Elaborations on the String Dual to N=1 SQCD
aa r X i v : . [ h e p - t h ] M a r Elaborations on the String Dual to N = 1 SQCD
Roberto Casero ∗ , Carlos N´u˜nez † and Angel Paredes ∗ ∗ Centre de Physique Th´eorique´Ecole Polytechniqueand UMR du CNRS 764491128 Palaiseau, France † Department of PhysicsUniversity of Swansea, Singleton ParkSwansea SA2 8PPUnited Kingdom.
Abstract
In this paper we make further refinements to the duality proposed between N = 1 SQCDand certain string (supergravity plus branes) backgrounds, working in the regime of com-parable large number of colors and flavors. Using the string theory solutions, we predictdifferent field theory observables and phenomena like Seiberg duality, gauge coupling and itsrunning, the behavior of Wilson and ’t Hooft loops, anomalous dimensions of the quark su-perfields, quartic superpotential coupling and its running, continuous and discrete anomalymatching. We also give evidence for the smooth interpolation between higgsed and confiningvacua. We provide several matchings between field theory and string theory computations.CPHT-RR 143.0907 [email protected] [email protected] [email protected] ontents ρ → ∞ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.1 N f < N c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.2 N f > N c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 IR behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.1 Type I expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.2 Type II expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.3 Type III expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 UV expansions revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 A comment on integration constants . . . . . . . . . . . . . . . . . . . . . . 22 N f = 2 N c case . . . . . . . . . . . . . . . . . . . . 284.4 Wilson Loop and QCD-string . . . . . . . . . . . . . . . . . . . . . . . . . . 294.5 t’Hooft loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.6 Back to the 1970’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.7 A comment on the running couplings . . . . . . . . . . . . . . . . . . . . . . 331.8 Global Anomaly matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.8.1 A different charge assignation . . . . . . . . . . . . . . . . . . . . . . 38 κ m Maldacena proposed the AdS/CFT Conjecture [1] almost ten years ago and since then,great advances have taken place to refine and extend the original duality (see [2],[3] andreferences to those articles). One of the lines of research that captured the interest of manyphysicists is the use of the ideas and framework presented in [1]-[3], to formulate duals tophenomenologically relevant field theories. The final aim of this line is to find a dual tolarge N c QCD, that according to early ideas of t’Hooft [4] should be described by a theoryof strings.A slightly less ambitious project (and perhaps a necessary step to overtake the case ofQCD) is to find duals to phenomenologically viable field theories with minimal SUSY. Again,remarkable advances on this area of research have been achieved in the last nine years, seefor example [5] and references to those works. One well-known limitation of the papersmentioned in [5] and successive work on that line is the fact that the dual field theory containsfields transforming only in the adjoint representation (also in bi-fundamental representationsfor the cascading theories), hence they could be thought of as SUSY versions of Yang-Mills.The rich dynamics added by the presence of fields transforming non-trivially under the centerof the gauge group, was absent in the examples of [5].To ameliorate this situation the introduction of ‘flavor’ branes (that is spacetime fillingbranes usually with higher dimensionality than the ‘color’ branes) was considered. Indeed,the flavor degrees of freedom need the presence of an open string sector, and this can only beachieved by placing in the background N f flavor branes, on which the SU ( N f ) symmetry willbe realized. This problem was first treated in the ‘probe’ approximation, where the smallnumber of N f flavor branes was not changing the geometry generated by the large numberof N c color branes. This non-backreacting approximation has a diagrammatic counterpartin the dual QFT, called the quenched approximation. The basic idea is that when funda-mentals run on internal loops of the diagram, the putative diagram is weighted by powersof N f /N c and therefore suppressed in this approximation. In other words, the quenchedapproximation boils down to considering the path integral of a theory with adjoints and fun-damentals and approximating it by the path integral of the theory with adjoints only, where2he fundamentals are only external states (or run on external lines). The approximation isgood if the quarks are very massive, hence difficult to pair produce, or if N f /N c → N c → ∞ , N f = f ixed . Indeed, one can see that this is the case. From experiencein lattice field theory, the quenched approximation is good to compute static properties (likethe spectrum of QCD), but works poorly when trying to address thermodynamical, phasetransitions or finite density problems (see for example [7]). Obviously quenching will be a badapproximation if the quarks are light and/or if the number of fundamentals is comparableto the number of adjoints.A natural problem is then to understand how to backreact with the flavor branes or, in thedual field theory, how to ‘un-quench’ the effects of the fundamentals. From the diagrammaticviewpoint, this unquenching appears when considering the Veneziano Topological Expansion[8] instead of the t’Hooft expansion [4].On the string side, this problem did not go unnoticed and the approach proposed was tofind solutions to an action that puts together the supergravity action (for the color branes)and the Born-Infeld-Wess-Zumino action (for the flavor branes). S = S gravity + S branes = S IIB/A + S BI + S W Z (1.1)Cases where the flavor branes dynamics, represented by S branes in eq.(1.1) are localizedby some delta-function (that is, finding solutions in pure supergravity for intersecting D p - D p +4 branes), have been discussed, see for example [9]. In these cases, typically there isa curvature singularity at the position of the delta-function, which may hide some of theinteresting dynamics arising from S branes .On the other hand, cases where S branes has support in the whole spacetime were firststudied in non-critical string set-ups [10] and later in type II supergravity where a varietyof examples were constructed, ranging from duals to a version of minimally SupersymmetricQCD (SQCD) [11] and [12], N = 2 version of SQCD [13] and the addition of fundamentals tocascading field theories/backgrounds [14]. Recently, a localized solution for the D − D − ¯ D N = 1 SQCDexample mentioned above can be found in [16] and applications to SUSY breaking in [17].From a more formal point of view it was shown in [18] that our intuitive way of adding flavorbranes can be put on a formal basis where SUSY is guaranteed whenever the flavor branesare calibrated, and the second-order Einstein equations are automatically satisfied once thefirst-order BPS and Maxwell equations and Bianchi identities with sources are solved.Notice that in the context of critical string theory, backreacting with the flavor branes as3ndicated in eq.(1.1) is a natural continuation of the approach in [6] and indeed the correctway to introduce the fundamental degrees of freedom with SU ( N f ) global symmetry.In this paper, we will study the dual to minimally SUSY QCD that was proposed by thepresent authors in [11]. Indeed, in [11] a set of solutions to type IIB supergravity plus braneshas been found and a large number of checks presented, where it was shown that the typeIIB backgrounds were capturing non-perturbative effects of this version of SQCD.We will present a new set of solutions to type IIB supergravity plus branes, dual to thefield theory in different vacua and will provide a large number of matchings to show thatour new solutions indeed capture new qualitative and quantitative information of the dualQFT.The organization of this paper is as follows: in section 2 we will review some materialfrom [11], set up the notation for the string backgrounds and dual SQCD. We also discusssome aspects of the SQCD-like theory that can be learned from the background withoutknowing the explicit solution, like Seiberg duality and the matching of R-symmetry anoma-lies, and we discuss some aspects of the field theory that will be relevant for the rest ofthe paper. In section 3 we present the new solutions that motivate this paper. We showthat an expansion for large values of the radial coordinate (the UV of the dual SQCD)can be smoothly connected with the IR expansion of the solution. We discuss in detailthe numerics and present plots to clarify ideas. In section 4 we study the predictions thatthese new solutions make for the physics of the SQCD theory; they include definitions ofgauge coupling, anomalous dimensions, quartic coupling and computation of the beta func-tion, continuous and discrete anomaly matching, potential between quark-antiquark andmonopole-antimonopole pairs as computed by Wilson and ’t Hooft loops and different as-pects of the QFT that we learn from the string solution. We close with some conclusionsand future work to appear. We complement our presentation with an appendix on a nicetechnical subtlety. In this section, we will go over some points of [11] that will be relevant in the rest of thepaper. First, we will review the two types of backgrounds that follow from the work [11].After that, we will discuss some aspects of the dual QFT.
We presented in [11] a string dual to a version of N = 1 SQCD. The construction proceededas explained around eq.(1.1) by adding flavor branes to a given supergravity solution andfinding a new background where the effects of flavor branes were encoded. The unflavored N = 1 SYM in [19](the solution was originally found in 4d gauged supergravity in [20]). We considered thebackreaction of the flavor branes, that in this particular case are N f D A and solutions of type N .Backgrounds of type A look quite simple, consist on a metric, a Ramond three-form, adilaton and they read, in Einstein frame and setting α ′ = g s = 1 (we have slightly changedthe notation with respect to [11]), ds = e φ ( ρ )2 h dx , + 4 Y ( ρ ) dρ + H ( ρ )( dθ + sin θdϕ ) + G ( ρ )( d ˜ θ + sin ˜ θd ˜ ϕ )+ Y ( ρ )( dψ + cos ˜ θd ˜ ϕ + cos θdϕ ) i F (3) = − h N c θd ˜ θ ∧ d ˜ ϕ + N f − N c θdθ ∧ dϕ i ∧ ( dψ + cos ˜ θd ˜ ϕ + cos θdϕ ) ,φ = φ ( ρ ) , (2.1)where we have used the coordinates x = ( x µ , ρ, θ, ϕ, ˜ θ, ˜ ϕ, ψ ). The functions in eq.(2.1), satisfya set of BPS eqs (that solve the Einstein, Maxwell and Bianchi eqs). The derivation of theseBPS eqs was discussed in detail in [11]. They read: H ′ = 12 ( N c − N f ) + 2 Y (2.2) G ′ = − N c Y (2.3) Y ′ = −
12 ( N f − N c ) YH − N c YG − Y ( 1 H + 1 G ) + 4 Y (2.4) φ ′ = − ( N c − N f )4 H + N c G (2.5)On the other hand, solutions of type N are slightly more involved and can be seen as ageneralization of the type A backgrounds. The dilaton is a function of the radial coordinate φ = φ ( ρ ), while the metric and Ramond three form read, ds = e φ ( ρ )2 h dx , + e k ( ρ ) dρ + e h ( ρ ) ( dθ + sin θdϕ ) ++ e g ( ρ ) (cid:0) (˜ ω + a ( ρ ) dθ ) + (˜ ω − a ( ρ ) sin θdϕ ) (cid:1) + e k ( ρ ) ω + cos θdϕ ) i F (3) = N c " − (˜ ω + b ( ρ ) dθ ) ∧ (˜ ω − b ( ρ ) sin θdϕ ) ∧ (˜ ω + cos θdϕ ) + b ′ dρ ∧ ( − dθ ∧ ˜ ω + sin θdϕ ∧ ˜ ω ) + (1 − b ( ρ ) ) sin θdθ ∧ dϕ ∧ ˜ ω − N f θdθ ∧ dϕ ∧ ( dψ + cos ˜ θd ˜ ϕ ) . (2.6)5here ˜ ω i are the left-invariant forms of SU (2)˜ ω = cos ψd ˜ θ + sin ψ sin ˜ θd ˜ ϕ , ˜ ω = − sin ψd ˜ θ + cos ψ sin ˜ θd ˜ ϕ , ˜ ω = dψ + cos ˜ θd ˜ ϕ . (2.7)For the case of backgrounds of type N we also found a set of BPS eqs and solutions to it,by expanding in series near ρ = 0 (the IR of the dual QFT) and ρ → ∞ (the UV of the dualQFT) and then showing that the two expansions can be numerically matched in a smoothway. It was for the backgrounds of type N that we presented a considerable number ofchecks showing how they captured non-perturbative aspects of SQCD [11].In this paper, we will focus on backgrounds of type A above, finding a new set of solutionsfor the functions H, G, Y, φ and we will study the strong-coupling effects that they predictfor the dual QFT. This kind of solutions was first overlooked in [11], because at that timewe could only find badly singular type A solutions. We believe the new set of backgroundswe derive in the present paper, which are characterized by a good IR singularity, nicelycomplement the overall picture.After having set up the stage, it is now convenient to discuss some subtleties about thedual theory, that as we anticipated above, is a version of N = 1 SQCD. As it is well known, all duals to non-conformal gauge field theories relying on the supergravityapproximation like the ones in [5], [19] or any other model available, are characterized by thepresence of extra modes or some other kind of UV completion . This UV completion is notnecessarily an ugly feature, one could perhaps be interested in the field theory with the UVcompletion that string theory provides (like in the case of cascading theories) or perhaps,there are ways to distinguish when a given field theory correlator is affected or not by theextra modes. In the particular case of interest in this paper, the field theory dual to thebackground in [19] [20] was proposed in the paper [19] to be N = 1 SYM plus an infiniteset of KK modes, which is just another way of saying that the UV completion is a higherdimensional theory. The dynamics of these extra modes was studied in detail in [21] and away to disentangle the effects of the UV completion from the Super-Yang-Mills modes wasproposed in [22]. In summary, the picture of the dual field theory of [19] is clear by now.Simplifying things a bit, one could say that the field theory without flavors is described by This can be avoided in the non-critical string approach, but in that case the supergravity approximationis not reliable. L = − T r [ 14 F µν + iλDλ − X k ( D µ φ k ) − M KK φ k + ψ k ( iD − M KK ) ψ k + V ( φ k , ψ k , λ )] == Z d θ X k Φ † k e V Φ k + Z d θ (cid:16) W α W α + X k W (Φ k ) (cid:17) + h.c. (2.8)where, schematically, we wrote a lagrangian describing N = 1 SYM coupled to an infinitetower of SUSY KK modes, with a given superpotential W (Φ k ) (for a detailed expressionsee [21]).We proposed in [11] that the addition of the flavor branes introduces two chiral superfields Q, ˜ Q transforming in the fundamental and anti-fundamental of the SU ( N c ) gauge group,respectively. The dynamics and the coupling of the quark superfields with the alreadyexisting adjoints was proposed to be of the form, L = Z d θ ( Q † e V Q + ˜ Q † e V ˜ Q ) + Z d θ ˜ Q Φ k Q + h.c. (2.9)where Φ k is a generic adjoint field. So, putting together (2.8) and (2.9), we have a lagrangianthat in superspace will schematically read, L = Z d θ (cid:16) Q † e V Q + ˜ Q † e V ˜ Q + X k Φ † k e V Φ k (cid:17) + Z d θ (cid:16) W α W α + X k ˜ Q Φ k Q + W (Φ k ) (cid:17) + h.c. (2.10)Let us consider the simplest case in which the superpotential contains only a mass term W (Φ k ) = µ k Φ k . The occasional presence of higher powers of the chiral multiplet of KKmodes will be irrelevant for the IR dynamics as we will see below. Since we are interested instudying the field theory at low energies, we can integrate out the scalar KK modes. Thisshould be taken with some care because the mass scale of the KK modes is comparablewith the scale of strong coupling which we are interested in, so the integration-out of theseadditional modes is not totally clean. After integrating over the dynamics of the scalarmultiplet, we get a dynamics for the quarks and vector superfields of the form L = Z d θ (cid:16) Q † e V Q + ˜ Q † e V ˜ Q (cid:17) + Z d θ (cid:18) W α W α + 1 µ h T r ( ˜ QQ ) − N c ( T r ˜ QQ ) i + O ( ˜ QQ ) (cid:19) (2.11)So, we arrive to a theory with the field content of N = 1 SQCD, but with an importantdynamical difference: the flavor symmetry is SU ( N f ). The root of this difference is basicallythat the ‘mother theory’ (2.10) has already SU ( N f ) and not SU ( N f ) l × SU ( N f ) r becauseof the term ˜ Q Φ k Q . This same qualitative difference from SQCD or QCD will genericallyappear when adding flavors to field theories that could be thought of as coming from an N = 2 theory. On the other hand, when color and flavor branes are codimension six, as for7nstance in the last paper of ref.[6] and [23, 15], this kind of coupling is not present. Anotherinteresting point is that if the adjoint superfield transforms in U ( N c ) instead of SU ( N c ),then the double trace term in the superpotential is absent.In the following, only the quartic coupling for the fundamentals will be considered. Insection 2.2.2, we will argue that higher order couplings are irrelevant. We present here aheuristic complementary argument which favors the appearance of the quartic coupling andnot the higher ones upon integrating-out the adjoint scalars. The superpotential for theadjoint scalars is of the form [21], W (Φ k ) = T r h Φ [Φ , Φ ] + X i µ i i . (2.12)The “democratic” way in which the N f flavor branes are treated suggests that the ˜ QQ matrixis proportional to the identity (notice that ˜ QQ here is not a meson operator, gauge indicesare not contracted). When we integrate out the adjoints, and insert back their value in W (Φ k ), the higher order contributions involving ˜ QQ vanish using conmutation relations.We would like to mention at this point that, in order to get our BPS eqs and their solu-tions, we introduced in [11] a smearing procedure . This has some effect on the superpotential,as discussed in that work. We will not insist here on the fact that qualitative and quanti-tative aspects of the localized solution are well captured by the smeared solution. For morediscussions, we refer the interested reader to our forthcoming paper [24].In the paper [11], backgrounds of type N in eq.(2.6) above were shown to capture a varietyof non-perturbative aspects for a field theory like (2.11). In the rest of this paper, we willshow how backgrounds of type A also encode interesting dynamics of (2.11). We now turnto study some aspects of the theory defined in eq. (2.11) that will be relevant for the rest ofthe paper. We want to explain briefly some nice subtleties that arise when considering the theory (2.11).Let us start with the superpotential in (2.11), W tree = 1 µ h T r ( ˜ QQ ) − N c ( T r ˜ QQ ) i (2.13)It is known that when analyzing the space of vacua for this theory the quantum mechanicalsuperpotential differs from the classical one in eq. (2.13). Indeed, if N f < N c an Affleck-Dine-Seiberg superpotential [25] is induced W = W tree + ( N c − N f ) (cid:16) Λ N c − N f det( ˜ QQ ) (cid:17) Nc − Nf (2.14)8hich changes qualitatively the dynamics, for example, when looking for solutions to theF-term eqs, not allowing the (diagonalized) meson field M = ˜ QQ to have zero eigenvalues.When N f ≥ N c a different quantum superpotential is induced. Actually, for the cases N f = N c , N f = N c + 1 and N f > N c , three different superpotentials are induced that presentdifferent space of vacua. Here, we will content ourselves with stating that for the case of N f ≥ N c , the field theory presents two possible behaviors, one in which the meson matrixhas non-zero eigenvalues and another branch where the meson matrix has zero determinant(the meson matrix can always be diagonalized). For details, see [26], [27].We will explain in section 3 how these two possible behaviors are matched by the super-gravity solutions of type A and type N .The second interesting subtlety refers to the large ρ behavior of the solutions we found.As we will see in section 3.4, the careful study of the UV (large ρ expansion) of our solutionssuggests that in the dual field theory an operator of dimension six is getting a VEV. Probablyas a consequence of this, some well-known or expected behaviour of the IR dynamics of thetheory (2.11), see [29], will not be realized by our string background, that in turn is dual tothe field theory but in a particular vacuum. Indeed, in the following we will check that thestring solutions seem to agree with the interpretation of SQCD with quartic superpotentialin the UV (large ρ ) region but we will observe that in general the expected IR fixed pointis not attained. Of course, this depends on the precise vacuum on which the theory isrealized. As a matter of fact, if the theory flows to the IR Seiberg’s fixed point, i.e. , thequartic coupling flowed to zero, there is an enhancement of the flavor symmetry at the IRfixed point. It is hard to see how a brane construction of the kind we are using can reflectthis fact. Thus, there must be something in the field theory dual to the presented solutionspreventing the vanishing of this coupling. It would be very interesting to understand thispoint better. In section 4.6 we will comment on the actual IR behaviour which can be readfrom the geometry. We know that in an N = 1 field theory, there are lots of simplifications for the beta functionsof different couplings. For the gauge coupling, the well known NSVZ-Jones beta functionreads β g = − (cid:16) g / (16 π )(1 − g N c π ) (cid:17)(cid:16) N c − N f (1 − γ Q ) (cid:17) (2.15)where γ Q is the anomalous dimension of the quark superfields. In the following, we willbe interested in the wilsonian beta function, for which the denominator in (2.15) is absent.Suppose that the superpotential for our model is (here we neglect the double trace terms ofthe form N c ( T r ˜ QQ ) , that will not change the beta functions) W = κ ( Q ˜ Q ) + λ ( Q ˜ Q ) + z ( Q ˜ Q ) + ... (2.16)9et us remind the standard result in four-dimensional SUSY gauge theories; for a superpo-tential of the form W = hQ n A m if the fields have (classical) dimensions [ A ] = mass a , [ Q ] = mass q we can form the quantity ∆ h = 3 − nq − ma and the adimensional coupling ˜ h = hµ − ∆ h ( where µ is some mass scale). Then, the beta function for the coupling ˜ h is β ˜ h = ˜ h [ − ∆ h + n γ Q + m γ A ] (2.17)where γ A , γ Q are the anomalous dimensions of each field. Let us apply this to our case (2.16), β ˜ κ = ˜ κ [1 + 42 γ Q ] , β ˜ λ = ˜ λ [3 + 62 γ Q ] , β ˜ z = ˜ z [5 + 82 γ Q ] (2.18)We will argue in section 4.1.1 that in the solutions for N f < N c , near ρ → ∞ , one finds: γ Q = − − ǫ , ( N f < N c ) (2.19)with ǫ some small correction in the UV (but at lower energies it may become large). Thus,when N f < N c the quartic coupling ˜ κ is relevant (becomes marginal at N f = 2 N c ), but theothers are irrelevant. When N f > N c , we have γ Q > − and all couplings are irrelevant(their beta functions are all positive). So we see that dropping ( ˜ QQ ) and higher ordersfrom the lagrangian (2.11) is well motivated if we are interested in the IR dynamics. Also,the argument around eq.(2.12) suggests that the higher order operators do not occur at all,so in the following we will consider the case of quartic superpotential only. In section 4.1.1,we will show how solutions of type A reproduce the beta functions for the gauge couplingin (2.15) and the anomalous dimensions in (2.19).Let us now focus on another interesting property of the field theory (2.11), Seiberg duality. A well known property of N = 1 SQCD-like theories is Seiberg duality [28]. In the particularcase of field theories like the one in eq.(2.11), this equivalence is specially nice as explainedfor example in [29], where it was shown that the duality can be exact and not just an IRequivalence of two theories. In [11], we explained how the duality proposed by Seiberg ismanifest in our backgrounds of type N . Here, we will concentrate on showing how type A backgrounds reflect this duality.Indeed, without knowing the solution to the BPS system of equations (2.2)-(2.5), we canobserve that the equations and hence their solutions are invariant under: H ↔ G, N c → N f − N c (2.20)with φ, Y, N f unchanged. This change has the correct numerology to act as Seiberg duality.Indeed, we can see that the transformation (2.20) is idempotent.10et us comment on the correspondence between Seiberg duality and the transformation(2.20). The reader may want to think about it in this way: if we take backgrounds of theform (2.1), characterized by BPS eqs.(2.2)-(2.5), then it is observed that a solution for thefunctions H, G, Y, φ , dual to a field theory with N f flavors and N c colors, automaticallyproduces ‘another’ solution for the functions G, H, Y, φ that we can associate with the fieldtheory with N f flavors and N f − N c colors. The fact that the two solutions are the samesolution (up to a coordinate redefinition), indicates that the two field theories are actuallythe same theory. This nicely matches with the exactness of Seiberg duality for theories like(2.11), as discussed in detail in [29].One natural question that may appear at this point is whether the Seiberg duality change(2.20) is only a property of the SUSY system or, on the contrary, all solutions (SUSY andnon-SUSY) to the eqs of motion that dictate the dynamics of type A backgrounds have theproperty above. To sort this out, notice that not only the BPS equations, but also the type A ansatz (2.1) is invariant under (2.20) if we relabel: θ ↔ ˜ θ , ϕ ↔ ˜ ϕ (2.21)We can see that Seiberg duality is basically the interchange of the two-manifolds labelledby ( θ, ϕ ) and (˜ θ, ˜ ϕ ). Thus, even non-supersymmetric solutions of the resulting equations ofmotion, would have a Seiberg-dual counterpart.This kind of reasoning should nevertheless be taken carefully. Indeed, Seiberg proved [28]that in dual pairs there was matching of global anomalies and that deformations of modulispaces could be put in correspondence. It is therefore necessary to study how anomalies forthe R-symmetry are manifest in each theory in the dual pair. The prescription to computeR-symmetry anomalies was studied in [11] and, in following sections, we just apply thatinformation for the matching of R-symmetry anomalies on both dual pairs.Summarizing, we propose the following interpretation: since a gravity solution correspondsto a field theory in a given vacuum, the fact that two different field theories correspond to thesame gravity solution means that they are the same theory, they contain the same information(they are Seiberg dual). However, the way of extracting gauge theory information such ascouplings or rank of the gauge group from the geometry can depend on the interpretation,on how one wants to understand -in view of the change (2.21)- the role of the different cyclesin the geometry among other things. As usual for anomalies, we look into the Ramond fields, in this case C . One might beconfused, since we write a potential C when eq. (2.1) implies that dF = 0 (that is theBianchi identity indicates the presence of sources). The point that we made in [11] was that11ne should first restrict the background to the cycle,Σ = (cid:16) θ = ˜ θ, ϕ = 2 π − ˜ ϕ, ψ (cid:17) , (2.22)since this is where color branes are wrapped around. On this sub-space F is exact and onecan write a potential C . For the theory with N f < N c , that we call e -theory (for electric),and for the Seiberg dual theory, with N f > n c , that we call the m -theory (for magnetic) ,the potentials are, C e = ( ψ − ψ ) N c − x ) sinθdθ ∧ dϕC m = ( ψ − ψ ) n c x − sin ˜ θd ˜ θ ∧ d ˜ ϕ (2.23)We have defined x = N f /N c and ¯ x = N f /n c as in [11]. Notice that both expressions in (2.23)are the same expressed in electric and magnetic variables respectively.Imposing that the instanton that is represented by a Euclidean D1-brane wrapping thecontractible two-cycle (2.22) is not observed in Bohm-Aharonov experiment [30], leads tosome particular values of the ψ angle (or selects some possible translations in ψ , if we make ψ → ψ + 2 ǫ ). Indeed, remember that ψ → ψ + 2 ǫ (with ǫ in [0 , π ]) was the change ofR-symmetry in the unflavored case [19] and the unobservability of the shrinking instantonimplies for the partition function of the putative Euclidean brane, Z = Z [0] e i π R C → e iǫN c (2 − x ) e i π R C × Z [0] = Z (2.24)where Z [0] is the partition function for this probe including the Born-Infeld part; this inturn leads to, ǫ e = 2 πn N c − N f , ǫ m = 2 πkN f − n c (2.25)The relation in eq.(2.25) should be interpreted as the matching of the anomaly in the correla-tor of two gauge currents and the R-symmetry current, < J ( R ) J ( N c ) J ( N c ) > on the theoriesof the dual pair. We can see that ǫ e = ǫ m as k = (1 , ....N f − n c ) and n = (1 , ... N c − N f ).This will work in the same way for the type N backgrounds, since it is the same RR field C that is relevant. This is not one of the diagrams alluded by the ’t Hooft criteria, thosewill be discussed in section 4.8.The fact that the R-symmetry is broken by anomalies to the discrete subgroup Z N c − N f suggest that the R-charge for the quark superfield is R [ Q ] = . Indeed, as we will see insection 4.1.1 and in section 4.8 the result of eq.(2.25) implies that the R-charge of the quarksuperfield is R [ Q ] = 1 / γ Q = − , in agreement with what weobserved around eq.(2.19). The relation is the usual n c = N f − N c . .4 A jewel of the 1970’s Before leaving this field theory section, we would like to remind the readers about nice workdone three decades ago.Several different suggestions have been made to address the problem of confinement. Theidea of ’t Hooft and Mandelstam [31] was to think about confinement as a dual Meissnereffect, where a ‘vortex’ of chromoelectric flux joined together two quarks giving an area lawfor the Wilson loop. The ‘abelian-projection monopoles’ proposed to be dual to the quarks[32] paved the way to interpret the phenomena of confinement in QCD as a dual Meissnereffect. The following possible phases manifest in a non-abelian gauge theory: • Free phase: in analogy with QED; for example, if the number of flavors is very largein QCD. • Coulomb phase: the IR theory is conformal. • Higgs phase: here the monopoles are confined and quarks are free. • Wilson/Confining phase: here the quarks are confined and monopoles are free. • Oblique confined phase: quarks and monopoles are confined, dyons condense.In QCD, SQCD or in any QCD-like theory with fundamental matter, one can move smoothly(without phase transitions) between the different phases mentioned above [33]. We will seehow the solutions of type A that we will find in section 3 realize these phases and theirsmooth interpolation.Let us now present the new solutions for the backgrounds of type A in eqs.(2.1)-(2.5). In this section we present the new solutions for backgrounds of type A described in eq.(2.1).Let us first explain why we are interested in solutions of type A . It is well known that evenwhen singular, certain supergravity solutions can faithfully describe the dynamics of a dualgauge theory at strong coupling. The question of which singularities are allowed inspired avariety of criteria to discard those singular spacetimes that cannot be proposed to be dualto a given gauge theory and accept those that can. Among these criteria, we will choosethe one in [34], that basically imposes on a singular spacetime to have finite value of g tt atthe singularity (in this discussion we are alluding to IR singularities, the UV singularities ofthese kind of theories are resolved in a stringy way). Needless to say, this regards the problem of confinement on Yang-Mills; in QCD, the presence of funda-mental matter allows to interpolate between the Higgs and the confining phase. A are singular with a so called “badsingularity”, that is, the putative background cannot be interpreted as being dual to a fieldtheory, see for example [19].In our case, backgrounds of type A will have associated a value of g tt = e φ (in Einsteinframe). If we analyze the BPS eq. for the dilaton- see the last eq of (2.5)- we immediatelysee that the dilaton has positive derivative if N f ≥ N c . Hence, type A backgrounds presenta “good singularity” when the number of flavors is bigger or equal to the number of colors.This restriction very nicely matches what we explained in section 2.2.1 regarding the twopossible vacua for N f ≥ N c . In other words , backgrounds of type A describe N=1 SQCDwith the lagrangian (2.11) when the gaugino condensate and the VEVs of the meson matrixvanish. Conversely, we propose that backgrounds of type N studied in detail in [11] describethe same field theory, in a vacuum with non-vanishing meson VEV and gaugino condensate(we remind the reader that, in the unflavored case, it is well established that the function a ( ρ ) in (2.6) is, roughly, the dual of the gaugino condensate and that type A solutions areobtained from type N by seeting a ( ρ ) = 0).In summary, for N f ≥ N c , type A backgrounds can be thought of as good duals to our N = 1 SQCD of eq.(2.11) in a vacuum with vanishing meson VEV and gaugino condensate.To describe these new solutions, we now present in detail the integration of the BPS eqs.(2.2)-(2.5) for the type A backgrounds (2.1).We start by noting that the difference H − G can be integrated immediately, G = H + N f − N c ρ − C (3.1)where C is a constant of integration. The function Y can also be written in terms of H , Y = 12 ∂ ρ H + 14 ( N f − N c ) (3.2)This allows us to write the system as a single second order differential equation for H , ∂ H∂ρ = (cid:18) ∂ ρ H + 14 ( N f − N c ) (cid:19) " − ∂ ρ H + N f − N c H − N f + 2 ∂ ρ HH + N f − N c ρ − C + 8 (3.3)and the dilaton that can be obtained as, φ ( ρ ) = φ + Z h N f − N c H + N c G i dρ (3.4)There is of course, one simple solution for N f = 2 N c (the case N f = 2 N c is the ”conformalsolution” studied in [11]) that reads H = N c /ξ , G = N c / (4 − ξ ), Y = N c /
4, but even in This interpretation was developed in discussions with Francesco Bigazzi and Aldo Cotrone. N f = 2 N c there are more solutions that we will not discuss here [35].Unfortunately we did not succeed in finding other exact solutions to (3.3). Nonetheless wederive below the asymptotic expansions, near the UV and the IR, of the solutions and thenshow that a smooth interpolation between them exists. ρ → ∞ ) As explained above, we are interested in solutions to (2.2)-(2.5) with asymptotically lineardilaton. There are two distinct possibilities, depending on whether N f is bigger or smallerthan 2 N c . N f < N c There is a one-parameter family (depending on the constant C that enters (3.1)) of UV(large ρ ) asymptotic solutions. For the different functions we have: H = (cid:18) N c − N f (cid:19) ρ + (cid:18) C + N c (cid:19) + N c N f N c − N f ) ρ − + . . .G = N c N c N f N c − N f ) ρ − + N c N f ( N f − N c − C )32(2 N c − N f ) ρ − + . . .Y = N c − N c N f N c − N f ) ρ − − N c N f ( N f − N c − C )32(2 N c − N f ) ρ − + . . .∂ ρ φ = 1 − ρ − + 8 C + 2 N c + 2 N f N c − N f ) ρ − + . . . (3.5)This coincides with the asymptotic expansion found in equation (4.19) of [11] for the solutionsof type N , except for the fact that here there is an extra free parameter C , which in thosesolutions seems to be fixed to C = ( N f − N c ) / N f > N c Here again a one-parameter family of solutions depending on C is found, H = N f − N c N f ( N f − N c )16( N f − N c ) ρ − + N f ( N f − N c )(4 C + 2 N c − N f )32( N f − N c ) ρ − + . . .G = N f − N c ρ + N f − N c − C N f ( N f − N c )16( N f − N c ) ρ − + . . .Y = N f − N c − N f ( N f − N c )32( N f − N c ) ρ − − N f ( N f − N c )(4 C + 2 N c − N f )32( N f − N c ) ρ − + . . .∂ ρ φ = 1 − ρ − + − C − N c + 3 N f N f − N c ) ρ − + . . . (3.6)15hich coincides with the expansion for the type N backgrounds found in eq.(4.20) of [11] ifwe set C = ( N f − N c ) / Let us now present the solutions to the BPS eqs. at small ρ , as they appear from solving(3.3) together with (3.1), (3.2) and (3.4). We remind the reader that, as discussed in thebeginning of section 3, we require N f ≥ N c so it is guaranteed that the quantity g tt ∼ e φ does not diverge in the IR. The space is singular at the origin, but we will call it a goodsingularity and proceed assuming that correct physics predictions for the IR of our theoryof interest (2.11) can be obtained by doing computations with the type A backgrounds.There is quite a rich structure since we actually found three possible IR expansions thatcan be smoothly connected with the UV expansions of section 3.1. The three possible IRbehaviors will be labelled as type I, type II and type III. Let us study them in turn. An obvious solution of (3.3) is H = ( N f − N c )( c − ρ ), where c is an integration constant.But eq. (3.2) would yield Y = 0. Thus, this simple solution cannot be physical. Nevertheless,there are solutions that asymptotically approach this behaviour in the IR (as ρ → −∞ ).The deviation from the simple linear solution is given by exponentially suppressed terms (as ρ → −∞ ) in the following way, H ( ρ ) = N f − N c c − ρ ) + X k ≥ P k ( ρ ) e kρ ( ρ → −∞ ) , (3.7)The P k ( ρ ) are order k + 1 polynomials in ρ . The first one is P = c (cid:20) ρ − ρ ( c + c + 12 ) + c c + 14 ( c + c ) + 18 (cid:21) (3.8)and the higher ones can be obtained iteratively. Notice that one of the integration constantscan be reabsorbed by shifting the origin of ρ and that there is a relation among the differentconstants C = c ( N f − N c ) − c N c . The leading IR behaviour of the rest of the functionscan be readily obtained G ( ρ ) = N c c − ρ ) + O ( e ρ ) Y ( ρ ) = 4 c e ρ ( c − ρ )( c − ρ ) + O ( e ρ ) ( ρ → −∞ ) e φ = e φ p ( c − ρ )( c − ρ ) + O ( e ρ ) (3.9)This IR asymptotics can be numerically connected to the UV behaviours described in section(3.1). We depict some sample plots in section 3.3.16 .2.2 Type II expansions Differently from the type I expansion we presented in the previous subsection, equation (3.3)also admits solutions where H , G , or both vanish at a finite value of ρ , which we take tobe ρ = 0 without loss of generality. The way H and G reach the origin depends essentiallyon the value of the constant C in (3.3). When C is strictly negative H goes to zero like ρ whereas G goes to a constant. When C is strictly positive, H and G exchange roles: H goesto a constant while the leading behavior of G is ρ . We call both these kinds of solutionstype II. When C = 0, instead, H and G vanish at the same time, and their leading term goeslike ρ ; we call this solution type III and postpone its presentation until the next subsection.More explicitly, for C < H = h ρ + (cid:18) h C + N c − N f (cid:19) ρ + h C + 20 h + 3 C (10 N c − N f )72 C ρ + . . .G = − C + h ρ + (cid:18) h C − N f (cid:19) ρ + h C + 20 h + 3 C (10 N c − N f )72 C ρ + . . .Y = h ρ + 112 (cid:18) h C + 3 N c − N f (cid:19) + 34 h C + 20 h + 3 C (10 N c − N f )72 C ρ + . . .φ = φ + N f − N c h ρ + 3 C ( N f − N c ) − h (2 N c + N f )12 Ch ρ + . . . (3.10)whereas for C >
0, we have the following power series H = C + h ρ − (cid:18) h C + N f (cid:19) ρ + h C + 20 h + 3 C (10 N c − N f )72 C ρ + . . .G = h ρ − (cid:18) h C + N c (cid:19) ρ + h C + 20 h + 3 C (10 N c − N f )72 C ρ + . . .Y = h ρ − (cid:18) h C + N c (cid:19) + h C + 20 h + 3 C (10 N c − N f )96 C ρ + . . .φ = φ + N c h ρ + 3 C + h (3 N f − N c )12 Ch ρ + . . . (3.11)The constants h , C and φ are arbitrary. In particular φ can be fixed to any value. Forwhat concerns C and h , instead, what happens is more interesting. While any choice ofthem gives a solution to equations (3.1) - (3.4), requiring that a particular solution behavesin the UV as the functions in section 3.1 imposes an additional condition that enforces arelation between the useful values of C and h . Figure 1 is the plot of such a relation fortwo fixed values of N f N c . All other possible choices still give a solution to (3.1) - (3.4), but thefunctions either reach a singularity at some finite value of ρ or reach the UV with a differentasymptotical behaviour, in which we are not interested here (analogous to the one studiedin section 8 of [11]). 17 Figure 1: We represent here the relationship between C and h that ensures that the evolutionof a given type II expansion goes asymptotically to the behaviour we presented in section3.1. On the left we are considering negative values of C , corresponding to expansion (3.10)while on the right positive values are considered, which correspond to (3.11). Dashed linescorrespond to N f N c = 1 . N f N c = 2 . When the constant C in (3.1) vanishes, type II expansions are not allowed anymore, but anew branch of solutions opens up, which we will call type III. In this case both H and G vanish at the origin of space. The expansion reads as follows H = h ρ + 110 (5 N c − N f ) ρ + 23 h ρ + . . .G = h ρ −
110 (5 N c + 2 N f ) ρ + 23 h ρ + . . .Y = h ρ − − N f
10 + 49 h ρ + . . .φ = φ + 3 N f h ρ + 3 10 N c − N c N f + 7 N f h ρ + . . . (3.12)Again the constants h and φ are arbitrary, but to obtain a solution that asymptotes to thebehaviour of section 3.1, we have to choose a particular value of h for any fixed value of N f N c . The expansions above are essential in the analysis of solutions to the system (3.1)-(3.4),but their knowledge is not enough to guarantee the existence of an overall solution thatconnects the given UV and IR behavior. We have checked that such connecting solutionindeed exists for the three kinds of IR behaviours. In this section, we plot some samplenumerical computations.Figure 2 gives some examples of the numerical integration for the type I IR behaviour.18 Ρ @ Ρ D -4 -2 2 4 Ρ @ Ρ D -4 -2 2 4 Ρ @ Ρ D -4 -2 2 4 Ρ -1123456 Φ @ Ρ D Figure 2: Type I solutions obtained numerically. We have fixed C = 0, N c = 1 and defined ρ = 0 as the point where H ( G ) reaches a minimum for N f < N c ( N f > N c ). We plotthe solutions for different values of N f N c : dotted, dashed, dot-dashed and continuous linescorrespond respectively to N f = 1 .
2, 1 .
8, 2 .
2, 3 N c .As we have already anticipated, requiring that the type II and III expansions are connectedwith the UV asymptotics of section 3.1 imposes an additional condition, fixing the value ofthe integration constant h in terms of C and N f N c . This additional condition produces theplots in Figure 1. Given that we choose the suitable value of h , it is then possible tonumerically integrate equations (3.1)-(3.4), and find a complete solution for the type A backgrounds. For type II and type III solutions, we present examples of these solutions inFigures 3 and 4, for different values of C and N f N c . The reader may wonder how it is possible that different IR behaviours asymptote to thesame UV when it seems that the UV expansions (3.5), (3.6) have no free parameters apartfrom C . Indeed, those expansions are not complete and there is an extra free parameter inthe UV if one considers the possibility of exponentially suppressed terms: we can write, H = Q ( ρ ) + e − ρ ρ − α Q ( ρ ) + O ( e − ρ ) (3.13)19 Ρ Ž H Ρ L Ρ Ž H Ρ L Ρ Ž H Ρ L Ρ Ž H Ρ L Ρ Ž H Ρ L Ρ Ž H Ρ L Ρ Φ H Ρ L Ρ Φ H Ρ L Figure 3: Type II solutions derived numerically. In these plots we present the results for C = − N f N c : dotted, dashed, dot-dashed and continuous lines correspondrespectively to N f = 1 .
2, 1 .
8, 2 .
2, 3 N c . The graphs on the left-hand side represent thesolutions for the background on a large range of ρ , while the plots on the right-hand columnfocus on the behaviour of the functions close to the origin. Tilded function are rescaled by N c as ˜ H = HN c , etc. 20 Ρ Ž H Ρ L Ρ Ž H Ρ L Ρ Ž H Ρ L Ρ Ž H Ρ L Ρ Ž H Ρ L Ρ Ž H Ρ L Ρ Φ H Ρ L Ρ Φ H Ρ L Figure 4: Type II and III solutions derived numerically. In these plots we present thebehaviour of type III solutions, and compare it with small- C type II solutions. Clearly thetransition through C = 0 is almost continuous. Here we have fixed N f = 1 . N c . Dottedlines correspond to the type III solution, while dot-dashed and continuous lines correspondto type II backgrounds with respectively C = − . C = 0 . Q are the expansions (3.5), (3.6) with: α = 2 N c − N f N c − N f ) ( N f < N c ) α = 2 N c + N f N c − N f ) ( N f > N c ) (3.14)The Q i ( ρ ) are polynomials such that Q i ( ρ ) = Q i, + Q i, ρ − + Q i, ρ − + . . . . Q , is a freeparameter and the rest are fixed in terms of it. We have, thus (once C, N c , N f are fixed), a oneparameter family of UV solutions, labelled by Q , . Different choices Q , should modify theIR behaviour and, in particular, make the system fall into one of the qualitatively differentIR possibilities. However, checking this numerically requires great numerical precision andwe could not bear out this expectation explicitly.It would be interesting to know the physical interpretation of this parameter Q , . Since itcomes multiplying a term with e − ρ , it must be related to the vacuum expectation value of anoperator of UV dimension 6, here we anticipate the UV radius-energy relation log[ µ/ Λ] = ρ that we will use in equation (4.6). The simplest guess is an operator of the type ( Q ˜ Q ) (sincein the UV γ Q → − , at least when N f ≤ N c ). A single vev of this type would spontaneouslybreak the R-symmetry, apparently in contradiction with the described solution. However,since there are infinitely many smeared flavored branes in the construction, it is natural tothink that, if in fact such vevs are turned on, they may also be smeared effectively restoringthe U (1) R . This is similar to the discussion in [13] where, considering smeared massesfor hypermultiplets, the R-symmetry that would be broken for a single mass is effectivelyrestored. The different expansions are written in terms of some integration constants which are relatedamong them. In order to avoid confusion, we clarify here the number of parameters on whichthe different families of solutions depend.We have four first order equations (2.2)-(2.5). Let us fix N f /N c (notice that, mantainingthis ratio fixed, N c can be reabsorbed by simultaneously rescaling H, G, Y ). Then a solutionof the system depends on four integration constants. One is an additive constant for φ thatdoes not affect the integration of the other functions. Since the BPS eqs do not dependexplicitly on ρ , a second integration constant is just a shift of ρ , which we can fix withoutloss of generality. In type I, we can define ρ = 0 as the point where H ′ (or G ′ ) vanishes, asin figure 2. For type II and III, we have defined ρ = 0 as the point where H (and/or G )vanish. The third integration constant is fixed by requiring that the system goes to the lineardilaton UV (so there are not growing exponentials in H ). From the IR point of view, thisis, for instance, what fixes h in terms of C in the type II IR expansions, see the discussionin section 3.2.2. 22hus, for fixed N f /N c and fixing also the additive constant of φ , we are left with the fourthintegration constant, which gives a one-parameter family of type I and type II solutions. Fortype III, where we further require C = 0, there is a single solution (a limiting point of thetype II family).After these discussions on the solutions, we will focus on obtaining physics predictionsfrom them. To this we turn now. In this section, we study some non-perturbative aspects that can be learned from the solu-tions presented in section 3. We will discuss expressions for the gauge coupling and anoma-lous dimensions of the quark superfields, give predictions for their values and compute thebeta function in the UV of the theory (matching results known from field theory). The pre-dicted anomalous dimension for the quark superfield will be in agreement with the resultsof eq.(2.19) and those spelled out in section 2.3.1. We will present similar results for thebeta function of the quartic coupling (for which we will provide a geometrical definition).We will study the Wilson and ’t Hooft loops and address in detail the issue of (continuousand discrete) anomaly matching.Let us remind the reader that we take units so that α ′ = g s = 1. With this choice, thetension of a D5 brane is T = π ) . We will give an expression for the theta-angle and gauge coupling that can be simply derivedby considering the action for a D5 brane on the submanifold ( t, ~x, θ = ˜ θ, ϕ = 2 π − ˜ ϕ ), atconstant radial coordinate. This is a well known definition, originally proposed in [36]. In[11] we defined the gauge coupling using an instanton, whose action is computed using a D1brane wrapping the two-cycle ( θ = ˜ θ, ϕ = 2 π − ˜ ϕ ). This two-manifold is special, being theonly shrinking closed manifold in the geometry. As expected, the results using D5 or D1probes do coincide.We consider the dynamics of a D5 with gauge fields A µ excited on the R , directions,obtaining (in string frame), S D = − T Z d xe − φ p − det( g + 2 πF ) + 12 (2 π ) T Z C ∧ F ∧ F (4.1)The induced metric and two-form on the probe D5 brane read, in string frame, ds ind = e φ h dx , + ( H + G )( dθ + sin θdϕ ) i , ( C ) ind = 2 N c − N f ψ − ψ ) sin θdθ ∧ dϕ (4.2)23ollowing the procedure of [36], we can expand (4.1) up to second order in the gauge fields(this is the QFT part of the α ′ expansion), perform the integrals over the two-cycle andcompare to the usual Yang-Mills action in order to obtain the couplings.For the theta-angle, we get: Θ = (2 N c − N f )2 ( ψ − ψ ) . (4.3)Since changes in Θ are associated with traslations ψ → ψ + 2 ǫ then only some values of ǫ = kπ N c − N f are allowed and they correspond to changes in Θ → Θ + 2 kπ , which nicelymatches what we wrote about anomalies between eq.(2.23)-(2.25).The gauge coupling in our SQCD theory takes the form,8 π g = 2( H + G ) . (4.4)For type I solutions, the series expansions of section 3.2.1 show that the gauge couplingvanishes in the IR. The expansions of section 3.2.2 for type II solutions show a versionof ‘soft confinement’ (with bounded gauge coupling in the IR), while for type III moreconventional confinement is obtained (with divergent gauge coupling in the IR). Figure 5shows the behavior of the coupling as a function of the radial coordinate. We can compute the beta function of the gauge coupling as defined in (4.4). As we statedbelow eq.(2.15), we will be interested in the wilsonian beta function, that takes the expres-sion, β π g = ∂ (cid:16) π g (cid:17) ∂ log( µ/ Λ) = 3 N c − N f (1 − γ Q ) . (4.5)Now, we need a relation between the radial coordinate ρ and the scale µ in order to performthe derivative in (4.5). In section 5.5 of [11] we proposed that the radius-energy relation, inthe UV of the gauge theory, that is for large values of the radial coordinate, is:log[ µ/ Λ] = 23 ρ , (UV , ρ → ∞ ) (4.6)This relation was proved in the unflavored backgrounds by identifying the gaugino condensate(an operator of dimension 3) with a deformation of the background that asymptoticallybehaves as e − ρ , see [36, 37]. Since in the flavored backgrounds, the same kind of deformation(leading to the type N solutions) still has the same exponential behaviour [11], (4.6) istrustable in the present setup. 24 Ρ Ρ Ρ Ρ Figure 5: These plots represent the behaviour of the gauge coupling as defined in equation(4.4) in type II (top) and type III (bottom) backgrounds. For the top plots we have taken C = − N f N c . The correspondence between line styles and N f N c is thesame as in figure 3. For the bottom plots, we have taken N f = 1 . N c and C = − . . C = 0 (type III). For the correspondence between values of C and line styleswe refer the reader to the legend of figure 4. The plots on the right are a zoom around ρ = 0of the plots on the left.We now proceed doing the computation (4.5) in the UV (for ρ → ∞ ). After substituting(4.4), (4.6) and (2.2), (2.3), one obtains an expression for the anomalous dimension of thequark superfield γ Q in terms of the background functions, γ Q = 12 N f Y − − N c N f , (UV , ρ → ∞ ) (4.7)Inserting (3.5), we find the UV behaviour of the anomalous dimension for N f < N c : γ Q ∼ − − N c N c − N f ) ρ , ( N f < N c ) (4.8)This expression is quite nice, because it matches the fact that the anomalous dimensionsof the quark superfield approaches γ Q = − to have the R-charge R [ Q ] = 1 /
2, see thediscussion in section 4.8. 25ne may wonder what happens in theories with N f > N c (which are Seiberg dual tothose with N c < N f < N c ). Introducing in (4.7) the expansion of Y for N f > N c , eq.(3.6), we find: γ Q ∼ − N c N f − N f − N c )8( N f − N c ) ρ , ( N f > N c ) (4.9)So γ Q > − , as anticipated in section 2.2.2. We will now analyze similar issues for the quartic coupling in the superpotential of (2.11).Let us start with the beta function.
Let us go back to the field theory expression of the beta functions as we discussed them ineq.(2.18). We have, for the (dimensionless) quartic coupling: β ˜ κ = ∂ ˜ κ∂ log( µ/ Λ) = ˜ κ (1 + 2 γ Q ) (4.10)Using the radius-energy relation (4.6) and the value of the anomalous dimension computedin (4.7), we find: ∂ log ˜ κ∂ρ = 23 (1 + 2 γ Q ) = 16 YN f − N c N f , (UV , ρ → ∞ ) (4.11)So, in view of (2.3), we can identify:log ˜ κ = c + 8 GN f , (UV , ρ → ∞ ) (4.12)where c is some constant.Let us analyze what happens with the quartic coupling when N f < N c and N f > N c .Whether it is relevant or irrelevant depends on the sign of 1 + 2 γ Q = N f (4 Y − N c ).When N f < N c : 1 + 2 γ Q = − N c N c − N f ) ρ − < N f > N c , 1 + 2 γ Q = 6 N f ( N f − N c ) > .2.2 Geometric interpretation In principle, one could expect that the quartic coupling could be read as data of the geometry, i.e. as some quotient of the volumes of the different cycles of the manifold. In that sense,(4.12) is not very satisfactory.Without a definite understanding, we speculate in this section about a definition of thequartic coupling that makes use of the geometry of the internal manifold. Since (4.12) istrustable in the UV, we require that our guess behaves similarly in the large ρ region. Westart with the N f < N c in which the coupling is relevant and bounded in the UV:˜ κ = e c + GNf ≈ N c N c − N f ) ρ − + . . . , ( N f < N c ) (4.15)where we have conveniently fixed c (an overall factor) to make the expression simpler.We will now propose a definition for the quartic coupling ˜ κ in terms of the geometry. First,let us notice that, for fixed ρ , we have one well defined two-cycle, a well defined one-cycleand a couple of well defined three-cycles. The two-cycle is the one mentioned before andgiven by ˆ S = [ θ = ˜ θ, ϕ = 2 π − ˜ ϕ ] , V ol [ ˆ S ] = 4 πe φ ( H + G ) . (4.16)The one-cycle, along the worldvolume of the flavor branes, is parametrized by the angle ψ and its volume is S = [ ψ ] , V ol [ S ] = 4 πe φ/ √ Y . (4.17)It is possible to define two different 3-cycles in the internal manifold S = ( θ, ϕ, ψ ) , V ol [ S ] = (4 π ) e / φ H √ Y , ˜ S = (˜ θ, ˜ ϕ, ψ ) , V ol [ ˜ S ] = (4 π ) e / φ G √ Y (4.18)We define the quartic coupling as a quotient of the volume of the two-cycle times the one-cycle divided by the volume of one of the three-cycles. This definition makes sense since thecoupling we are interested in (relevant for matchings between string and gauge theory) mustbe adimensional : ˜ κ = V ol [ S ] × V ol [ ˆ S ] V ol [ S ] (4.19)Then: ˜ κ = 1 + GH = 1 + N c N c − N f ) ρ − + . . . (4.20)When studying the Seiberg dual magnetic theory, one should change S → ˜ S according tothe prescription (2.21). This amounts to the interchange H ↔ G .When N f > N c , the quartic coupling grows exponentially towards the UV and thedefinition (4.19) should be taken carefully. This is perhaps not surprising since the fieldtheory description with a quartic term breaks down and needs some UV completion. Werelegate further discussion to Appendix A. 27 .3 A nice matching with the N f = 2 N c case In this section, we will show a connection between our solutions for N f < N c with the onefor N f = 2 N c and the one for N f > N c , and analyse it in view of the properties of thequartic coupling discussed above. We will need to use that all of the solutions are of theform (2.1), with the expansions in eqs.(3.5)-(3.6) and that for N f = 2 N c an explicit solutionis given by [11] ( ξ is a free parameter valued in 0 < ξ < H ( ρ ) = N c ξ , G ( ρ ) = N c − ξ , Y = N c , φ ( ρ ) = φ + ρ. (4.21)We now consider the solution with N f < N c in the far UV ( ρ → ∞ ), and change the numberof flavors towards N f → N − c with a particular scaling limitlim ρ →∞ , N f → N − c (2 N c − N f ) ρ → ∞ , (4.22)so we have that the functions in eq.(3.5) behave as H ≈ (cid:16) N c − N f (cid:17) ρ → ∞ , G ≈ N c , φ ≈ ρ. (4.23)According to (4.12), the quartic coupling takes some finite value at this point.We see that we can match this with the N f = 2 N c solution (4.21) if we take ξ = 0. Aftermatching, we move freely in the parameter ξ (since in the N f = 2 N c case, ξ represents amarginal coupling [11]), from ξ = 0 towards ξ = 4. At the point ξ = 4 the functions takethe values G → ∞ , H = N c , φ = ρ (4.24)and the quartic coupling diverges. We can match (4.24) with the expansion for N f > N c in eq.(3.6) if we scale similarly as above,lim ρ →∞ , N f → N + c ( N f − N c ) ρ → ∞ . (4.25)The interpretation is the following: since when N f > N c , the quartic is irrelevant, wecan only connect those theories with the end of the N f = 2 N c marginal line in which thiscoupling is infinite. On the other hand, when N f < N c , the quartic is relevant so thesetheories approach the conformal N f = 2 N c point by the opposite limit of the marginal line,where the quartic takes its smallest value. At N f = 2 N c , we can tune ξ (accordingly changingthe quartic coupling) to move from one end of the line to the other. The process continuouslyconnects, in the space of theories, the N f < N c case to the N f > N c one.This “flow in the number of flavors” provides a nice quantitative check and suggests thatour discussion about the quartic coupling in section 4.2.1 is sensible. It would be nice to28nderstand if similar matchings and flows may happen for different solutions with N f = 2 N c [35].Let us now study the potential between static quark-antiquark and monopole-antimonopolepairs, using Wilson and ’t Hooft loops. In order to get some intuition of what will happen to the quark-antiquark pair when sep-arated, we can analyze a quantity associated with the tension of the putative QCD-stringthat forms between them. The functional form of this tension (that can be thought of asthe tension of an F1-string when taken to the IR of the background) is given by, T Q ¯ Q ∼ √ g tt g xx | IR = e φ | IR T Q ¯ Q,T ypeI → T Q ¯ Q,T ypeII → e φ T Q ¯ Q,T ypeIII → e φ (4.26) T Q ¯ Q gives a qualitative way of evaluating the tension in the IR dynamics. The expressions(4.26) tell us that type I solutions will have vanishing QCD-string tension (indicating thatquarks are free), type II and type III solutions will show finite tension, indicating confine-ment. This is in agreement with the behavior of the gauge coupling in eq.(4.4). Of course,the presence of dynamical quarks will produce screening (string breaking) when the energystored in the QCD-string (4.26) is of the order of the lightest meson in the theory.Let us now compute the Wilson loop following the standard prescription of [39]. Considerthe action for an F1 string parametrized by t = τ, x = σ, ρ = ρ ( σ ) (4.27)with x taking values in [0 , L ] (such that ρ (0) = ρ ( L ) = ∞ ) and τ in [0 , ∞ ). The inducedmetric on this string is, ds ind = e φ h − dτ + (1 + 4 Y ρ ′ ) dσ i (4.28)So, the Nambu-Goto action for this string is S = 12 πα ′ Z dσe φ q Y ρ ′ (4.29)This is in the form to follow the usual treatment for Wilson loops in [40], and write expressionsfor the (renormalized) energy E and the separation L . There is a one-parameter family of29tring profiles depending on the minimal value of ρ reached by the string, which we call ρ . E ( ρ ) = 12 πα ′ (cid:20) e φ ( ρ ) L + 2 Z ∞ ρ √ Y ( p e φ − e φ ( ρ ) − e φ ) dρ − Z ρ √ Y e φ dρ (cid:21) L ( ρ ) = 4 e φ ( ρ ) Z ∞ ρ r Ye φ − e φ ( ρ ) dρ (4.30)These integrals can be studied numerically for our solutions. We will present an example infigure 6 at the end of the next section. It is interesting to perform a qualitative analysis similar to the one around eq.(4.26). Inthis case, the effective tension for the monopole-antimonopole pair is given by the effectivetension of the magnetic string, represented by a D3-brane wrapping the two-cycle (4.16) andextended along ( x, t ) stretched at the bottom of the geometry. See eq.(4.32) and below fordetails, T m ¯ m ∼ e φ ( H + G ) | IR T m ¯ m,T ypeI → e φ N f T m ¯ m,T ypeII → e φ CT m ¯ m,T ypeIII → T m ¯ m gives a qualitative way of evaluating the tension of the monopole-antimonopole pair inthe IR dynamics Intuitively, eq. (4.31) indicates that type I and type II solutions confinethe monopoles, while type III solutions exhibit free monopoles.Let us now study in detail the t’ Hooft loop that can be thought of as the “Wilson loop formagnetic monopoles”(the rigorous definition is in terms of the commutation relations withthe Wilson loop operator). It is a similar computation to the one for the Wilson loop in theprevious section, as we explained above, for a extended wrapped D3 brane on the cycle Σ ,Σ = (cid:16) t = τ, x = σ, ρ ( σ ) , θ = ˜ θ, ϕ = 2 π − ˜ ϕ (cid:17) . (4.32)This gives an induced metric that in string frame reads, ds ind = e φ h − dt + (1 + 4 Y ρ ′ ) dσ + ( H + G )( dθ + sin θdϕ ) i (4.33)and the action (per unit time) for the ‘effective string’ obtained after integrating out thetwo-cycle is, S = 4 πT Z dσ q ˆ f + ˆ g ρ ′ , ˆ f = e φ ( H + G ) , ˆ g = 2 ˆ f √ Y (4.34)30e observe that this expression is “Seiberg invariant” according to the definition (2.20), andanalogous formulas to those in eqs. (4.30) can be used. They read L ( ρ ) = 2 Z ∞ ρ ˆ g ˆ f ˆ f ( ρ ) q ˆ f ( ρ ) − ˆ f ( ρ ) dρE = 4 πT " ˆ f ( ρ ) L + 2 Z ∞ ρ ˆ g ( ρ )ˆ f ( ρ ) (cid:16)q ˆ f ( ρ ) − ˆ f ( s ) − ˆ f ( ρ ) (cid:17) dρ − Z ρ ˆ g ( ρ ) dρ (4.35)An example of numerical analysis for the expressions (4.35) using the functions (4.34) isplotted in figure 6. Figure 6: As a typical example, we plot E vs. L graphs for N f = 1 . N c . The dotted linecorresponds to the type I IR behaviour, the dashed line to type II (with C = −
1) and thesolid line to type III. The graph in the left is for Wilson loops as discussed in section 4.4while the one in the right corresponds to the ’t Hooft loops introduced in section 4.5Our interpretation of figure 6 is as follows: first, it is interesting to notice that the graphsfor the backgrounds with different IR behaviours share some qualitative properties, suggest-ing a smooth interpolation among the different solutions. In all cases, there is a maximumlength reached by the loop which we interpret as string breaking due to pair creation. Moreover, in both graphs, there is a line (the type III one for the Wilson loop and the onecorresponding to type I for the ’t Hooft loop) which stretches more than the rest indicatinglinear confinement (before the string is eventually broken). This qualitatively suggests achange of roles of quarks and monopoles between type I and type III backgrounds. We willelaborate on this in the next section. Finally, let us point out that there are also minimallengths in the graphs. Presumably, this is due to the growing linear dilaton, which signalsthe failure of the supergravity description we are using at asymptotically large ρ . With suchan UV completion, there should be a minimal length ( L min ) of the field theory that can be In [41], it was argued in a similar situation that such kind of behaviour did not correspond to stringbreaking. ρ when ρ is taken large). Thus, we discard this part of suchlines as unphysical. The fact that the type III line of the ’t Hooft loop graph only followsthis unphysical behaviour indicates that in the corresponding field theory, monopoles arefree (or they are screened at a length shorter than L min ).We close this section with a technical comment about the obtention of the graphs in figure6. In order to compare the loops probing the different solutions, one has to ensure that theyasymptote to the same UV (once we fix the additive constant of integration φ for one of thesolutions, the φ of the rest have to be fixed accordingly). It is also necessary to take intoaccount that having C = 0, the UV behaviour of H is different from the C = 0 case (seeeq(3.5)). Thus, in order to compare with other solutions, the definition of the coordinate ρ has to be shifted ρ → ρ − C N c − N f (so the origin of space is not at ρ = 0 any more). Briefly, we want to make contact with the material in section 2.4.The intuitive analysis for the string tensions in eq.(4.26) and eq.(4.31), shows that in thethree types of IR solutions the behavior for the potential between the (non-dynamical) quark-antiquark and monopole-antimonopole pair is quite diverse. Indeed, the intuitive analysispresented around eqs.(4.26)-(4.31) shows that for small separations compared to the mass ofa pair, type I solutions confine monopoles and let quarks move freely. The opposite occursin solutions of type III, while type II solutions present a situation in which quarks andmonopoles experience a growing force when separated .Another important observation is the fact that one can smoothly interpolate between thesolutions of types I, II and III by changing the values of the integration constants. Indeed,one can see that in the limit of vanishing constant C in type II solutions (see section 3.2.2),the expansions change into those of type III in section 3.2.3 (this was repeatedly stressedwhen plotting the functions). The connection with type I solutions in section 3.2.1 mustbe thought in a more heuristic way, by noticing that if the constants C and h in the typeII solutions were absent the expansions could continue past ρ = 0 to smaller values of theradial coordinate and the leading term in the expansion would go like the type I solutions(this connection could be made more precise in terms of the UV constants of integration C, Q , , see section 3.4, but this is hard to see in the numerical analysis).This ‘heuristic matching’ resembles the fact that in theories with fundamental fields thereare no phase transitions in moving between a confining and a higgsed situation [33]. Thedifferent VEV’s of the operator of dimension six mentioned in section 3.4 is what should We emphasize again that for long distances pair creation gives place to string breaking and the lawchanges. i.e. with the worldvolume electricfield turned on. We leave this computation for the future.The qualitative picture described above is quite appealing. However, we should write herea word of caution. The tensions (4.26) and (4.31) depend on a combination of functionsat the singularity; even when the singular behavior cancels from the expressions of (4.26)and (4.31) -a manifestation of the ‘good’ character of the singularity- these results shouldbe taken with care. It could happen that the criterion we are using for ‘good singularities’is not enough and that not all the discussed solutions should be accepted as physical.None of the described behaviours corresponds to an IR conformal fixed point. We remindthe reader that SQCD with a quartic superpotential flows in the IR to Seiberg’s conformalfixed point of SQCD [29] (at least in the ratio N f /N c appropriate to fall in the conformalwindow). We believe that this does not show up in our set-up since, as explained in section2.2.1, the theory dual to the string solution differs from the usual SQCD+quartic near theIR, presumably due to a non-trivial VEV. Coming back to the expressions for the Wilson and ’t Hooft loops in eqs.(4.29)-(4.34), weobserve that they are invariant under the Seiberg duality transformation (2.20). Even more,one can easily see that the gauge coupling defined in eq.(4.4) is also invariant. This maypuzzle the reader, who may expect that while the electric theory is strongly coupled in theIR, the magnetic theory should be IR free and vice versa. This is not what we observe; oursolutions predict that both theories are strongly coupled (or both have weak g Y M coupling)towards the IR. This situation is not totally original, it is indeed what happens inside theconformal window in Seiberg’s SQCD. In contrast, in our case, we do not have a conformalwindow, but we will see that having two theories with similar IR behavior is not ruled out.Let us discuss this issue using the expressions for the wilsonian beta function in eq.(4.5)and the values for the anomalous dimensions we obtained in eqs.(4.8)-(4.9) for the N f < N c and N f > N c cases respectively. First, we introduce the notation: the theory e is the onewith gauge group SU ( N c ) and N f flavors. We will assume that in the e theory N f < N c ,and the superpotential is, W e = κ e (cid:16) ˜ QQ ˜ QQ − N c ( ˜ QQ ) (cid:17) (4.36)We will denote by m theory the one obtained by Seiberg duality from the theory e . The33 theory will have gauge group SU ( n c ) with N f flavors, of course the relation is n c = N f − N c .This, together with the fact that in the e theory we set N f < N c implies that N f > n c inthe theory m . The superpotential in the m theory is (we will ignore, since it plays no role,the double trace term), W m = ˆ κ m M M + 1 µ ˜ qM q (4.37)where we have used that the new fields in the Seiberg dual theory are the meson M ij (notcharged under the SU ( n c ) gauge field w α ), the quarks q and the antiquarks ˜ q .Now, let us compute the beta functions in each theory. For this, we will need to know theanomalous dimensions of the fields. Our solutions predict (in the UV) anomalous dimensionsfor the e quarks γ Q and the m quarks γ q given by eqs.(4.8) and (4.9) γ Q = 12 N f Y − − N c N f ∼ − − N c (64 N c − N f ) ρ γ q = 12 N f Y − − n c N f ∼ − n c N f − N f − n c )(8 N f − n c ) ρ . (4.38)Let us neglect the small O ( ρ − ) correction in the far UV ( ρ → ∞ region), keeping only theleading terms. Now, we compute the beta functions. We use the values for the anomalousdimensions in eq.(4.38), obtaining that the wilsonian beta functions for the two theories aregiven by, β π g e = 3 N c − N f (1 − γ Q ) = 32 (2 N c − N f ) ,β π g m = 3 n c − N f (1 − γ q ) = 32 ( N f − n c ) , (4.39)This shows that either both the electric and magnetic theories confine or they both are IRfree. The result in eq.(4.39) is precisely the one we obtain if we use the UV expansions for H ( ρ ) , G ( ρ ) in section 3.1 together with eqs.(4.7)-(4.9). We now move to the matching ofanomalies. In a field theory like SQCD or ours in eq.(2.11) there are two types of currents, global(denoted below by j g ) and local or gauge currents, denoted by J L . Hence, there are fourpossible 3-point correlators of currents one may compute < J L J L J L >, (4.40) < j g j g J L >, (4.41) < j g J L J L >, (4.42) < j g j g j g > . (4.43)34he first and second correlators (4.40)-(4.41) must be zero, or the gauge symmetry is anoma-lous and the theory is inconsistent.The correlator with two local currents in eq.(4.42) if nonzero, indicates the presence of ananomaly in the global current < ∂ µ j µg > = 0. While for non-anomalous symmetries in thesense of eq.(4.42), the correlator for three global currents in eq.(4.43) must match in the twodescriptions of the theory. This is the ’t Hooft anomaly matching condition.We will start by studying the anomaly for the R symmetry current, as expressed by thecorrelator in (4.42), and related to the results of sections 2.3.1 and 4.1. This is what appearsin the changes for the Θ angle. A general expression for this anomaly is, ψ → e iǫγ ψ, ∆Θ = ǫ X r n r T ( r ) R [ ψ ] (4.44)where the ψ denotes the massless fermions of the theory, n r is the number of particles inthe r representation of the gauge group, T ( r ) is the weight of the representation (we remindthe reader that T ( adj ) = 2 N c , T ( f und ) = 1) and R [ ψ ] is the R-charge of the correspondingfermion. We need to assign R-symmetry charges to the superfields. The gaugino has R [ λ ] = 1and we want to determine the R-charges of the fundamental superfields. They cannot beread directly from the supergravity solution, but they can be extracted indirectly from theΘ-angle transformation. Let us start looking at the electric e -theory (with N f < N c ). From(4.44), we read (using R [ ψ Q ] = R [ ψ ˜ Q ], required by the symmetry):∆Θ e = ǫ e (2 N c + 2 N f R [ ψ Q ]) (4.45)The values of ǫ e that leave Θ e invariant (modulo 2 π ) are written in (2.25) and immediatelysuggest R [ ψ Q ] = − . Thus, we are led the following assignation (for the fermions in themultiplet) under the global symmetry SU ( N f ) V × U (1) B × U (1) R , for the electric e -theory: ψ Q = ( N f , , − / , ψ ˜ Q = ( ¯ N f , − , − / , λ = (1 , ,
1) (4.46)This implies that for the scalars of the quark multiplets R [ Q ] = R [ ˜ Q ] = . Notice that this isconsistent with the existence of the quartic superpotential preserving the R-symmetry at theclassical level. Let us also discuss the relation of this R-charge to the anomalous dimensionof the quark field. At a conformal point dimO = 32 R [ O ] (4.47)Using this relation and the UV value of the anomalous dimension of the squark (4.8), wefind R [ Q ] = dim [ Q ] = (1 + γ Q ) = , in agreement with the discussion above. Therelation (4.47) can be used here because our theory has an asymptotic UV fixed point (thebeta-function for the gauge coupling becomes trivially zero since the gauge coupling itselfgoes to zero and the one for the quartic coupling also vanishes due to γ Q = − , see (4.10)).35e now turn to the dual magnetic m -theory with n c = N f − N c colors such that N f > n c .We denote the magnetic quarks with lower case q . In this case, (4.44) yields:∆Θ m = ǫ m (2 n c + 2 N f R [ ψ q ]) (4.48)Comparison to (2.25) again suggests R [ ψ q ] = − , which we believe is the charge relevantto the backgrounds discussed in this paper. However, there is another logical possibilitywhich we find interesting and we will discuss it in section 4.8.1. Therefore, we consider thefollowing charges for the magnetic m -theory: ψ q = ( ¯ N f , N c n c , −
12 ) , ψ ˜ q = ( N f , − N c n c , −
12 ) ψ M = ( N f × ¯ N f , , , ˜ λ = (1 , ,
1) (4.49)This implies that for the scalars of the quark multiplets R [ q ] = R [˜ q ] = and for the mesonmultiplet R [ M ] = 1, again allowing the superpotential to preserve the R-symmetry. However,given (4.9), one can see that the relation (4.47) is not satisfied. This should not come as asurprise, because the quartic coupling grows in the UV of the magnetic theory, see aroundeq.(4.14) and the UV theory is far from a conformal point.Indeed, now that the assignations in eqs.(4.46),(4.49) have been given, the next step is tocheck whether ’t Hooft matching in the sense of eq.(4.43) holds. Since the R-symmetry isanomalous, the continuous global symmetry is SU ( N f ) V × U (1) B . The triangles that shouldbe matched are, < SU ( N f ) V >, < SU ( N f ) V U (1) B >, < U (1) B >, < U (1) B T T > (4.50)where the last triangle representes the ‘gravitational anomaly’ with two insertions of T µν .Let us quote the results in the e theory with the values of eq.(4.46), we denote d ABC = T r ( T A [ T B , T C ]), < SU ( N f ) V > = d ABC N f N c (1 + ( − ) = 0 < SU ( N f ) V U (1) B > = tr ( T A T B )2 N f N c (1 −
1) = 0 < U (1) B > = N f N c (1 + ( − ) = 0 < U (1) B T T > = N f N c (1 −
1) = 0 (4.51)while for the m theory, with values (4.49), we have < SU ( N f ) V > = d ABC N f n c (1 + ( − ) = 0 < SU ( N f ) V U (1) B > = tr ( T A T B )2 N f n c ( N c n c − N c n c ) = 0 < U (1) B > = N f n c [( N c n c ) + ( − N c n c ) ] = 0 < U (1) B T T > = N f n c ( N c n c − N c n c ) = 0 (4.52)36e see that there is anomaly matching between the electric and magnetic descriptions,satisfying ’t Hooft criteria. Up to here, the computation is identical to that for the usual N = 1 SQCD without superpotential. In [42], it was shown in that triangles involvingdiscrete symmetries should also be matched. It is therefore a non-trivial consistency check tocompute the values involving the remnant non-anomalous R-symmetry Z N c − N f = Z N f − n c .We now explicitly check the matching of discrete anomalies between the electric and mag-netic description. Following the paper [42], we can compute the values for the trianglesinvolving the R-symmetry. There is a technical subtlety explained in [42]. We need tohave integer values for the different charges to match discrete anomalies, so they have to berescaled. Thus, following the prescription in [42], we consider a Z k = Z N c − N f R-symmetry.The baryonic and R-charges are ‘rescaled’ with respect to the values in the e -theory eq.(4.46)for the fermion in the multiplet and read: ψ Q = ( N f , N f − N c n , − , ψ ˜ Q = ( ¯ N f , − ( N f − N c ) n , − , λ = (1 , ,
2) (4.53)while for the dual theory m , rescaling in the same fashion the values in eq.(4.49), we have ψ q = ( ¯ N f , N c n , − , ψ ˜ q = ( N f , − N c n , − ψ M = ( N f × ¯ N f , , , ˜ λ = (1 , ,
2) (4.54)The integer n is defined as the greatest common divisor of N c and N f : n ≡ GCD ( N c , N f ) (4.55)One has to study the following triangles: < Z k T T >, < Z k U (1) B >, < Z k >, < SU ( N f ) Z k > (4.56)We will consider the difference between the triangles in the electric and the magnetic theory.They should match up to multiples of k = 4 N c − N f [42]. We obtain the following results, < Z k T T > e = 2 N c N f ( −
1) + 2( N c − , < Z k T T > m = 2 n c N f ( −
1) + 2( n c − , ∆( e − m ) = 0 (4.57)which is a multiple of k = 4 N c − N f as required by [42].For the second triangle involving three ‘discrete’ currents, < Z k > e = 2 N c N f ( − + 8( N c − , < Z k > m = 2 n c N f ( − + 8( n c − , ∆( e − m ) = 6 N f (2 N c − N f ) . (4.58)37gain, the difference is a multiple of k . For the triangle involving two baryonic and a‘discrete’ current, < Z k U (1) B > e = 2 N c N f (cid:18) N f − N c n (cid:19) ( − , < Z k U (1) B > m = 2 n c N f ( − (cid:18) N c n (cid:19) , ∆( e − m ) = 2 N c N f n ( N f − N c )(2 N c − N f ) , (4.59)we obtain the correct matching modulo k ; finally for the triangle involving the non-abelianglobal symmetry < SU ( N f ) Z k > e = − N c N f < SU ( N f ) Z k > = − n c N f ∆( e − m ) = − N f (2 N c − N f ) , (4.60)We see that the assignation of charges appropriately rescaled (4.53), (4.54) satisfies thediscrete anomaly matching up to multiples of k = (4 N c − N f ) as explained in [42]. Thisnice match indicates that the value for the R-charge suggested by the string background isconsistent with the field theory computations using the charges in eqs. (4.46),(4.49). We again consider the charges for the electric theory written in (4.46), but give differentR-charges for the magnetic theory. Coming back to (4.48), a second possibility to match thesupergravity result (2.25) is to take R [ ψ q ] = − n c N f .Our second assignation of charges for the magnetic theory under the symmetries is, thus(like before, we give the charge of the fermions in the multiplet): ψ q = ( ¯ N f , N c n c , − n c N f ) , ψ ˜ q = ( N f , − N c n c , − n c N f ) ψ M = ( N f × ¯ N f , , − n c N f ) , ˜ λ = (1 , ,
1) (4.61)The nice thing about this second assignation is that it satisfies the relation in eq.(4.47), ascan be checked by using (4.9) (although, as explained above, there is no reason to expect(4.47) to hold out of a conformal point). The downside is that this implies that the couplingˆ κ m in eq.(4.37) must be charged, so the R-symmetry is completely broken at the classicallevel in the magnetic theory. This could only correspond to the dual of an electric theoryin which the R-symmetry is spontaneously broken and, thus, it does not seem relevant tothe type A backgrounds discussed in this paper. We believe that this second assignationcould describe a situation like what happens for type N background in eq.(2.6) or thosesolutions found in [11], which will have the same UV as our type A backgrounds. Since inthis case there is no remnant R-symmetry, an anomaly matching computation as the one in(4.56)-(4.60) does not apply. 38n summary, to match in field theory the R-symmetry anomalies to the ones computed bythe string solutions, we proposed values for the charges of the multiplets under the globaland discrete symmetries. Notice that the R-charges are checked by the string background inthe sense that the anomaly is encoded in the background via the potential C (2) (see section2.3.1). Both charge assignations we have given are consistent with this computation. Onthe other hand, the charges under SU ( N f ) and U (1) B written above are suggested by fieldtheory considerations but not from the actual string solution. However, it is remarkablethat the string solution indirectly knows about them (at least in the case of assignation(4.46),(4.49)) throught the discrete anomaly matching computation displayed in eqs. (4.56)-(4.60). The results in this section together with the previous checks, make us very confidentof the interpretations that we have put forward for the string background. Let us briefly summarize the different topics elaborated in this paper. We have found a setof solutions for backgrounds of type A in eq.(2.1), proposed to be dual to a version of N = 1SQCD (2.11). With the new solutions, we predicted the behavior for different field theoryobservables and phenomena like Seiberg duality, gauge coupling and its running, Wilsonand ’t Hooft loops, anomalous dimensions of the quark superfields, quartic coupling, itsrunning and anomalies (continuous and discrete). Based on those string theory predictions,we provided a large number of matchings between field theory expectations and string theoryresults. Interestingly, our solutions seem to realize the Fradkin-Shenker result for the non-existence of phase transitions when moving between higgsed and confining phases of a theorywith fundamental fields [33].This work leaves some open questions for future research. Indeed, it should be nice, amongother things, to understand if the same ‘phase structure’ occurs in type N backgrounds ineq.(2.6) and the behavior of the quartic coupling in that case. The picture of Seiberg dualityfor type N backgrounds could be understood on a similar way as we did in this paper. Thecase of massive flavors, either for type A or N backgrounds also presents interest because itgives a resolution of the IR (good) singularity and its many applications. These and othertopics will be dealt with in [24].It should also be interesting to apply our treatment for quartic couplings to cascadingtheories. Sorting out other dynamical aspects of the backgrounds presented here, like for ex-ample, the spectrum of glueballs, mesons and baryons [43] is of main interest. The Venezianoexpansion we are adopting indicates that there will be interactions leading to decays betweenthe first two types of excitations and suggests ways to study baryons as solitons in the BI-WZaction.It should also be interesting to extend all this story to the type II A case, using backgrounds39ike the ones in [44]. Extensions to lower dimensional field theories may prove interesting forcondensed matter problems. Finally, it should be nice to think about possible high energyphenomenology applications of this line of research. Acknowledgments:
In the last year or so, discussions with different colleagues have shaped our understandingand the presentation of the topics in this paper. For those discussions, we would like tothank: Gert Aarts, Riccardo Argurio, Adi Armoni, Francesco Benini, Francesco Bigazzi,Aldo Cotrone, Felipe Canoura, Stefano Cremonesi, Justin Foley, Seba Franco, Simon Hands,Prem Kumar, Biagio Lucini, Asad Naqvi, Alfonso V. Ramallo and Diego Rodriguez G´omez.RC and AP are supported by European Commission Marie Curie Postdoctoral Fellowshipsunder contracts MEIF-CT-2005-024710 and MEIF-CT-2005-023373. RC and AP’s workwas also partially supported by INTAS grant, 03-51-6346, RTN contracts MRTN-CT-2004-005104 and MRTN-CT-2004-503369, CNRS PICS
APPENDIXA Beta function for κ m The computation of the beta function for the quartic coupling in the magnetic case seemsto be tricky, so, here we will write the results we get and our interpretation.If we Seiberg dualize the coupling κ e in eq.(4.20) we get the magnetic version κ m = 1 + HG .We compute the beta function from string theory using the radius-energy relation eq.(4.6)and we obtain 1 κ m β ˜ κ m = ∂ log[˜ κ m ] ∂ log( µ Λ ) = 32 ∂∂ρ log (cid:16) HG (cid:17) ≈ −
34 ( N f − n c )( N f − n c ) ρ . (A.1)Now, let us look at the QFT results. The superpotential after Seiberg duality is given by, W m = ˆ κM M + ˆ z ˜ qM q (A.2)The classical dimensions for fields and couplings are,[ q ] = m, [ M ] = m, [ˆ κ ] = m, [ˆ z ] = 1 (A.3)where we used that the “meson” M is a fundamental field in the magnetic theory and not acomposite of electric quarks. The R-charges of fields and couplings, obtained from the main40art of the paper eqs.(4.61) are, R [ q ] = ( 32 − n c N f ) , R [ M ] = (4 n c N f − , R [ˆ κ ] = 4(1 − n c N f ) (A.4)and the anomalous dimensions from the main part of the paper (for the meson it is computedassuming that the relation between dimension and R-charge is the one of a fixed point), γ q = 12 N f Y − − n c N f ≈ − n c N f −
38 ( N f − n c )( N f − n c ) ρ , γ M = − n c N f (A.5)We can now compute β ˆ κ = ˆ κ [ − γ M ] = ˆ κ (12 n c N f − , β ˆ z = ˆ z [ γ q + 12 γ M ] = −
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