton-like directions.
This branch is the piece of the Higgs branch contin-uously connected to the mesonic directions discussed above. The vacua are in one-to-one correspondence with a similar Higgs branch in the N = 2 case, indeed anysolution to the N = 2 C / Z ADHM equations (3.9) with ξ L C = ξ R C = µ/ √ h and ξ L R = ξ , ξ R R = ξ solves the N = 1 equations (5.13), (5.14). Thus all instantons ofthe N = 2 C / Z theory are present in the N = 1 conifold theory as well, and thetwo spaces have equal dimension.The instanton-like vacua have a block diagonal form and can be of the “Left” orthe “Right” type. They are parametrized by two integers ( n c ≥ , n f ≥
2) whichdefine the size of the blocks:Left : n f × n c × n c × , Right : 0 × n c × n c × n f . (5.26)Moreover solutions for the blocks exist for any choice of ξ , . To calculate ˜ QQ onewould need to know an explicit solution, but the VEV of | Q | − | ˜ Q | follows immedi-ately from the equations (5.14) and the block size: | Q | − | ˜ Q | = n c ( ξ + ξ ) . (5.27)41epending on the ranks of the unbroken gauge symmetry, ξ and − ξ can be zero,equal to each other, linearly dependent or arbitrary. The vacua with ξ + ξ = 0correspond to the noncommutative instantons on C / Z in the N = 2 case. The instanton-like vacua describe D3-branes dissolved inside the D7-branes. Ingeneral the D3-branes can become point-like instantons and leave the D7s, so thesedirections touch the mesonic directions (but not the baryonic one).
Higgsed mesonic directions.
Other disconnected branches of vacua exist in whichthe two gauge ranks are broken by an unequal amount. Such vacua have a blockdiagonal form n Lf × n c × n c × n Rf with n c = n c , and in the classical theory theyonly exist for µ = 0. They are disconnected from the mesonic and baryonic branchesdiscussed before, and for each value of n c − n c we get a different disconnected branch.Below we focus on the cases with n Lf + n Rf = 1, which correspond to the Abelianinstantons (the more general directions are obtained by “adding” non-Abelian instan-tons). In this case either Q L , ˜ Q L or Q R , ˜ Q R acquire VEV and this generically forces A α , B ˙ α to acquire VEV as well. We parametrize the blocks by an integer r ∈ Z , andtheir dimension is (compare with (3.13) and (3.14))Left: 1 × r × r ( r − × × r ( r + 1) × r × . (5.28)The case r = 0 coincides with the mesonic flat directions. The quivers (5.28) can beobtained from the r = 0 case via a chain of Seiberg dualities discussed in section 6.Notice that the left gauge rank minus the right gauge rank equals r , and there isa symmetry that flips left and right and maps r → − r . We can parametrize both leftand right blocks by an integer n ∈ Z defined as n = r − r right ⇔ r = h n + 12 i − (5.29)where [ x ] − is the highest integer equal or smaller than x . For r = 0, sign n = sign r .The number n is what appears in the supergravity description. The blocks containcoefficients a , . . . , a K − , and we will define K such that formally a K ≡
0. The left Although the field theory of section 3.1 admits ξ + ξ = 0 only when all gauge symmetry isbroken, in the conifold theory more general situations are possible, for instance ξ + ξ = 0 is foundon the baryonic branch. r ≥ × r × r ( r − ×
0) are given by A = βα a U T . . .a L a U T . . . a L . . . ... ... . . . B T = βα a U T . . . − a L a U T . . . − a L . . . ... ... . . . A = βα a L T . . . − a U a L T . . . − a U . . . ... ... . . . B T = βα a L T . . .a U a L T . . . a U . . . ... ... . . . ˜ Q L = Q T L = α (cid:16) . . . (cid:17) , ˜ Q R = Q R = 0 . (5.30)The unknowns are a , . . . , a r − , α , β and K = 2 r −
1. The right blocks for r ≥ × r ( r + 1) × r ×
1) are given by A = βα a U a L T . . . a U a L T . . . a U . . . ... ... ... . . . B T = βα − a U a L T . . . − a U a L T . . . a U . . . ... ... ... . . . A = βα − a L a U T . . . − a L a U T . . . − a L . . . ... ... ... . . . B T = βα a L a U T . . . a L a U T . . . a L . . . ... ... ... . . . ˜ Q L = Q L = 0 , ˜ Q R = Q T R = α (cid:16) . . . (cid:17) . (5.31)The unknowns are a , . . . , a r − , α , β and K = 2 r . The blocks of one kind with r ≤ − r ≥ A α , B ˙ α and exchanging Q L ↔ Q R , ˜ Q L ↔ ˜ Q R . The left blocks for r ≤ − × | r | × | r | ( | r | + 1) ×
0) have a , . . . , a | r |− and K = 2 | r | . The right blocksfor r ≤ − × | r | ( | r | − × | r | ×
1) have a , . . . , a | r |− and K = 2 | r | −
1. In allcases r = 0 the number of a j ’s is (cid:12)(cid:12) n + (cid:12)(cid:12) − while K = (cid:12)(cid:12) n + (cid:12)(cid:12) − . The number of unknowns is really one less, because we could reabsorb β into a j . We will fixthis redundancy later. a j . From the F-terms we get equa-tions that fix a j ’s through the recursive relation0 = j a j − a j +1 − ( j + 3) a j +2 for j = 1 , . . . , K − , a K ≡ . (5.32)With some choice of normalization the solution is a j = (2 K + 1) − ( − j + K (2 j + 1) j ( j + 1) (5.33)and all a j ’s are positive. The other unknowns are give by β = sign( r ) a + 3 a = 14 r , α = ( − n +1 µ √ h a + 3 a a − a = − µ √ h r . (5.34)From here we can extract the VEV of the quark bilinear˜ Q i Q i = α = − µ √ h r (5.35)where i = L, R depending on the block, while | Q | − | ˜ Q | = 0.Let us note that the explicit solutions above and the ones in appendix A.1 (dis-cussed below) also solve the N = 2 C / Z = 2 ADHM equations (3.9).The Higgs vacua break the theory at scale ( µ/ √ h ) / . Each block reduces colorand flavor ranks according to its dimension (5.28). Below the breaking scale the low-energy theory SU ( ˜ N ) × SU ( ˜ N ) can have mesonic or, if ˜ N = k ( ˜ N − ˜ N ), baryonicdirections. In the massless µ = 0 case all these vacua collapse to the origin of themesonic directions i.e. since α ∼ µ all fields are zero. We will see that in thequantum theory the vacua described above do not degenerate in the µ → Higgsed mesonic directions with resolution.
The Higgsed blocks discussedabove can be modified to solve the vacuum equations with generic parameters ξ and ξ . Possible constraints on ξ , will come from the remaining components of theD-term equations along the directions with unbroken gauge symmetry. The explicit44olutions generalizing (5.30) and (5.31) can be found in appendix A.1. However theVEV of | Q | − | ˜ Q | follows directly from (5.14) and the size of the blocks:L: Q † L Q L − ˜ Q L ˜ Q † L = r ξ + r ( r − ξ , R: Q † R Q R − ˜ Q R ˜ Q † R = r ( r + 1) ξ + r ξ . (5.36)Notice that the result is independent of µ , and indeed such vacua remain non-trivialin the µ → SU ( N ) × SU ( N ) one can turn on ξ = − ξ in thelow energy theory causing the VEV Q † Q − ˜ Q ˜ Q † = r ξ . (5.37)If it is SU (( k + 1) M ) × SU ( kM ) the low energy theory develops a baryonic branchwith ( k + 1) ξ = − kξ and the VEVL: Q † L Q L − ˜ Q L ˜ Q † L = r − ( k + 1) rk + 1 ξ R: Q † R Q R − ˜ Q R ˜ Q † R = r − krk + 1 ξ . (5.38)Let us comment on k -dependence in (5.38). Different k correspond to differentsteps along the cascading RG flow of the same theory, therefore well-defined physicalquantities should not depend on k . The reason why (5.38) is k -dependent is thatthe definitions of U (1) fL and U (1) fR are not invariant under Seiberg duality – as wesaw in section 5.1 – the precise relation being (5.12). It is a simple exercise to showthat the VEVs (5.38) are a consequence of the map (5.12). In short, as we go up inenergy and perform a Seiberg duality on the right node, k up = k + 1 and n up = n + 1(exchanging right and left flavors). Moreover ξ up2 = k +2 k +1 ξ , as follows analyzing thetheory below the Higgsing scale. N + N f L < N Here we start analyzing how quantum corrections modify the moduli space. In thisand the following sections we will consider M ≥
0. The case
M < N + N fL < N , in which case there are no baryonic45irections. The left node goes to strong coupling in the IR while the right node goesto weak coupling. The left node is parametrized by its mesons M = B A B A B Q L B A B A B Q L ˜ Q L A ˜ Q L A ˜ Q L Q L = M M N M M N ˜ N ˜ N Φ . (5.39)First, we study the dynamics of the left node alone as if the right node had zerocoupling, and then we gauge the SU ( N ) group and introduce the correspondingD-term equations.Along the moduli space of the left node there is a dynamically generated Affleck-Dine-Seiberg (ADS) superpotential [68] W ADS = ( N − N − N fL ) (cid:16) Λ N − N − N fL det M (cid:17) N − N − NfL . (5.40)The total effective superpotential is a sum of two terms W eff = W ADS + W , W = Tr (cid:20) h ( M M − M M ) − h (cid:16) ˜ N N + ˜ N N − µ √ h Φ (cid:17) + h − h ˜ Q R (cid:16) M + M − µ √ h (cid:17) Q R − h Q R Q R ˜ Q R Q R (cid:21) . (5.41)It will be convenient to introduce a matrix N , equal to the variation of the classicalsuperpotential with respect to the mesons N ij ≡ ∂W ∂ M ji = − h M + Q R ˜ Q R − M N − M M + Q R ˜ Q R N ˜ N ˜ N − µ √ h − Φ . (5.42)The F-term equations therefore are N = (cid:16) Λ N − N − N fL det M (cid:17) N − N − NfL M − , (5.43)0 = (cid:16) M + M + Q R ˜ Q R − µ √ h (cid:17) Q R = ˜ Q R (cid:16) M + M + Q R ˜ Q R − µ √ h (cid:17) . (5.44)Calling − ǫ the factor on the right hand side of the first equation and multiplyingby M on the left and on the right we get M N = N M = − ǫ N + N fL . This isa counterpart of the classical F-term equation with a dynamically generated term.46ince the right node is IR free it has a canonical K¨ahler potential and the D-termequation is [ M ˙ αα , M † ˙ αα ] + N ˙ β N † ˙ β − ˜ N † β ˜ N β + Q R Q † R − ˜ Q † R ˜ Q R = ξ N . (5.45)The solutions to these equations form a quantum deformed version of the mesonicand Higgsed mesonic directions of section 5.2. They have the same block diagonalform, each block describing dissolved D3 and D5-branes. To illustrate how it workswe will find the solutions for the quantum counterparts of the “Left” and “Right”Abelian Higgs vacua (5.28) (excluding some special cases, the generic non-Abelianinstanton-like directions cannot be presented in a closed form)Left: 1 × r ( r − × , Right: 0 × r × . (5.46)Here the ranks refer to U ( N fL ) × SU ( N ) × U ( N fR ) whilst SU ( N ) is confined. Theexplicit form of the matrices, for ξ = 0, is given in appendix A.2.The blocks 0 × × M ˙ αα whichsatisfy M M − M M = ǫ/h . In terms of the complex coordinates w a ˙ αα (5.17) wehave det ˙ αα w a ˙ αα = ǫ . (5.47)The D3-branes move on a deformed conifold with the deformation parameter ǫ .We are particularly interested in the quark bilinear. In the “Left” and “Right”cases it is given byΦ = − r µ / − ǫh (2 r − − µ √ h , or ˜ Q R Q R = − r µ / − ǫh r (5.48)where the branch cut has been chosen to match the ǫ → ǫ . In the massless case µ = 0, all 2 N + N fL components of M are of order p ǫ/h implying ǫ ∼ (cid:0) Λ N − N − N fL )1 h N + N fL (cid:1) N − N − NfL , (5.49)which agrees with the semiclassical computation (5.20). This result does not dependon the particular Higgsed vacuum. For large mass µ ≫ ǫ and in the trivial vacuum M has 2 N components of order p ǫ/h and N fL of order µ/ √ h , so that ǫ ∼ (cid:0) Λ N − N − N fL h N ( √ h µ ) N fL (cid:1) N − N , (5.50)47hich agrees with the semiclassical result (5.21).It is also possible to find the solutions to (5.43)-(5.45) for generic values of ξ , asis done in appendix A.3. The corresponding VEVs in the “Left” and “Right” casesare X ˙ α | N α | − X α | ˜ N α | = r ( r − ξ , or | Q R | − | ˜ Q R | = r ξ . (5.51)Both agree with the semi-classical computation (5.38) for k = 0 which makes perfectsense as we are considering the IR theory which corresponds to the last step of thecascade. The VEVs of the chiral operators ˜ Q R Q R and Φ are independent of ξ . N + N f L ≥ N When 2 N + N fL ≥ N we have to consider three different cases. Case 2 N + N fL = N . The left node, which runs to strong coupling in theIR, has baryons and a quantum deformed moduli space, whilst the right node is IRfree. We construct the baryons A = A N Q N fL L , B = B N ˜ Q N fL L which are singletsof SU ( N ) and SU (2) AB . The quantum deformed moduli space is described by thesuperpotential W = W + X (det M − AB − Λ N ) (5.52)where X is a Lagrange multiplier. Besides the constraint det M − AB = Λ N , wealso get the F-term equations 0 = N + X (det M ) M − , 0 = X A = X B together with(5.44).There are two separate branches. The mesonic branch (characterized by X = 0)where A = B = 0 and therefore det M = Λ N . The solutions along this branchare the same as in the previous section (with the identification ǫ = X det M ). Thedynamically generated scale ǫ follows the same formulæ: (5.49) in the massless case,and (5.50) in the case of large mass and trivial vacuum.The baryonic branch is characterized by A , B 6 = 0, while X = 0 and N = 0. In particular M = M = N ˙ α = ˜ N α = 0, M = M = − Q R ˜ Q R and Φ = − µ √ h M to be either 0 or − µ √ h , The matrix (det M ) M − is the matrix of cofactors of M and therefore is a smooth functionof M . M ≤ N fR . For N fR < N – which will be our focus – this also impliesdet M = 0 and therefore AB = − Λ N . Let us compute the dynamically gener-ated scale in the massless case. Above the scale Λ , the second group SU ( N ) has2 N − N fR flavors and the instanton factor Λ N − N − N fR = Λ − N − N fL − N fR . Becauseof confinement of SU ( N ) below the scale Λ it has 4 adjoints and 2 N fL + N fR fla-vors (plus singlets), and the same instanton factor. All mesons receive mass h Λ ,therefore the low energy theory is a quartic SQCD N ,N fR with the instanton factorΛ N +4 N fL Λ − N − N fL − N fR h N +2 N fL and an effective superpotential W eff ∼ (cid:0) Λ N +8 N fL Λ − N − N fL − N fR h N +4 N fL + N fR (cid:1) N − NfR . (5.53) Case 2 N + N fL = N + 1. The moduli space of the strongly coupled left nodeis described by mesons and baryons with a superpotential. The baryons ¯ A and¯ B are in the fundamental and anti-fundamental representation of the flavor group U (2 N + N fL ), and we can decompose them as ¯ A = ( A α , ˜ F ) and ¯ B = ( B ˙ α , F )respectively. The superpotential is W = W − N − N − N fL (cid:0) det M − ¯ BM ¯ A (cid:1) . (5.54)The F-term equations are N = Λ − (3 N − N − N fL )1 ( M − det M − ¯ A ¯ B ), 0 = M ¯ B = ¯ AM and (5.44).The moduli space has two separate branches. On the mesonic branch det M 6 = 0,therefore ¯ A = ¯ B = 0 and N = det M Λ N − N − NfL M − . The solutions have been describedin section 5.3, and the scale ǫ is as in (5.49) and (5.50).There is another branch where ¯ A , ¯ B 6 = 0 and det M = 0. The F-term equationset N = − N − N − N fL ¯ A ¯ B . (5.55) For instance, for N fR = N there are vacua with | det M| = ( µ/ √ h ) N + N fL . By suitablytuning µ one could obtain AB = 0, that is no deformation. Indeed this corresponds in supergravityto a configuration with singular D7 embedding µ = ǫ .
49n particular, defining A α = ǫ αβ A β , B ˙ α = ǫ ˙ α ˙ β B ˙ β , we have M ˙ βα + δ ˙ βα Q R ˜ Q R = h Λ N − N − N fL A α B ˙ β , N ˙ α = h Λ N − N − N fL δ ˙ αα A α F , − µ √ h − Φ = ˜
F F , ˜ N α = h Λ N − N − N fL δ α ˙ α ˜ F B ˙ α , (5.56)which formally coincide with (5.11). The superpotential has the form W ∼ h Λ N − N − N fL (cid:2) det α ˙ α A α B ˙ α + . . . (cid:3) where the missing terms reproduce W . We get a ˆ N fL × ˆ N × ˆ N × ˆ N fR theory butwith ranks N fR × N × × N fL . If 2 + N fR < N we can borrow the results from section 5.3. Let us consider the µ = 0 case. First we need to match the scales. To that end we canonically normalizethe baryons ˆ¯ A = ¯ A / Λ N − , ˆ¯ B = ¯ B / Λ N − getting the coefficient in front of thesuperpotential ˆ h = 1 / ( h Λ ). Then we match the scale of SU ( N ). Above Λ it hasthe instanton factor Λ N − N − N fR . Below Λ it has 4 adjoints and 2 + 2 N fL + N fR fundamentals, with the instanton factor Λ − N − N fL − N fR low ∼ Λ N − N − N fR Λ − . The 4adjoints and 2 N fL fundamentals get mass h Λ , so that the scale of the SU ( N ) factoris ˆΛ N − − N fR ∼ Λ N +4 N fL − Λ − N − N fL − N fR +22 h N +2 N fL . Eventually we can plug the hatted quantities in (5.49): ǫ ∼ (cid:0) Λ N +4 N fL − N fR − Λ − N − N fL − N fR +2)2 h N +4 N fL + N fR − (cid:1) N − − NfR . (5.57) Case 2 N + N fL > N +1. All the remaining cases are very similar to the previousone. First, the mesonic branch of the SU ( N ) node (which might or might not bestrongly coupled) is described by the effective Affleck-Dine-Seiberg superpotential(5.40). Considering W eff = W + W ADS plus the D-term equations of SU ( N ) we getthe same type of solutions – (Higgsed) mesonic directions – as in section 5.3. Thedeformation scale is again given by (5.49) or (5.50).The analysis above however does not exhaust all set of vacua. To find the remain-ing ones, we dualize the SU ( N ) node to a SU (2 N + N fL − N ) gauge group with50esons M , dual quarks ˆ A , ˆ B and a superpotential W = W + 1Λ ˆ B M ˆ A , where W is expressed in terms of M and Q R , ˜ Q R , and the role of the scale Λ isexplained in [69]. For generic values of M the dual quarks are massive, the SU (2 N + N fL − N ) group can be integrated out and we reproduce W eff = W + W ADS and themesonic branch above.On the other hand if we integrate out the massive mesons as in section N fR × N × (2 N + N fL − N ) × N fL . This theory has its own (Higgsed) mesonic vacua, plus possibly other vacua obtainedby further dualizations. Notice that in the process the quiver is flipped, and mesonicoperators are mapped as in (5.11). Therefore a vacua labeled by ˆ n in the dual theoryhas a VEV for ˜ QQ corresponding to n = ˆ n + 1. We will see in section 6 what is thesupergravity counterpart of this fact.To get the dynamically generated scale on the mesonic vacua we proceed to matchthe scales. The left node SU ( N ) is dualized to SU ( ˆ N ), with ˆ N = 2 N N fL − N .Choosing the normalization scale Λ = Λ , we simply have ˆΛ = Λ . Integrating outthe mesons and rewriting the superpotential in terms of ˆ A , ˆ B , we get ˆ h = 1 /h Λ .The right node SU ( N ) is untouched, so that ˆ N = N . However its dynamicalscale gets modified. Above the scale Λ its instanton factor is Λ N − N − N fR . Belowthe scale Λ the gauge group has 4 adjoints and 4 N + 4 N fl − N + N fR flavors,with the instanton factor Λ N − N − N fL − N fR low ∼ Λ N − N − N fR Λ N − N − N fL )1 . Theadjoints and 2 N fL fundamentals get mass h Λ , so that the low energy SU ( ˆ N ) grouphas an instanton factor ˆΛ N − N − N fL − N fR ∼ h N +2 N fL Λ N Λ N − N − N fR . Bringingall together we haveˆ N fL = N fR , ˆ N = N , ˆ N = 2 N + N fL − N , ˆ N fR = N fL , ˆ h = 1 h Λ , ˆΛ N − N − N fL − N fR ∼ h N +2 N fL Λ N Λ N − N − N fR , ˆΛ = Λ . (5.58) In section 5.1 we dualized the right node, going “up in energy”, and then flipped the quiver.Proceeding backwards we go “down in energy”, as here. Besides we used a different normalizationof the mesons M . ǫ in all other vacua of the theories dualized multiple times (unfortunatelywe could not find a closed formula). Notice in particular that at each dualization theparameters of the low energy theory are related to those of the high energy theory byˆ I = I ( h Λ ) N fR , ˆ L = L I ( h Λ ) N fR . (5.59) vs field theory The map between the supergravity solutions presented in section 3 and the vacua ofthe field theories discussed in section 5 starts with the UV identification. The pa-rameters that identify the field theory, at some energy scale, are the gauge ranks N , and the number of flavors N fL,R . In supergravity one can compute the Page charges Q D3 , Q D5 , count the number N f ( r ) of D7-branes (we will suppress the dependence on r in the following) and measure the Wilson line ˆ ρ at some cut-off radius representingthe UV scale.The relation between the supergravity charges and the field theory ranks is foundwith a dictionary. The mutually BPS probe branes on the conifold are: two types offractional D3-branes (a D5-brane wrapped on the conifold’s S and an anti-D5-braneon S with − F ) each giving rise to one color (vectormultiplet) in the quiver, and two types of fractional D7-branes (both wrapping the S , one without and one with − C / Z N = 2 orbifold casein [64] (see our section 3.1). In the conifold case this matches with the expected RGflow (section 6.3). The Page charges for different D-branes areD3 D3 D7 fL D7 fR Q D3 Q D5 − − Q D7 N fL = Tr(1I − ˆ ρ ) / Q D3 = N + 14 N fL N = Q D3 + Q D5 + 14 N fL N fR = Tr(1I + ˆ ρ ) / Q D5 = N − N − N fL N = Q D3 − N fL (6.2)and we also define N f = N fL + N fR .This dictionary identifies the field theory description at some energy scale. Itis valid only if the NSNS potential b defined after equation (2.11) is in the range b ∈ [0 , If this condition is not met, we can perform a large gauge transformation b → b − [ b ] − which however shifts the Page charges.Let us compute (see appendix B.1) how Page charges shift under a large gaugetransformation B → B + α ′ πω , i.e. b → b + 1 Q ′ D5 = Q D5 − N f , Q ′ D3 = Q D3 − Q D5 + N f . (6.3)Since F = P [ B ] + 2 πα ′ F is gauge-invariant, the large gauge transformation shifts F → F − P [ ω ] and it affects the Wilson lines ˆ ρ → − ˆ ρ . We can then compute themodification of gauge theory ranks associated with such a shift. Using (6.2) we findthat the theory with ranks ( N fL , N , N , N fR ) is mapped to one with ( N fR , N , N + N fL − N , N fL ). It is known [70, 16, 61] that a large gauge transformation in the bulkcorresponds to a Seiberg duality in field theory. Indeed the shift of ranks agrees withSeiberg duality on the SU ( N ) IR strongly coupled node and a flip of the quiver.Further evidence of our dictionary between the Page charges and the ranks comesfrom the study of the RG flow in section 6.3.We have identified the field theory (more precisely, the effective description atsome energy scale) dual to the gravity background. Now we want to identify thecorrect vacuum using the following argument. Consider the background with D7-branes, no worldvolume flux and 0 < b <
1. Such a configuration corresponds tothe field theory with flavors on the right in its trivial vacuum with ˜ Q R Q R = 0. Let us now crank up the value of b , that is we change the gauge couplings. When If the B-field is outside the range [0 , e.g. the holo-graphic formulæ do not give real-valued gauge couplings). One possibility is to construct a correcteddictionary, another is to perform a large gauge transformation of B , as proposed in the text. This is in agreement with the naive holographic map since there is no worldvolume flux. = 1 the right node is at infinite coupling and we can move to a dual description byperforming Seiberg duality on the right node ( b → b − Q L Q L = − µ/ √ h .This is the Higgsed vacuum labeled by n = 1 in section 5.2.Seiberg duality corresponds to a large gauge transformation in supergravity whichincludes a shift F → F + P [ ω ]. The new background has one unit of worldvolumeflux on each of the D7s while in the dual field theory the flavors are on the left. Welearn that the background with n = 1 units of worldvolume Abelian flux correspondsto a theory in the n = 1 Higgsed vacuum. Repeating the argument (possibly in theopposite direction as well) we recover the quiver dimensions (5.28) and the VEVs of˜ QQ (5.35), and conclude that a background with n units of Abelian worldvolumeflux F corresponds to the Higgsed vacuum labeled by n (5.29). Let us compare the expectation values of protected operators from the flavor sectorcomputed in field theory and in supergravity. Here we restrict for the moment to thecase with N f = 1.We start with the singular conifold, discussed from the gravity point of viewin section 3.7. The D7-brane affects the background above µ , while below µ thebackground is unperturbed and the low-energy theory is on the mesonic branch. Thevacua of the theory above µ are the classical (Higgsed) mesonic vacua of section 5.2,with ˜ Q i Q i = − µ √ h r , Q † i Q i − ˜ Q i ˜ Q † i = 0 , where r = (cid:2) n +12 (cid:3) − and i = R ( L ) for n even(odd). In the case of even n we havean exact matching with the supergravity computation (3.77) up to an overall nor-malization factor h − / . This is a universal factor for the operator ˜ QQ in all vacua.Such normalization factors are anyway unavoidable as the kinetic term of Q is notexplicitly known in field theory.In the case of odd n we cannot directly compare with supergravity: the back-ground has a non-trivial worldvolume connection A ( i.e. Z Wilson line ˆ ρ = −
1) at Notice that to derive the VEVs (5.48) of the quantum theory one would have to include thenon-perturbative superpotential W ADS in the analysis of section 5.1. N ↔ N , N fL ↔ N fR and ˜ QQ → − ˜ QQ . On the gravity side this corresponds tochanging the sign of F , B and F . Moreover, since − b is outside the range [0 , − b → − b + 1. As a result n → n ′ = − n − n into even n ′ and r → r ′ = − r . Since for even n ′ field theory andsupergravity match, we have established agreement for odd n as well.Let us try to understand what exactly happens when n is odd. In this case thegravity computation gives a result which is shifted, compared to the field theory VEV,by an n -independent number˜ Q L Q L (cid:12)(cid:12)(cid:12) field theory = ˜ Q L Q L (cid:12)(cid:12)(cid:12) gravity − µ √ h . (6.4)The interpretation is that, for odd n , the gravity field F is dual to a mix of the operator˜ Q L Q L with the unity operator multiplied by µ/ √ h , which has the same dimensionand R-charge. We saw in section 5.1 that Seiberg duality mixes the operators: onehas to introduce the shift above to make this mixing compatible with the large gaugetransformation in the bulk.In the resolved conifold case the field theory VEVs are˜ Q i Q i = − µ √ h r , Q † i Q i − ˜ Q i ˜ Q † i = rξ whilst the gravity result is Q † Q − ˜ Q ˜ Q † = a n/ n we have agreement,up to a universal overall coefficient, as the resolution parameter a is proportional to ξ . We have agreement for odd n as well, by flipping the quiver and noticing that itmaps ξ ↔ ξ = − ξ .As before, for odd n the gravity computation differs from the field theory VEVby an n -independent constant shift Q † L Q L − ˜ Q L ˜ Q † L (cid:12)(cid:12)(cid:12) field theory = Q † L Q L − ˜ Q L ˜ Q † L (cid:12)(cid:12)(cid:12) gravity + a . (6.5)Therefore the operators ˜ Q L Q L and | Q L | − | ˜ Q L | , which in the N = 2 case forman SU (2) R triplet, mix with the deformation/resolution parameters µ and a ( ξ ) ofΣ = C / Z . The flip transformation maps A ↔ B , and invariance of the superpotential 5.1 with η L,R = √ h requires h → − h , ˜ QQ → − ˜ QQ and √ h µ → −√ h µ , so that µ/ √ h is invariant.
55n the deformed conifold case, depending on whether or not there are mobileD3-branes, below µ the low-energy theory is either on the mesonic branch or at the Z -invariant point of the baryonic branch. The quantum analysis of sections 5.3 and5.4 gives us for the theory above scale µ (5.48)Φ = − r µ / − ǫh (2 r − − µ √ h , or ˜ Q R Q R = − r µ / − ǫh r and | N α | − | ˜ N α | = 0 or | Q R | − | ˜ Q R | = 0. For even n we have a perfect agreement withthe gravity result (3.87): remarkably the non-perturbative field theory effects encodedin ǫ are precisely reproduced by the geometry of Σ embedded in the deformed conifold.For odd n we cannot flip the quiver, because the quantum analysis of sections 5.3,5.4 assumes that the left node goes to strong coupling. On the other hand we canexploit the dictionary (6.4), derived for the classical vacua, which is an identificationin the UV that does not rely on the IR effects. Again we find a perfect agreement.Finally, the resolved deformed conifold (or BGMPZ background) describes theKS theory on the baryonic branch. From the gravity computation (3.95) the VEV of | Q | − | ˜ Q | grows as n , as opposed to the linear growth (3.82) in the resolved conifoldcase. The quadratic in n behavior is indeed what we find in (5.51) from the quantumfield theory analysis (for odd and even n correspondingly) X ˙ α | N α | − X α | ˜ N α | = n − ξ , and | Q R | − | ˜ Q R | = n ξ , (6.6)where in the left case we again find an n -independent shift in the gravity result.The gravity computation is done in the gauge such that b = 0 at the tip of thedeformed conifold. Hence this calculation refers to the lowest step in the cascade ofSeiberg dualities, k = 0. Indeed the quantum VEVs above match the semi-classicalcomputation (5.38) with k = 0. N f > and noncommutative instantons When N f >
1, the moduli space includes the instanton-like directions (section 5.2),which represent the mobile D3-branes dissolved into the D7s forming the conventionalgauge instantons with continuous moduli. On the field theory side this picture isbacked by the fact that any solution to the N = 2 C / Z ADHM equations – i.e. vacuum equations in field theory – is also a solution to the classical F- and D-term56quations in the N = 1 conifold case. On the quantum level this relation holds in the ǫ → C / Z , hence the modulispaces of instantons in the two cases share the same complex structure. Thereforethe parallel with the N = 2 C / Z theory provides a good qualitative understandingof these vacua.It is interesting however to consider D7-branes embedded in the BGMPZ back-ground. In this case the SUSY condition for the worldvolume gauge field is notanti-self-duality, but rather a non-linear deformation of it (see section 3.10) so thata parallel to N = 2 instantons is less straightforward. On the other hand the clas-sical field theory analysis of section 5.2 is not sensitive to the low-energy theoryand is valid in all cases. Since the baryonic vacua of the low energy theory require( k + 1) ξ + kξ = 0, i.e. ξ + ξ = 0, the “non-linear instantons” in the BGMPZ back-ground are related to the noncommutative instantons on C / Z . This relation helpsunderstanding why the non-linear instantons in question cannot shrink to zero sizeand leave the D7s and become the mobile D3-branes in the bulk. We know that thisindeed must be the case, because the mobile D3-branes are not SUSY on the BGMPZbackground [21] (see sections 2.2 and 5.2 for a field theory explanation), but this isnot apparent from the SUSY condition (3.88) itself. The relation to noncommuta-tive instantons partially clarifies this point, as the noncommutative instantons cannotshrink to zero size and leave the larger brane as well [71]. It would be interesting tostudy the moduli space of the non-linear instantons satisfying (3.88) and provide anexplicit map, in the spirit of the Seiberg-Witten map [71], to the noncommutativeinstantons on Σ.Another interesting question is related to the vacua that completely break thewhole gauge symmetry of the field theory. These vacua admit generic values of ξ , ξ and the form of the F- and D-term equations (or at least the N = 2 ADHMequations) suggest that we are dealing with the noncommutative instantons. Thiscomment equally applies to the N = 2 C / Z orbifold theory and to the N = 4theory broken to N = 2 by flavor. It is tempting to attribute the appearance ofthe noncommutative instantons to the presence of a self-dual B -field in the bulk [72],however such B -field is not normalizable [54] and therefore cannot describe a branchof vacua of the field theory. In fact since the whole gauge symmetry is broken, i.e. all D3-branes are dissolved in the D7s, the probe approximation for D7s breaks down57nd we have no control over the geometric description. Eventually we want to compare the RG flow of gauge couplings in the large N limit, computed in field theory with the NSVZ beta-function formula [32], withthe backreacted supergravity solutions of section 4. Those solutions represent an SU (2) × SU (2) × U (1) invariant smeared distribution of D7-branes, which describea precise large N field theory dual [15]. In the Veneziano limit N f /N c = fixed,the number N f of D7-branes is large. Let us parametrize them with a flavor index U ∈ U (2) which takes N f values in U (2). Each D7-brane has a different embeddingand correspondingly a different superpotential, e.g. in the “Right” case:Σ U = { U α ˙ α w ˙ αα = µ } W ⊃ − h ˜ Q UR (cid:16) U α ˙ α B ˙ α A α − µ √ h (cid:17) Q UR . (6.7)In the N f → ∞ limit the index U becomes continuous and, if we uniformly distributethe N f values on U (2), the theory acquires an extra U (2) ∼ = SU (2) × U (1) sym-metry. Since the running of gauge couplings does not depend on the details of thesuperpotential, it will be the same as in the original theory in (5.1).At large N , the field theory is quasi-conformal and the anomalous dimensions (atscales much larger than √ h µ ) are fixed by the quartic superpotential (5.1) togetherwith charge conjugation symmetry γ [ A ] = γ [ B ] = γ [ Q R,L ] = γ [ ˜ Q R,L ] = − . (6.8)At scale Λ where the effective description has ranks N fL × N × N × N fR the NSVZformula ∂∂ log Λ 8 π g = 3 T [ G ] − X chiral i T [ r i ](1 − γ i ) (6.9)gives ∂∂ log Λ 8 π g = 3 (cid:16) N − N − N fL (cid:17) , ∂∂ log Λ 8 π g = 3 (cid:16) N − N − N fR (cid:17) . (6.10)Let us extract the RG flow from the backreacted supergravity solution. First,consider the massless solution ( µ = 0) characterized by the dilaton e φ (4.4) and the58-field b ( ρ ) (4.6). The gauge couplings can be extracted with the holographic formulæ(2.11) 8 π g = 2 πe φ b , π g = 2 πe φ (1 − b ) . (6.11)We obtain ∂∂ρ π g = − N f c , ∂∂ρ π g = 3 N f ( c − . (6.12)The holographic formulæ can be applied in a gauge where b ∈ [0 , B should be shifted by a large gauge transformation to meet the condition, andthe Page charges shift accordingly. In such a gauge, from (4.7) and (6.2) we obtain − N f c / N − N − N fL in terms of the effective description. After identifying ρ = log Λ, supergravity precisely reproduces the NSVZ beta-function. We stress thatthe fully backreacted solution is necessary to reproduce the exact NSVZ result.Finally consider the backreacted supergravity solution for massive D7-branes with n units of worldvolume flux (in the KT approximation) detailed in appendix C. Againwe need the dilaton (C.8) and the B-field (C.14). Below the scale µ the dilatonis constant, the B-field is logarithmically running and supergravity reproduces thebeta-functions ∂∂ρ π g , = ± N − N ). At the scale µ (that we called ρ = ρ ) thepage charges in (C.15) jump by δQ D5 = n and δQ D3 = n , according to the breakingpattern of the Higgsed vacuum. Above the scale µ , we can use the holographicformulæ (6.11) writing the result first in terms of the Page charges (C.15) and then interms of the ranks using the dictionary (6.2), to exactly reproduce the NSVZ result(6.10). In this paper we studied the supersymmetric vacua of the N = 1 SU ( M + N ) × SU ( N )theory with bifundamental and flavor matter. In the limit N f ≪ N + M we usedthe dual geometries with probe D7-branes and worldvolume gauge configurations(“instantons”) to describe various Higgs vacua. In the N = 1 case, as opposed to N = 2, supersymmetry is not powerful enough to prevent quantum corrections to theHiggs branch. On the gravity side in most cases the quantum corrections arise fromthe deformations of the geometry and of the K¨ahler potential in the bulk, affectingthe VEVs of the protected operators from the flavor sector. On the field theory side,instead, the quantum corrections arise from the non-perturbative contributions to the59uperpotential and the change of degrees of freedom: when a gauge group confines,the original microscopic flavor degrees of freedom are not relevant anymore and onehas to use the low energy meson variables.In the N = 2 case there is a direct relation between the bulk description ( i.e. the world-volume instantons) and the field theory description ( i.e. the F- and D-term equations), given by the ADHM construction. Clearly this relation does notrely on the AdS/CFT correspondence. The opposite is also true: the AdS/CFTduality predicts a one-to-one correspondence between the field theory vacua and theconfigurations in the bulk, but does not outline in details how to construct the map.The fact that such a direct relation is known in the N = 2 case is a nice bonus. Itis not immediately clear if such a relation can be found for N = 1 theories: althoughthe instantons in the bulk do have some version of the ADHM construction, thecorresponding matrix equations are different from the quantum version of the vacuumequations in field theory.So far we mainly discussed the quantum corrections from the field theory point ofview. In fact the N = 1 case can be drastically different from the N = 2 case in thebulk as well. When the underlying N = 1 background has a complicated structure,the SUSY condition for the wold-volume gauge fields becomes nonlinear [73, 23]. Wesaw that when this happens on the conifold, the resulting nonlinear instantons arerelated to noncommutative instantons on the same space. It would be very interestingto study the moduli space of these nonlinear instantons systematically and investigateif one can find such configurations with some sort of matrix equations in the spirit ofthe ADHM construction. Acknowledgments
We thank F. Fucito, S. Kachru, I. R. Klebanov, Z. Komargodski, J. Maldacena,N. Nekrasov, N. Seiberg, Y. Tachikawa, H. Verlinde, and E. Witten for numerousdiscussions. F.B. would like to thank the Aspen Center for Physics, the GalileoGalilei Institute for Theoretical Physics in Florence, the High Energy group at theWeizmann Institute and the Simons Center for Geometry and Physics, for hospitalityand support during the course of this work. A.D. would like to thank the Aspen Centerfor Physics, and the High Energy group at the Tel-Aviv University for hospitality. Thework of F.B. was supported in part by the US NSF under Grants No. PHY-084482760 = βα c a U T . . .c a L c a U T . . . c a L . . . ... ... . . . B T = βα c − a U T . . . − c − a L c − a U T . . . − c − a L . . . ... ... . . . A = βα c a L T . . . − c a U c a L T . . . − c a U . . . ... ... . . . B T = βα c − a L T . . .c − a U c − a L T . . . c − a U . . . ... ... . . . Q T L = αc (cid:16) . . . (cid:17) , ˜ Q L = αc − (cid:16) . . . (cid:17) , ˜ Q R = Q R = 0 . Table 2: Higgsed mesonic vacua with resolution, left blocks.and PHY-0756966. The research of A.D. was supported by the DOE grant DE-FG02-90ER40542, by Monell Foundation, and in part by the grant RFBR 07-02-00878 andthe Grant for Support of Scientific Schools NSh-3035.2008.2.
A Higgsed vacua
In this appendix we give the explicit form of various vacua discussed in the main text.
A.1 Classical Higgsed mesonic directions with resolution
The blocks (5.30), (5.31) of the classical Higgsed mesonic directions can be generalizedto incorporate arbitrary parameters ξ and ξ . The left blocks for r ≥
1, of dimension1 × r × r ( r − ×
0, are in table 2. The variables a , . . . , a r − ( K = 2 r − α , β are the same as in the ξ , = 0 case (5.33) and (5.34) and the unknowns are c , . . . , c r − . The right blocks for r ≥
1, of dimension 0 × r ( r + 1) × r ×
1, are in table3. Again, a , . . . , a r − ( K = 2 r ), α , β are the same as before and the new unknownsare c , . . . , c r − . The blocks of one kind with r ≤ − r ≥ A α , B ˙ α and by exchanging Q L ↔ Q R , ˜ Q L ↔ ˜ Q R . 61 = βα c a U c a L T . . . c a U c a L T . . . c a U . . . ... ... ... . . . A = βα − c a L c a U T . . . − c a L c a U T . . . − c a L . . . ... ... ... . . . B T = βα − c − a U c − a L T . . . − c − a U c − a L T . . . c − a U . . . ... ... ... . . . B T = βα c − a L c − a U T . . . c − a L c − a U T . . . c − a L . . . ... ... ... . . . ˜ Q L = Q L = 0 , Q T R = αc (cid:16) . . . (cid:17) , ˜ Q R = αc − (cid:16) . . . (cid:17) . Table 3: Higgsed mesonic vacua with resolution, right blocks.The F-term are solved by (5.33), (5.34) for any choice of c j ’s. It is convenient todefine the quantities x j ≡ | αβ | a j ( c j − c − j ) . (A.1)From the D-term equations we get for a left (right) block the recursive equations ξ = j x j + ( j + 2) x j +1 j even(odd) , ξ K ≡ , − ξ = j x j + ( j + 2) x j +1 j odd(even) , ξ = | α | ( c − c − ) ± x . (A.2)The solution is x j = ( − j + K (2 K + 1) − (2 j + 1)8 j ( j + 1) ( ξ − ξ ) − ( − j + n K ( K + 1) − j ( j + 1)4 j ( j + 1) ( ξ + ξ ) , (A.3)and the resulting quark bilinears areL: Q † L Q L − ˜ Q L ˜ Q † L = | α | ( c − c − ) = ξ − x = r ξ + r ( r − ξ R: Q † R Q R − ˜ Q R ˜ Q † R = | α | ( c − c − ) = ξ + 2 x = r ( r + 1) ξ + r ξ . (A.4)Besides solving the vacuum equations in the N = 1 case, the matrices above solvethe N = 2 ADHM equations and describe the noncommutative Abelian instantonson C / Z (also see [74]). 62 = β α − η U T U − η U T L T . . .η L U η L L T − η U T U − η U T L T . . . η L U η L L T − η U U T . . . ... ... ... . . . M = β α η U T L − η U T U T . . . − η L L η L U T + η U T L − η U T U T . . . − η L L η L U T + η U L T . . . ... ... ... . . . M = β α η L T U η L T L T . . .η U U η U L T + η L T U η L T L T . . . η U U η U L T + η L U T . . . ... ... ... . . . M = β α − η L T L η L T U T . . . − η U L η U U T − η L T L η L T U T . . . − η U L η U U T − η L L T . . . ... ... ... . . . ˜ Q R = Q T R = α (cid:16) . . . (cid:17) N i = ˜ N i = Φ = 0 . (A.5)Table 4: Higgsed mesonic directions with deformation, right blocks. A.2 Quantum deformed Higgsed mesonic directions
When the left gauge group goes to strong coupling, we describe it using gauge-invariants. For 2 N + N fL < N there are only mesons, defined in (5.39), whilefor 2 N + N fL ≥ N there are also baryons. Mesons are always good coordinates onmesonic branches. The right blocks, of dimension 0 × r ×
1, are in table 4. Let us take r >
0, although the same ansatz gives the solution for both r and − r . The unknownsare η , . . . , η r − , η , . . . , η r − , r − , β , α . Setting η i = a i and η ij = a i a j we simplyhave M ˙ αα = B ˙ α A α and the mesons solve the underformed equations. That wouldcorrespond to the classical theory, where mesons are products of elementary fields. Inthe quantum theory – as a result of confinement – the mesons are independent fields.63he D-term equations (5.45) with ξ = 0 are identically solved. From the F-termequations we find0 = j η j − η j +1 − ( j + 3) η j +2 j = 1 , · · · , r −
30 = ( η k + η k +1 ) − (2 k − η k − , k + (2 k + 3) η k +1 , k +2 k = 1 , · · · , r − ≡ η r − , r − η η + 3 η − µ √ h α β = 1 η + 3 η ǫ = − hα η η − η ( η + 3 η ) . (A.6)The recursive equation for η j is the same as in the classical case, but the last equationfor j = 2 r − µ = 0 case we have to impose η − η = 0, and after arbitrarilyfixing a multiplicative constant by η + 3 η ≡
1, we get η j = j ( j +1) and η k − , k asgiven below with C = 1, C = 0. Fixing α in terms of ǫ we get α = − ǫh − (2 r ) . Inthe case with generic µ we proceed as follows: The general solution to the recursiveequations is η j = C + C (cid:2) − ( − j (2 j + 1) (cid:3) j ( j + 1) η k − , k = C r − k r k (4 k − . (A.7)Then we determine α and ǫ : α = − µ √ h η + 3 η η − η ǫ = − µ η η − η ( η − η ) . (A.8)Notice that only the ratio C /C affects the solution, while the overall normalizationdrops out. We should fix C /C to match ǫ , and then determine the full solution and α as a function of ǫ . However one can directly verify that α = − ǫ + µ / h (2 r ) = − ǫ + µ / h n . (A.9)We take the branch cut in the square root such that˜ Q R Q R = α = − r − ǫ + µ / h r (A.10)which matches with the ǫ → | r | , the equations have two solutions correspondingto r and − r . 64 = β α η U U T − η L T L − η L T U T . . .η U L η U U T − η L T L . . . ... ... . . . M = β α η U L T + η L T U − η L T L T . . . − η U U η U L T + η L T U . . . ... ... . . . M = β α η L U T + η U T L η U T U T . . .η L L η L U T + η U T L . . . ... ... . . . M = β α η L L T − η U T U η U T L T . . . − η L U η L L T − η U T U . . . ... ... . . . ˜ N = N T = βα (cid:16) ζ U T . . . (cid:17) Φ = α (cid:16) (cid:17) ˜ N = N T = βα (cid:16) ζ L T . . . (cid:17) ˜ Q R = Q R = 0 . (A.11)Table 5: Higgsed mesonic directions with deformation, left blocks.The left blocks, of dimension 1 × r ( r − ×
0, are in table 5. Let us take r > r and − r + 1. The unknownsare η , . . . , η r − , η , . . . , η r − , r − , ζ , β , α . Setting η i = a i , η ij = a i a j , ζ = a wehave M ˙ αα = B ˙ α A α , ˜ N α = ˜ Q L A α , N ˙ α = B ˙ α Q L and Φ = ˜ Q L Q L as in the undeformedequations.The D-term equations with ξ = 0 are identically solved. From the F-term equa-tions we find0 = j η j − η j +1 − ( j + 3) η j +2 j = 1 , · · · , r −
40 = ( η k +1 + η k +2 ) − k η k, k +1 + (2 k + 4) η k +2 , k +3 k = 1 , · · · , r −
20 = 1 − η η + 3 η + µ √ h α β = 1 η + 3 η ≡ η r − , r − ǫ = − hα η η − η − η ( η + 3 η ) ζ = ( η + η ) + 4 η , η + 3 η . (A.12)65n the massless µ = 0 case we have to impose η − η = 0, and after arbitrarily fixinga multiplicative constant by η + 3 η ≡
1, we get η j = j ( j +1) and η k, k +1 as givenbelow with C = 1, C = 0. Fixing α in terms of ǫ we get α = − ǫh − (2 r − .In the case with generic µ we first write down the general solution of the recursiveequations: η j = C + C (cid:2) − ( − j (2 j + 1) (cid:3) j ( j + 1) , η k, k +1 = ( C + 2 C ) r ( r − − k ( k + 1)(2 r − (2 k + 1) ( k + k )(A.13)Then we determine α and ǫ : α = µ √ h η + 3 η η − η ǫ = − µ η η − η − η ( η − η ) . (A.14)One can verify the following relation: (cid:16) α + µ √ h (cid:17) = − ǫ + µ / h (2 r − . (A.15)We take the square root asΦ = α = − r − ǫ + µ / h (2 r − − µ √ h (A.16)which matches the ǫ → (cid:12)(cid:12) r − (cid:12)(cid:12) + , the equations have two solutions corresponding to r and − r + 1. A.3 Quantum deformed Higgsed directions with resolution
The blocks of the previous section can be generalized to solve the D-term equation(5.45) with generic ξ .The right blocks ( n even) are constructed by taking the ansatz (A.5) and addingnew variables c ij in front of η ij below the diagonal, c − ij in front of η ij above thediagonal, c in front of Q R and c − in front of ˜ Q R . The new variables cancel out ofthe F-term equations. Let us define˜ x k − , k ≡ | αβ | η k − , k (cid:0) c k − , k − c − k − , k (cid:1) . (A.17)From the D-term equations we obtain the system: ξ = (2 k − k ˜ x k − , k − (2 k + 2)(2 k + 3) ˜ x k +1 , k +2 k = 1 , · · · , r − ξ = | α | ( c − c − ) − x , ˜ x r − , r ≡ . (A.18)66he solution is ˜ x k − , k = ξ r − k k (4 k −
1) (A.19)from which we extract | Q R | − | ˜ Q R | = r ξ .The left blocks ( n odd) are constructed by taking the ansatz (A.11) and addingnew variables c ij in front of η ij below the diagonal, c − ij in front of η ij above thediagonal, c in front of N i and c − in front of ˜ N i . Let us define˜ x = | α βζ | ( c − c − ) , ˜ x k, k +1 = | αβ | η k, k +1 (cid:0) c k, k +1 − c − k, k +1 (cid:1) . (A.20)From the D-term equations we get the system: ξ = 2 k (2 k + 1) ˜ x k, k +1 − (2 k + 3)(2 k + 4) ˜ x k +2 , k +3 k = 1 , · · · , r − ξ = ˜ x − x , ˜ x r − , r − ≡ . (A.21)The solution is ˜ x k, k +1 = ξ r ( r − − k ( k + 1)4 k ( k + 1)(2 k + 1) (A.22)from which we extract P i =1 , (cid:0) | N i | − | ˜ N i | (cid:1) = r ( r − ξ . B Page charges
Here we compute the Page D3- and D5-charges on the D7-brane. At some fixed radius r in the bulk the Page charges are given by (we keep g s = 1 everywhere in text) Q D3 ( r ) = 1(4 π α ′ ) Z T , at r F − B ∧ F + 12 B ∧ B ∧ F ,Q D5 ( r ) = 14 π α ′ Z S at r F − B ∧ F . (B.1)It will be useful to call the integrands J D3 and J D5 “Page currents”. Using the Bianchiidentities dF = δ D72 dF = 4 π α ′ δ D54 + H ∧ F + F ∧ δ D72 dF = (4 π α ′ ) δ D36 + H ∧ F + 12 F ∧ F ∧ δ D72 , (B.2)67n the absence of non-dissolved D3 and D5-branes we find dJ D3 = (2 πα ′ ) F ∧ F ∧ δ D72 , dJ D5 = (2 πα ′ ) F ∧ δ D72 , dJ D7 = δ D72 . (B.3)Here δ D72 is a delta 2-form localized on (and orthogonal to) the D7s.The D3-charge is given by the integral of J D3 on T , and using Gauss law itreduces to N D3 = 18 π Z Σ F ∧ F .
An important observation is that for any functions ξ ( t ), λ ( t ) from (3.60,3.58) F I ∧ F II (cid:12)(cid:12) Σ = 0 (B.4)and therefore the integral splits into two parts. The first part is a full derivative thatcan be computed at the boundary: π R Σ F I ∧ F I = 4 a ξ (cid:12)(cid:12) r = ∞ r = r min . Since ξ → r − forlarge r , the contribution at infinity is zero. If we require regularity of ξ at the tip, thecontribution at r = r min vanishes as well because a ( t min ) = 0. The only exception isthe case z = µ/ √ a ≡
1. Then the integral gives 4 ξ (0) . In the deformedconifold case ξ (0) must vanish because g is not well-defined at the tip; in the resolvedconifold case only the combination dg + ω is regular at the tip, hence ξ (0) = n .To calculate the second part we notice that F II ∧ F II n | z − ǫ | ( r − | z | + | z − ǫ | ) dz ∧ d ¯ z ∧ dz ∧ d ¯ z | z | , (B.5)and using (3.20) and (3.22) and integrating over t we get π R Σ F II ∧ F II = n . Theonly exception is the resolved conifold case ǫ = 0 with z = 0. In this case F II vanisheseverywhere except at the tip and, as follows from (B.5), the second part is zero. Weconclude that, in all cases, N D3 = n . (B.6)The D5-charge is given by the integral of J D5 on S ⊂ T , and using Gauss lawit reduces to N D5 = 12 π Z Γ F ,
Γ = Σ ∩ ( S × R + ) . Γ is a two-submanifold inside Σ whose radial sections are circles S = Σ ∩ S . Since F = F I + F II = d ( A I + A II ), we easily compute 2 πN D5 = R ∂ Γ ( A I + A II ). Theboundary ∂ Γ is the difference between an S at large radius and an S at the tip r min ,where S shrinks into a point. R ∂ Γ A I vanishes at infinity because ξ goes to zero, and68ince ξ is regular at r min , the contribution there vanishes as well (with the exception ǫ = z = 0). Similarly R ∂ Γ A II does not contribute at infinity, but it does at r min .Although S shrinks, the potential A II is singular. To get the answer we computethe integral R S A II at radius r and take r to r min . To do that we need to define S more explicitly. We use the coordinates (3.24) but now on the deformed conifold X = (1 + ǫ r − ), Y = (1 − ǫ r − ), X · Y = 0. We define S at the given radius r as follows: we take a point ( X, Y ) and consider its orbit under the global symmetry SU (2) L . There are many different S corresponding to different initial points, butsince F , is closed, N D5 will not depend on the choice of S . To understand how S intersects Σ, let us start with ( X, Y ) that actually belongs to Σ i.e. z = µ/ √ U (1) ⊂ SU (2) L that keeps z invariant, and its orbit is thedesired S which is the homologically non-trivial path on S / Z . Such U (1) acts on z i as a rotation around the constant vector n i , i = 1 , , dz i = ǫ ijk n j z k | ~n | dφ , n i = − i ( z i ¯ z − ¯ z i z + ǫ ijk z j ¯ z k ) , | ~n | = r − | ǫ | . (B.7)Indeed dn i = 0. Using the explicit form of σ we get A II = − n z p ¯ z − ¯ ǫ p r − | ǫ | dφ + c.c. (B.8)Hence the integral over S at the minimal radius r = | z | + | z − ǫ | gives N IID5 = n/ z = ǫ = 0 when (B.8) vanishes.Now we can return back to the contribution of R ∂ Γ( r min ) A I . Using (B.7) we find g = 2 [( r − | z | ) − | ǫ − z | ] r p r − | ǫ | dφ . (B.9)The integral of A I = ξ ( r ) g over S parametrized by φ located at the minimal radius r min vanishes, unless z = 0 in which case the expression for R S A I takes the form4 πξ ( r ) q r −| ǫ | r and in the resolved conifold case ǫ = 0 we simply get N ID5 = 2 ξ (0).Taking into account that ξ (0) = n/ N IID5 vanish in thiscase, we get in all cases N D5 = n . (B.10)69 .1 Shift of Page charges Let us compute the shift of Page charges under B → B + πα ′ ω (accompanied by F → F − P [ ω ] i.e. n → n − Q D5 ( r ) − Q D5 (0) = 14 π α ′ Z S × I dJ D5 = N f π Z S × I F ∧ δ D72 = N f π Z Σ ∩ ( S × I ) F (B.11)where I is the interval [0 , r ] in r , and we have included the dependence on the numberof branes N f . For r → δQ D5 (0) = 0. We concludethat δQ D5 ( r ) = − N f π Z Σ ∩ ( S × I ) ω = − N f r → ∞ limit. Then Q D3 ( r ) − Q D3 (0) = 1(4 π α ′ ) Z T , × I dJ D3 = N f π Z Σ ∩ ( T , × I ) F ∧ F (B.13)and its variation under a large gauge transformation is δQ D3 ( r ) − δQ D3 (0) = N f π Z Σ ∩ ( T , × I ) (cid:16) − ω ∧ F + 14 ω ∧ ω (cid:17) . (B.14)However this time the variation at r = 0 does not vanish. Using the fact that forevery closed g form, R T , ω ∧ g = 4 π R S g , we get δQ D3 (0) = − π α ′ Z T , ω ∧ F = − π α ′ Z S F = − Q D5 (0) . (B.15)Finally we use that for every closed g form with compact support on Σ, R Σ ∩ ( T , × I ) ω ∧ g = 4 π R Σ ∩ ( S × I ) g . Therefore δQ D3 ( r ) = − Q D5 ( r ) + N f π Z Σ ∩ ( T , × I ) ω ∧ ω = − Q D5 ( r ) + N f . (B.16)Again, in the last equality we took the r → ∞ limit. C Backreacted solution with massive flavors andworldvolume flux
We can generalize the solutions of section 4 to the case of a massive embedding µ = 0,possibly with worldvolume flux F (the solution without flux has been found in [31]).70he SU (2) × SU (2) × U (1) invariant ansatz is the same as in (4.1), but the numberof flavors N f is substituted by a radial function N f ( ρ ) ds = h − dx , + h h e u (cid:16) dρ + 19 g (cid:17) + e g X (cid:0) dθ i + sin θ i dϕ i (cid:1)i F = N f ( ρ )4 π g , B = α ′ πb ( ρ ) ω , H = α ′ πb ′ ( ρ ) dρ ∧ ω . (C.1)The unwarped metric is K¨ahler and hence a SUSY embedding must be holomorphic.To construct holomorphic coordinates on the backreacted background (C.1) we pro-ceed as follows. From the K¨ahler form J and the metric in (4.1) we construct thecomplex structure and the holomorphic projector J = 12 J ab dx a ∧ dx b , g = g ab dx a ⊗ dx b , J = J g − , P = J + i i . (C.2)One can check that given an expression for holomorphic coordinates z j ( r, ψ, θ i , ϕ i ) onthe usual singular conifold, the substitution r → e ρ provides holomorphic coordinateson the backreacted background that satisfy P dz i = dz i and P d ¯ z i = 0.The embeddings we consider are z = µ/ SU (2) × SU (2) × U (1). Let us compute the smeared charge distribution. Thesymmetries dictate the form of F and therefore δ smeared2 = dF = N ′ f ( ρ )4 π dρ ∧ g + N f ( ρ )4 π X sin θ i dθ i ∧ dϕ i . (C.3)To determine the function N f ( ρ ), we consider a single localized embedding in theensamble, e.g. z = µ/
2, and integrate an invariant 4-form, e.g. ω ∧ ω , on it up toradius ρ . We get Z ρ D7 ω ∧ ω = 8 π (cid:0) − | µ | e − ρ (cid:1) = Z ρ log 2 | µ | π | µ | e − ρ dρ . (C.4)On the other hand, integrating the same 4-form with the charge distribution δ smeared2 we get Z ρ ω ∧ ω ∧ δ smeared2 = Z ρ π N ′ f ( ρ ) dρ . (C.5)Comparing and solving the differential equation (and multiplying by the number ¯ N f of D7-branes) we get N f ( ρ ) = ¯ N f (cid:0) − | µ | e − ρ (cid:1) ≡ ¯ N f (cid:0) − e − ρ − ρ ) (cid:1) . (C.6)71e defined ρ = log 2 | µ | , which is the tip of D7-branes in the coordinate ρ .The SUSY equations are the same as before. For dilaton and metric we find φ ′ = 3 N f ( ρ )4 π e φ , u ′ = 3 − e u − g − N f ( ρ )8 π e φ , g ′ = e u − g . (C.7)The solution for the dilaton with the boundary condition φ ( ρ → − ) = + ∞ is e φ = 4 πf ( ρ ) , (C.8)we we introduced the function f ( ρ ) = ¯ N f (cid:2) − ρ + e ρ − e − ρ − ρ ) (cid:3) for ρ ≤ ρ < N f (cid:2) − ρ − e ρ (cid:3) = const ≡ f for ρ ≤ ρ (C.9)Notice that f ( ρ ) ≥ ρ ≤
0, and f (0) = 0. Moreover f ′ ( ρ ) = − N f ( ρ ), sothat f ′ ( ρ ≤ ρ ) = 0 and f ( ρ ) is continuous with its first derivative, while its secondderivative jumps. For e − ( ρ − ρ ) ≪ e ρ ≪ f ( ρ ) ≃ − N f ρ .Also u and g can be analytically solved e u = c − ρ + 2 e ρ − e − ρ − ρ ) (cid:2) − ρ + 2 e ρ − e − ρ − ρ ) + e − ρ − ρ ) (cid:3) / e ρ for ρ ≤ ρ < e ρ for ρ ≤ ρ e g = c (cid:2) − ρ + 2 e ρ − e − ρ − ρ ) + e − ρ − ρ ) (cid:3) / e ρ for ρ ≤ ρ < e ρ for ρ ≤ ρ (C.10)even though we will not need them. We imposed e u = e g at ρ = ρ , whilst thereis still one multiplicative integration constant c which should be fixed by continuity.One can check that both functions are positive for ρ ≤ ρ <
0. For ρ ≤ ρ , u = g = ρ .From the SUSY equation H = e φ ∗ F we get F = α ′ f ( ρ ) b ′ ( ρ ) g ∧ ω . (C.11)Then the Bianchi identity dF = H ∧ F + F ∧ δ smeared2 , taking into account that F = n P [ ω ], gives 13 ( f b ′ ) ′ = N f b ′ + N ′ f ( b + n ) (C.12)72or ρ ≤ ρ <
0, where n is the number of flux units, and ( f b ′ ) ′ = 0 for ρ < ρ .The equation can be solved on both sides of ρ giving (here Θ is the Heaviside stepfunction Θ( x ) = 0 for x < x ) = 1 for x > b ± = c ± f ( ρ ) + c ± ρf ( ρ ) − n Θ( ± . (C.13)If we impose continuity of b and b ′ , we get c +1 = c − + n f and c +2 = c − . Now we puteverything together b ( ρ ) = ( c + c ρ ) f − (cid:0) f ( ρ ) − f (cid:1) nf ( ρ ) (C.14)For ρ ≤ ρ , b ( ρ ) = c + c ρ which coincide with the B-field of the KT solution[7]. Here c = 6 Q lowD5 /f is related to the integer number Q lowD5 of fractional D3-branesat the tip, while c is a free parameter related to the difference of gauge couplings(which imposes a constraint on the 5-form flux by integrality of the Page Q D3 ). For ρ ≤ ρ we can compute the Page charges Q D5 = c f n N f ( ρ ) , Q D3 = Q D3 ( ρ = ρ ) + n N f ( ρ ) (C.15)where partial integration and the SUSY equations have been used. References [1] I. R. Klebanov and E. Witten,
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