Charged particle-like branes in ABJM
aa r X i v : . [ h e p - t h ] S e p FPAUO-10/02CERN-PH-TH/2010-077
Charged Particle-like Branes in ABJM
Norberto Guti´errez a,b, , Yolanda Lozano a, , Diego Rodr´ıguez-G´omez c, a Department of Physics, University of Oviedo,Avda. Calvo Sotelo 18, 33007 Oviedo, Spain b Theory Group, Physics Division, CERN,CH-1211 Geneva 23, Switzerland c Centre for Research in String Theory, Department of Physics,Queen Mary, University of London,Mile End Road, London E1 4NS, UK
ABSTRACT
We study the effect of adding lower dimensional brane charges to the ’t Hooft monopole, di-baryon and baryon vertex configurations in
AdS × P . We show that these configurationscapture the background fluxes in a way that depends on the induced charges, and therefore,require additional fundamental strings in order to cancel the worldvolume tadpoles. Thestudy of the dynamics reveals that the charges must lie inside some interval in order tofind well defined configurations, a situation familiar from the baryon vertex in AdS × S with charges. For the baryon vertex and the di-baryon the number of fundamental stringsmust also lie inside an allowed interval. Our configurations are sensitive to the flat B -fieldrecently suggested in the literature. We make some comments on its possible role. We alsodiscuss how these configurations are modified in the presence of a non-zero Romans mass. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] Introduction
In the last two years important progress has been made towards understanding three dimen-sional superconformal field theories and their relation to the
AdS /CF T correspondence.Elaborating on the pioneering work of [1, 2], Aharony, Bergman, Jafferis and Maldacena(henceforth ABJM) proposed a theory conjectured to describe M2 branes probing a C / Z k singularity, where the orbifold acts with weights (1 , , − , −
1) [3]. The near horizon geom-etry is then
AdS × S / Z k . The ABJM theory is an N = 6 quiver Chern-Simons-mattertheory with gauge group U ( N ) k × U ( N ) − k and marginal superpotential. The high degree ofSUSY requires the superpotential coupling to be related to the CS level k . Since the latterrenormalizes trivially, the ABJM theory is expected to be exactly conformal. Being thesuperpotential coupling proportional to k − , an appropriate large k limit N / << k << N allows for a weak coupling regime. In turn, in the gravity dual, the appropriate descriptionis in terms of Type IIA string theory on AdS × P , with various RR fluxes [3].As standard in the AdS/CF T correspondence, the chiral ring of the field theory isto be identified with the KK harmonics of the dual geometry. While these operatorshave scaling dimensions of order one, there are in addition chiral operators whose scalingdimension is of order N . Typically they correspond to non-perturbative states in thegravity side realized as branes wrapping calibrated cycles. In the context of the morefamiliar AdS /CF T correspondence, a canonical example is the di-baryon [4, 5], whichcorresponds to a wrapped D3-brane. This brane looks like a particle in AdS , and therefore,from its mass in global AdS , one can read off its anomalous dimension. Yet another exampleof such class of operators still in
AdS /CF T is the baryon vertex [6]. It corresponds to aD5-brane wrapping the whole internal space. Since it captures the N units of 5-form flux,it requires N fundamental strings ending on it. These strings can be thought of as externalquarks (or Wilson lines in the fundamental representation) and this naturally suggests toidentify this wrapped D5-brane with the ǫ tensor of the SU ( N ) gauge group of the fieldtheory.The AdS /CF T case is less understood in general. Concentrating on the ABJM case,various particle-like branes of the N = 6 AdS × P background were already discussedin [3]. The P space has H q ( P ) = R for even q ≤
6. Thus, it is possible to have D D D D D D D D -field . More generally, these wrapped branes act as sources to vector fields in AdS arising from the reduction of RR potentials on topologically non-trivial cycles. In turn,vector fields in AdS admit quantization with either of the two possible fall-offs at theboundary [10], which amount to either a dynamical boundary gauge field or to a globalcurrent (discussions in this context have appeared recently in [11, 12, 13, 14]). Since adefinite quantization must be chosen, it follows that either magnetic or electric sources areforbidden for the corresponding bulk field [10]. This might shed some light on the roleof the B -field. Indeed, coming back to the D4-branes, the quantization allowing for theD4-branes to exist should correspond to that where the U (1)’s are non-dynamical. Underthat assumption, a determinant-like di-baryon dual operator would be gauge invariant byitself and it would have the right dimension to agree with the gravity result. On the otherhand, the quantization dual to dynamical U (1)’s would forbid the D4-branes, which mightsuggest that no B -field is needed. However, a full understanding of this very importantpoint is, at present, still lacking.Similar comments should hold for the remaining wrapped branes. It has been arguedthat the D0-brane corresponds to a di-monopole operator in the CFT side. The D6-brane,very much like the baryon vertex in AdS , requires N fundamental strings ending on it. Itsdual operator should then naturally involve the ǫ tensor of the gauge theory. On the otherhand, the D2-brane wrapped on the P ⊂ P develops a tadpole that has to be cancelledwith k fundamental strings. The dual operator is a monopole ’t Hooft operator, realizedas a Sym k product of Wilson lines [3]. As mentioned above, as of today, there is no fullysatisfactory understanding of the role of these branes and their dual operators.The gravitational configurations described above admit a natural generalization byallowing non-trivial worldvolume gauge fluxes. It is the aim of this paper to generalizethe spectroscopy of wrapped branes by adding such non-trivial worldvolume gauge fields.To that matter, we will assume that suitable boundary conditions are chosen in eachcase such that the discussed branes are possible. These generalized configurations are ofpotential interest for some AdS/CM T applications (see for instance [15, 16]), for exampleas candidates for holographic anyons in ABJM, as discussed recently in [17, 18].Allowing for a non-trivial worldvolume gauge field has the effect of adding lower di-mensional brane charges. This modifies how the branes capture the background fluxes in away that depends on the induced charges, such that, in some cases, additional fundamentalstrings will be required to cancel the worldvolume tadpoles. From that point of view, thegeneralized configurations are similar to holographic Wilson loops. We will see that theD2 and D6-branes do not differ much from the zero charge case, although they are stableonly if the induced charges lie below some upper bound. This situation is familiar from The original argument supporting this B -field in [8] concerns a detailed analysis of the supergravitycharges, while the analysis of the D4 worldvolume dynamics arises as a consistency check. For more detailswe refer to the original paper. AdS × S , studied in [19]. In these cases theenergy of the bound state increases with the charge that is being induced. However addingcharges allows to construct more general baryon vertex configurations. We will see thatfor the D6-brane the number of quarks that forms the bound state can be increased inthis manner. From this point of view adding flux provides an alternative mechanism tothat proposed in [20] for modifying the number of quarks. In turn, the D4-brane with fluxbehaves quite differently from the fluxless case, since it will require fundamental stringsending on it as opposed to the vanishing worldvolume flux case. As we will see, the studyof its dynamics reveals that the whole configuration is stable if the magnetic flux lies withina given interval, being maximally stable for an intermediate value, and reducing to freequarks at the boundaries.The paper is organized as follows. In section 2 we generalize the particle-like braneconfigurations studied in [3] to include a non-vanishing magnetic flux proportional to theK¨ahler form of the P . This magnetic flux induces lower dimensional charges in the braneworldvolume and, in some cases, modifies the charge of the tadpole. In section 3 we studythe dynamics of these configurations and show that there exist some bounds on the numberof branes that can be dissolved in the worldvolumes. In section 4 we show how the previousconfigurations are modified in the presence of a non-zero Romans mass. Finally in section5 we present our conclusions, where we try to relate the existence of the bounds for themagnetic flux to the stringy exclusion principle of [21]. AdS × P with magnetic flux In this section we generalize the particle-like brane configurations in [3] to include a non-vanishing magnetic flux. We analyze the various brane charges that are dissolved as wellas the charges of the different tadpoles induced. Following [8], an important observation isthat the dual gravity background might actually involve a non-vanishing but flat B field.It is possible to argue for such a shift by noting that the D4-brane with minimal flux (itwill turn out essential for the argument that this minimal flux has to be half-integer dueto the Freed-Witten anomaly [9]) should be dual to a di-baryon. In order to review thisargument, we will consider first the D4-brane case before turning to the D2 and D6 cases.Let us first start with a lightning review of the AdS × P background while collectingsome useful formulae. In our conventions the
AdS × P metric reads ds = 4 ρ L dx , + L dρ ρ + L ds P L (cid:16) ds AdS + ds P (cid:17) , (2.1)with L the radius of curvature in string units, L = (cid:16) π Nk (cid:17) / (2.2)This is a good description of the gravity dual to the U ( N ) k × U ( N ) − k CS-matter theory[3] when N / << k << N .It is well-known that for P one has H q ( P ) = R for even q . Indeed, parameterizing the P as (e.g. [22]) ds P = dµ + sin µ h dα + 14 sin α ( cos α ( dψ − cos θ dφ ) + dθ + sin θ dφ )+ 14 cos µ ( dχ + sin α ( dψ − cos θ dφ )) i (2.3)where 0 ≥ µ, α ≥ π , ≥ θ ≥ π , ≥ φ ≥ π , ≥ ψ, χ ≥ π , (2.4)there is a P at fixed θ , φ , and a P at µ = α = π/ χ , ψ .The K¨ahler form J = 12 d A , (2.5)where A is the connection in ds S = ( dτ + A ) + ds P , which in our coordinates reads: A = 12 sin µ (cid:16) dχ + sin α ( dψ − cos θ dφ ) (cid:17) , (2.6)satisfies Z P J = π , Z P J ∧ J = π , Z P J ∧ J ∧ J = π . (2.7)Therefore, 16 J ∧ J ∧ J = d Vol( P ) with Vol( P ) = π . (2.8)The AdS × P background fluxes can then be written as F = 2 Lg s J , F = 6 g s L d
Vol(
AdS ) , F = 6 L g s d Vol( P ) (2.9)where g s = Lk , L k = 32 π N . (2.10)The flux integrals read Z P F = 32 π N , Z P F = 2 π k . (2.11)5 .2 The di-baryon vertex Consider a D4-brane wrapping the P in P . This brane lives at fixed θ and φ , and since itdoes not capture any background fluxes it does not require any fundamental strings endingon it.Since the D P , which is not a spin manifold, it should carry a half-integer worldvolume gauge field flux through the P ⊂ P , due to the Freed-Witten anomaly[9]. Given that the gauge-invariant quantity in the worldvolume is F = B + 2 πα ′ F , thishalf-integer worldvolume flux can be cancelled through a shift of B . This motivated [8]to include a flat B -field in the dual IIA background: B = − π J (2.12)which should be considered in addition to the fluxes discussed in the previous section.We can now consider a more general configuration where we add extra worldvolumeflux F = N J on top of the F F W = J required to cancel the Freed-Witten anomaly, suchthat the total worldvolume flux is F T = ( N + 1) J with even-integer quantization (that is, N ∈ Z being N = 0 the minimal case). As noted above, the quantity appearing in thebrane worldvolume action is the combination F = B +2 π F T . Putting together the variousdefinitions, we have F = 2 π F , that is, the B shift and the extra half unit of worldvolumeflux cancel each other and we can effectively work as if we had no background B -field and F = N J .The DBI action is then given by: S DBI = − T g s Z d ξ p − det( g + 2 πF ) = − T g s Z d ξ p | g tt |√ g P (cid:16) L + 2(2 π ) F αβ F αβ (cid:17) = − π T g s (cid:16) L + (2 π N ) (cid:17) Z dt ρL . (2.13)Therefore, for non-vanishing magnetic flux the mass of the D4-brane satisfies m D L = N + k N / N = 0, the background B cancels the half-integer worldvolume flux induced by the Freed-Witten anomaly, such that m D L = N ; thus naturally admitting an interpretation as a di-baryon.The D4-brane with magnetic flux captures the F background flux through the coupling S CS = 12 (2 π ) T Z R × P P [ F ] ∧ F ∧ A = 2 (2 π ) T k N Z R × P J ∧ J ∧ A = k N T F Z dtA t (2.15)6herefore k N / N , this quantity is an integernumber. Moreover, the magnetic flux also dissolves D2 charge through the coupling: S CS = 2 π T Z R × P C ∧ F = N T Z C (2.16)Thus, the number of fundamental strings is k times the number of dissolved D2 branes.In fact, as we will see in the next subsection, a single D2-brane requires k fundamentalstrings ending on it. Thus, from this perspective, the fundamental strings ending on theD4 are cancelling the tadpole due to the dissolved D2-branes.We will see in the next section that the D4-brane with the k N / k N / F is proportional to the K¨ahler form on the P it satisfies that R P F ∧ F = N π . Therefore, it also induces D0-brane charge in the configuration, through thecoupling: S CS = 12 (2 π ) T Z R × P C ∧ F ∧ F = N T Z R C . (2.17)However, as noted in [8], there are relevant higher curvature corrections [23]∆ S ∼ Z C ∧ e F ∧ s ˆ A ( T )ˆ A ( N ) , (2.18)where ˆ A is the A -roof genus ˆ A = 1 − ˆ p
24 + 7 ˆ p − p · · · (2.19)and the Pontryagin classes are written in terms of the curvature of the correspondingbundle as ˆ p = − π Tr R ˆ p = 1256 π (cid:16) (Tr R ) − R (cid:17) (2.20)The relevant term in (2.18) is then∆ S = (2 π ) T Z C ∧
148 (ˆ p ( N ) − ˆ p ( T )) = − T Z C (2.21)Thus, the total D0 charge is (cid:16) N − (cid:17) T Z C (2.22)This equation shows that the D4-brane contains dissolved D0-brane charge even forthe minimal flux allowed. Note that the term k N / L N / N / AdS × P has been addressed recently in[18], although in the ansatz taken there the deformation of the D4-brane due to the electricfield is not taken into account. It would be interesting to check if spiky solutions exist forboth non-vanishing electric and magnetic fields. Given the topology of P it is possible to consider the KK reduction of the 5-form and7-form respectively on P and P giving rise to vectors in AdS . As discussed in [10] andfurther elaborated in a similar context in [11, 12, 13, 14], the two fall-offs are possible in AdS . Choosing one or the other amounts to the dual U (1) symmetry being gauged ornot. In turn, from the bulk perspective, this is seen as electric-magnetic duality (the so-called T -operation). It is possible to define a S -operation such that their combined actionforms an SL (2 , Z ) algebra, which then connects different boundary CFTs. The actionof such algebra is far from being understood. However, one particular implication wouldbe that depending on the boundary conditions that are chosen the allowed sources areeither the magnetic or the electric ones. From this point of view, one might argue that thequantization dual to dynamical boundary gauge fields forbids D4, D6 (which are electricallycharged under the 5-form and the 7-form respectively), which from the field theory pointof view would stand for the non-gauge invariance of the operators det A and ǫ . On theother hand, the boundary conditions allowing for the D6, D4 would be dual to a certain SU ( N ) version of the theory, in which the B field would presumably play an importantrole. Nonetheless, at this point this is no more than a speculation. In particular, the role ofthe higher curvature couplings, naively coupling the D4 to the 1-form potential (2.22) andthus endowing it with magnetic charge at the same time, remains to be clarified. It shouldbe pointed out that recently a detailed analysis of the field theory has been performed in It must be noted that, from an 11d perspective, the U (1) fields discussed here are not related to atopological symmetry as in [13, 14], which makes them more subtle. U (1) gauge fields suggests thatthe moduli space of the U ( N ) × U ( N ) gauge theory is a Z k cover of the a priori expectedSym N C / Z k , thus allowing for determinant-like operators to be gauge invariant [24] (seealso [25]). These operators are naturally dual to the wrapped D4, which suggests that the B field is turned on. It would be very interesting to clarify the role of the B field in thiscontext, and figure out whether a connection to the possibility raised above, namely thesubtle role of the quantization of abelian fields in AdS , is possible. Further studies of theseissues are well beyond the scope of this paper, and are postponed for further work.In this paper we will simply assume that suitable boundary conditions are chosenallowing for the corresponding wrapped objects, and, as we have done for the D4-brane, wewill include the effect of the (flat) B -field. The D4-brane with zero flux would be identifiedwith the di-baryon operator det A = ǫ i ...i N ǫ j ...j N A i j . . . A i N j N in the CFT side, being A oneof the bifundamentals in the field theory. It is also natural to ask what could be the dual ofthe D4-brane with non-minimal flux. Since once the worldvolume flux is turned on extra F1strings are required, we should expect such dual operator to involve n f = k N Wilson linesin the fundamental representation of U ( N ) × U ( N ). Indeed, the configuration is reminiscentof the D5 Wilson loop in AdS × S [26], which suggests that these fundamental indicesshould be antisymmetrized. We will see in the next section that dynamically a bound n maxf ∼ √ N k ∼ λ − N , where λ = N/k is the ’t Hooft coupling, in the number of suchfundamental indices appears, which is consistent with the antisymmetrization assumption.It would be interesting to elaborate further on this proposal, and in particular to understandthe dependence on the ’t Hooft coupling. We postpone such analysis for further work.
Let us now consider the D2-brane wrapping the P in P , identified in [3] with a (’t Hooft)monopole operator [27, 28, 29].Since this brane captures the F flux it requires fundamental strings in order to cancelthe worldvolume tadpole. Substituting (2.11) in the CS coupling S CS = 2 π T Z R × P P [ F ] ∧ A = k T F Z dtA t (2.23)we find that the number of fundamental strings must be q = k . Note in particular thatthis is the anticipated result from the di-baryon case, where the tadpole of a single D2 wasexpected to be k .We are now interested in adding worldvolume flux to this configuration. According tothe observation in the previous section, there is a background B field given by (2.12) [8]. The results for a vanishing B -field are simply obtained by tuning the extra worldvolume flux.
9t is then convenient to split the worldvolume flux as in the previous section F T = F + J ,with F = N J . (2.24)We should stress that the D2-brane, wrapping a spin manifold, does not capture the FreedWitten anomaly, and as such, the quantization condition for F T [30] is12 π Z F T = 12 ( N + 1) ∈ Z (2.25)Therefore, the case with minimal magnetic flux corresponds to N = − S DBI = − π T g s p L + (2 π N ) Z dt ρL . (2.26)Besides, there is D0-brane charge induced in the configuration, since S CS = 2 π T Z R × P C ∧ F = N T Z R C . (2.27)Note that even in the case of minimal magnetic flux, N = −
1, there is a non-zero D0-branecharge induced by the shifted B .In this case the charge of the worldvolume tadpole is not modified by the presence ofthe magnetic flux.In the next section we will study the dynamics of the configuration formed by theD2-brane plus the k fundamental strings, and show that adding magnetic flux allows toconstruct more general ’t Hooft monopole configurations with charge. This charge willhave to lie however below some upper bound for the configuration to be stable in the AdSdirection.In view of (2.27) we see that our system is actually formed by a D − D Thus, one might worry about thestability of the configuration with flux. Nevertheless, since both the cycle wrapped bythe brane and the worldvolume gauge field are topologically non-trivial, we expect theconfiguration to be stable, at least under small perturbations. As discussed in the previoussubsection, it is implicit in our probe brane approximation that the strings are uniformlydistributed over the D2 worldvolume. Grouping them together in a point would require toconsider their backreaction on the D2, which would deform it into a spike, which could inturn be unstable due to the lack of SUSY. Nevertheless, as long as we restrict to the probeapproximation, we expect the system to be perturbatively stable. For this reason conjecturing a dual operator seems much harder. .4 The baryon vertex Let us finally consider the D6-brane wrapping the whole P . This brane is the analogueof the baryon vertex in AdS × S [6]. In the absence of worldvolume magnetic flux thisbrane captures the F background flux, and it requires the addition of q = N fundamentalstrings: S CS = 2 π T Z R × P P [ F ] ∧ A = N T F Z dtA t . (2.28)Note however that once the shift in (2.12) has been taken into account, the aboveexpression is incomplete, since there are extra contributions to the F1 charge coming fromthe coupling R F ∧ B ∧ B . Nevertheless, once the higher curvature corrections are takeninto account, they cancel out so that the correct expression is actually (2.28). In the caseat hand the relevant term in (2.18) is∆ S = 32 (2 π ) T Z C ∧ F ∧
148 (ˆ p ( N ) − ˆ p ( T )) (2.29)As shown in [8] this term contributes to the D6-brane action inducing extra F1 charge as∆ S = −
18 (2 π ) k T Z dt A t , (2.30)and this precisely cancels the B contribution to (2.28).Let us now switch on a gauge flux, F T = N J . Note that this represents a slight changein the conventions compared to the previous sections, where we split F T into two piecesone cancelling B . In this case, due to the relevance of the curvature coupling in givingthe tadpole of N units in the unfluxed case, it turns out to be more convenient not to dothe spliting so that the argument as in [8] goes through. Since P is spin, the appropriatequantization condition is 12 π Z F T = N ∈ Z (2.31)The DBI action of the D6-brane becomes: S D = − π T g s (cid:16) L + (2 π ( N − (cid:17) / Z dt ρL . (2.32)In this case the magnetic flux modifies the number of fundamental strings that mustend on the D6, since it contributes to the worldvolume tadpole through the couplings S CS = 16 (2 π ) T Z R × P P [ F ] ∧ F T ∧ (cid:16) πF T + 3 P [ B ] (cid:17) ∧ A = k N ( N − T F Z dtA t . (2.33)Therefore q = N + k N ( N − /
8. Note that this is always an integer for the quantizationcondition (2.31). As for the D4-brane, this is the number of fundamental strings required11o cancel the tadpole of each of the D2-branes that are dissolved on the D6-brane by themagnetic flux and the B field, through the coupling: S CS = 12 T Z R × P C ∧ F ∧ F (2.34)In this coupling the term proportional to R C ∧ B ∧ B is precisely cancelled with thecontribution of the A-roof R C ∧ (ˆ p ( N ) − ˆ p ( T )). The other two terms give S CS = 12 (2 π ) T Z C ∧ F T ∧ F T + (2 π ) T Z C ∧ F T ∧ B = N ( N − T Z C (2.35)Note that the magnetic flux and the B field also induce D0-brane charge in the con-figuration.We will study the dynamics of the D6-brane with magnetic flux in the next section. Wewill see that, similarly to the D2-brane case, adding magnetic flux allows to construct moregeneral baryon vertex configurations in which the charge of the brane can be increased upto some maximum value. In this case, since the number of fundamental strings attachedto the D6-brane depends on the magnetic flux, the bound on the magnetic flux imposes aswell a bound on the number of F-strings that can end on the brane.As in the D2 brane case, induced D0 brane charge in a D6 suggests that the systemwill not be supersymmetric. However, again due to its non-trivial topology, we expect thesystem to be perturbatively stable. In this section we study the stability in the
AdS direction of the brane configurations thatwe have previously discussed. We follow the calculations in [20] and [31] (see also [19] forsimilar results for the baryon vertex with magnetic flux in
AdS × S ). We show thatthe energy of the various configurations is inversely proportional to the distance betweenthe quarks, as predicted by conformal invariance, and that the proportionality constantis negative, so that the configurations are stable against perturbations in ρ . As expected,we find the same non-analytical behavior with the square root of the ’t Hooft couplingthat was found for the baryon vertex in AdS × S [20] and the q ¯ q system [31, 32]. Thisrepresents a non-trivial prediction of AdS/CFT for the strongly coupled CS-matter theory.In order to analyze the stability in the ρ -direction we have to consider both the D p -brane wrapped on the P p/ and the q fundamental strings stretching between the braneand the boundary of AdS . The action is given by S = S Dp + S qF , (3.36)12here S Dp is of the form S Dp = − Q p Z dt ρL , with Q p = π p/ T p ( p )! g s ( L + (2 π N ) ) p/ , (3.37)and the action of the strings is given by S qF = − q T F Z dtdx r ρ L + ρ ′ , (3.38)where we have parameterized the worldvolume coordinates by ( t, x ) and the position in AdS by ρ = ρ ( x ). Following the analysis in [20] the equations of motion come in two sets:the bulk equation of motion for the strings, and the boundary equation of motion (as weare dealing with open strings), which contains as well a term coming from the D p -brane.One can show easily that these equations of motion are, respectively: ρ q ρ L + ρ ′ = c (3.39)with c some constant, and ρ ′ q ρ L + ρ ′ = 2 Q p L q T F , (3.40)where ρ is the position of the brane in the holographic direction and ρ ′ = ρ ′ ( ρ ). As in[20, 19] it is convenient to define p − β = 2 Q p L q T F , (3.41)where β ∈ [0 , ρ q ρ L + ρ ′ = 14 β ρ L . (3.42)Integrating the equation of motion we find that the size of the configuration is given by ℓ = L ρ Z ∞ dz βz p z − β , (3.43)where z = ρ/ρ . This expression has the same form as the size of the baryon vertex in AdS × S [20] and the q ¯ q system [31, 32], and can also be solved in terms of hypergeometric Note that for the D6-brane
N → N − Q p in order to account for the B field, consistently withthe quantization condition (2.31). We will take due care of this shift in section 3.3 below. And also that of the baryon vertex with magnetic flux constructed in [19]. ρ , and on L isalso the same, which is again a non-trivial prediction of the AdS/CFT correspondence forthe strong coupling behavior of the gauge theory.The total on-shell energy is given by E = E Dp + E qF = q T F ρ (cid:16) Q p L q T F + Z ∞ dz z p z − β (cid:17) = qT F ρ (cid:16)p − β + Z ∞ dz z p z − β (cid:17) . (3.44)The binding energy of the configuration can be obtained by subtracting the (divergent)energy of its constituents. When the D p -brane is located at ρ = 0 the strings stretchedbetween 0 and ∞ become radial, and therefore correspond to free quarks. At this locationthe energy of the D p vanishes. Therefore, the binding energy is given by: E bin = q T F ρ (cid:16)p − β + Z ∞ dz h z p z − β − i − (cid:17) . (3.45)This expression has again the same form than the corresponding expressions in [20, 19, 31,32]. Notice that for our configurations β is a function of the magnetic flux that is dissolvedon the D p -brane, since from (3.41) β = s − (cid:16) Q p L q T F (cid:17) . (3.46)In particular, in order to find a stable configuration we must have2 Q p L q T F ≤ . (3.47)This imposes a bound on the possible values of the magnetic flux, and therefore on thepossible charges that can be dissolved in the D p -brane. This situation is very similar to theone found in [19] for the baryon vertex in AdS × S with magnetic flux. Moreover, for thedi-baryon and baryon vertex configurations, for which the number of fundamental stringsrequired to cancel the tadpole depends on the magnetic flux, there is as well a bound onthe number of quarks that can form the bound state.For the values allowed by (3.47) the binding energy per string is negative and decreasesmonotonically with β . . Therefore, the configuration is stable, becoming less and less In this case we have added the on-shell energy of the D p -brane. Its behavior as a function of the magnetic flux depends on the specific function β ( N ) given by (3.46)We will analyze this behavior in the next subsections for the different branes. β decreases, with the binding energy reaching its maximum value at the bound, β = 0, where it vanishes. The configuration reduces then to q free radial strings stretchingfrom ρ to ∞ , plus a D p -brane located at ρ . Note that this configuration only exists whenthe magnetic flux is non-vanishing, since only then we can reach β = 0. When the D p -braneis charged the configuration corresponding to free quarks is therefore degenerate. It can berealized either as free radial strings stretching from 0 to ∞ plus a charged D p -brane, withthe charge satisfying (3.47), located at ρ = 0, or as free radial strings stretching from ρ to ∞ plus a D p -brane located at ρ , with a charge that has to satisfy the equality in (3.47).In this case the F1’s are less energetic due to the fact that they now stretch from ρ to ∞ but this is compensated by the energy of the brane at ρ , charged such that β = 0. Notethat the location of the D p -brane has become a moduli of the system. In both cases sincethe strings are radial the size of the configuration vanishes.Note that from (3.45) and (3.43) we have that for all values of the magnetic charge E bin = − f ( β ) ( g s N ) / ℓ (3.48)with f ( β ) ≥
0. Therefore dE/dℓ ≥ E bin as a function of the ’t Hooft coupling and the size of the configuration is the same as in AdS × S [31, 32, 20]. As in that case the fact that it goes as 1 /ℓ is dictated by conformalinvariance, while the behavior with √ λ is a non-trivial prediction for the non-perturbativeregime of the gauge theory. Note that the same non-analytic behavior with λ is predictedfor N = 4 SYM in 3+1 dimensions and for ABJM [33, 34, 35]. In fact, perturbativecalculations like those in [36, 37, 38] can explain this behavior when extrapolated to strongcoupling, as inferred in [39]. Further, the exact interpolating function between the weakand strong coupling regimes for 1/6 and 1/2 BPS Wilson loops was obtained in [40], usingtopological strings and the localization techniques applied in [41] to ABJM theories. We have plotted in Figure 1 the behavior of f ( β ) /qT F as a function of β . We can seethat when the number of strings does not depend on β , i.e. for the ’t Hooft monopolecase, the configuration becomes more stable as β increases. For the di-baryon and baryonvertex configurations the number of strings changes with the magnetic flux, and thereforethe stability of the configuration will vary with β in a way which depends on the specificfunction (3.46). We will analyze this behavior in the next subsections.We now discuss in some more detail the dynamics of the different configurations dis-cussed in the previous section. See also [42]. .0 0.2 0.4 0.6 0.8 1.0 Β H Β L (cid:144) qT F1 Figure 1: Stability of the configuration, for fixed number of strings, as a function of β . In this case Q = π T g s p L + (2 π N ) (3.49)and q = k , so that β = r − π (cid:16) π N L (cid:17) . (3.50)The behavior of the binding energy as a function of N is shown in Figure 2. The minimumbinding energy occurs for zero N , for which β = q − π , and β = 0 is reached for N max L = r − π , (3.51)for which the monopole is no longer stable and reduces to k radial F1’s, stretching from ρ to ∞ , plus a spherical D2-brane with D0-charge L q − π , located at ρ . As a functionof the ’t Hooft coupling (3.51) becomes N max = p λ (4 π −
1) (3.52)which is exactly the same behavior that was encountered in [19] for the maximum value ofthe magnetic flux that could be dissolved in the baryon vertex in
AdS × S . We will findthis same behavior for the di-baryon and baryon vertex with flux in the next subsections.Although dynamically the origin of the bound is quite clear, pointing at an instabilitywhen the magnetic flux makes the energy of the brane too large, its interpretation fromthe CFT side is not clear to us. We refer to the conclusions for a brief discussion.16 .2 0.4 0.6 0.8 1.0 N L - - - - - - E bin (cid:144) qT F1 Ρ Figure 2: Binding energy per string of the ’t Hooft monopole (in units of T F ρ ) as afunction of N /L . In this case Q = π T g s (cid:16) L + (2 π N ) (cid:17) (3.53)and q = k N /
2, so that β = r − L π N (cid:16) π N L (cid:17) . (3.54)This function has a maximum at N L = π , where it reaches β max = q − π . For this valueof the magnetic flux the binding energy per string is minimum. Note however that sincethe number of strings depends also on N this is not the value for which the configurationis maximally stable (if we define the stability in terms of the function f ( β ) in (3.48)). Theallowed values for the magnetic flux are those for which β ∈ [0 , β max ]:1 − r − π ≤ N L ≤ r − π . (3.55)At both ends N ± L = 1 ± q − π , β = 0, and the configuration turns into a collectionof q = k N ± / N /L is shown in Figure 3 (left). Since the total bindingenergy of the configuration depends on the number of strings, which is a function of themagnetic flux, the behavior of the binding energy is modified as shown in Figure 3 (right).The minimum energy occurs now for N = 1 . L . In Figure 4 we have plotted as well f ( β ) /T F (see (3.48)) as a function of the magnetic flux.As we have seen, the D4-brane with flux exhibits a very different behavior with themagnetic flux than the D2-brane. The main difference is coming from the fact that now And the D6-brane, as we will see next. .5 1.0 1.5 2.0 N L - - - - - - E bin (cid:144) qT F1 Ρ N L - - - - - E bin (cid:144) Π T F1 Ρ Figure 3: Binding energy per string (left) and total binding energy (right) of the di-baryon(in units of T F ρ and 2 π T F ρ √ kN respectively) as a function of N /L . N L H N L L (cid:144) Π T F1 Figure 4: f ( β ) for the di-baryon, in units of 2 π T F √ kN , as a function of the magneticflux.the magnetic flux induces a worldvolume tadpole in the D4-brane that was not present for N = 0, and this tadpole has to be cancelled by adding a number of F1’s proportional to N .Accordingly, the whole configuration of point-like D4-brane plus fundamental strings onlyexists for N 6 = 0, with the allowed interval for the magnetic flux, given by (3.55), implyingan allowed interval for the number of fundamental strings ending on the D4-brane:2 π √ kN (cid:16) − r − π (cid:17) ≤ q ≤ π √ kN (cid:16) r − π (cid:17) . (3.56)At the bounds the strings become radial and the configuration ceases to be stable. In this case Q = π T g s (cid:16) L + (2 π ( N − (cid:17) / (3.57)18 .5 1.0 1.5 2.0 2.5 3.0 N L - - - - - - E bin (cid:144) qT F1 Ρ N L - - - - - - - E bin (cid:144) T F1 Ρ N Figure 5: Binding energy per string (left) and total binding energy (right) of the baryonvertex (in units of T F ρ and T F ρ N respectively) as a function of N /L .and q = N + k N ( N − /
8, so that β = vuut − π (cid:16) π N ( N − L (cid:17) (cid:16) π ( N − L (cid:17) . (3.58)This function decreases monotonically with N , reaching its minimum value β = 0 when N L ∼ √ π − π . Therefore the allowed values of the magnetic flux are N L . √ π − π (3.59)We have plotted in Figure 5 (left) the binding energy per string as a function of N /L . Wecan see that the qualitative behavior is very similar to the D2-brane case, and also to thecharged baryon vertex in AdS × S [19]. In all these examples the binding energy per stringincreases with the magnetic flux till it becomes zero when the strings are radial and thebaryon size vanishes. Note however that in this case the tadpole induced in the worldvolumeof the D6-brane depends on the magnetic flux, and therefore the number of quarks thatcan form the bound state depends on N , as q = N + k N ( N − /
8. This modifies thebehavior of the total binding energy as shown in Figure 5 (right). Here we can see that theminimum energy configuration occurs for N /L ∼ .
01, and that the configuration losesstability till it reduces to free radial fundamental strings at N max /L ∼ √ π − π , for which q ∼ π N . The stability of the configuration as a function of the magnetic flux can beseen in Figure 6.The analysis in this section shows that the addition of magnetic flux to the D6-braneallows the construction of more general baryon vertex configurations in which the numberof quarks can be increased up to ∼ π N . A way to construct baryons with q < N number of quarks in AdS × S was considered in [20]. In this background q = N stringsare needed in order to cancel the tadpole in the worldvolume of the spherical D5-brane,19 .0 0.5 1.0 1.5 2.0 2.5 3.0 N L H N L L (cid:144) T F1 N Figure 6: f ( β ) for the baryon vertex, in units of T F N , as a function of the magnetic flux. N being the rank of the gauge group. It is however possible to find more general baryonvertex configurations with q < N quarks if N − q strings stretch between ρ and 0. Thestudy of the dynamics of these configurations reveals that they are stable if the numberof quarks satisfies 5 N/ ≤ q ≤ N . For the minimum value, q min = 5 N/
8, the strings areradial and the binding energy vanishes, exactly the same behavior that we have found atthe limiting values.A similar analysis to the one in [20] for the D6-brane wrapped in P shows that theboundary equation (3.40) has to be modified as ρ ′ q ρ L + ρ ′ = N πq (cid:16) π ( N − L (cid:17) / + 1 q (cid:16) N + k N ( N − − q (cid:17) , (3.60)from which we can conclude that the number of quarks has to satisfy:12 ( N + k N ( N − p − β ) ≤ q ≤ N + k N ( N − , (3.61)with β given by (3.58).Therefore we have found that by combining the addition of magnetic flux and theconstruction in [20] it is possible to find more general baryon vertex configurations inwhich the number of quarks differs from N in a way that depends on the magnetic fluxdissolved in the D6-brane and the number of strings that end on 0 instead of ∞ . Like forall the bounds found in this paper, the quarks are free for the minimum and maximumnumbers allowed, where the configuration ceases to be stable. In this section we briefly discuss how the results of the previous sections for the ’t Hooftmonopole, di-baryon and baryon vertex configurations are modified by the presence of a20on-zero Romans mass F .It was shown in [7] that the CS-matter theory dual to a perturbation of the previous AdS × P background by a mass term should be a perturbation of ABJM with levels k + k = F . The simplest way to see this is to realize that a D0-brane in this backgrounddevelops a tadpole through its CS coupling to the mass [43]: S CS = T Z dt F A t , (4.62)and therefore F fundamental strings should end on it. One can account for these extraindices in the fundamental by modifying the level of one of the gauge groups, such thatthe di-monopole operator dual to the D0-brane becomes O D = (Sym k ) i ...i k + F (Sym k ) j ...j k A i j . . . A i k j k . (4.63)It was shown in [7] that indeed ABJM can be deformed in different ways such that thelevels do not sum to zero. In all cases the deformed theory flows to a CFT, with differentamounts of supersymmetries and global symmetries. The theory that is obtained fromABJM by simply changing the CS levels such that k + k is small (in the precise wayshown in [7]) breaks all the supersymmetries, but flows to a CFT that respects the SO(6)R-symmetry. This is the theory that can be most simply identified as a deformation of the N = 6 solution by a Romans mass, and the one that we will consider in this section.The gravity dual of the N = 0 CFT with SO(6) global symmetry discussed in [7] can beconstructed as a perturbation of the N = 6 solution, with the usual Fubini-Study metricon P , by a small mass F << k, N . In that case the F and F fluxes are not modified,and the F flux that has to be introduced along with the mass (see [7]) can be compensatedwith the coupling of F with a closed B field. This B field will be conveniently absorbedin our definition of F . Note however that it contributes to higher order in the mass toexpression (4.64). Therefore we will ignore it in our analysis below. Moreover, as in [7],we will ignore the effect of the Freed-Witten anomaly. The CS coupling to the mass inthe D4-brane case, given by equation (4.66) below, suggests that a fractional number ofF-strings should be added to the D4-brane in order to cancel the tadpole induced by themass and the Freed-Witten worldvolume flux. Therefore including this effect requires amore careful study, that we hope to address in a future publication.In this massive background the D0-brane acquires a tadpole. This is however not thecase for the other particle-like branes , since the only modification in the action in themassive AdS × P background is the coupling to the mass [43] S CS = T p Z F X r =0 r + 1)! (2 π ) r A ∧ F r (4.64) If we ignore the effect of the Freed-Witten worldvolume flux, as in [7].
21n the CS part.Let us now add a magnetic flux as we did in the previous sections, F = N J . A D2-brane wrapped on S will now develop a tadpole proportional to the mass, given that inthe Chern-Simons action: S CS = 2 π T Z F A ∧ dA = F N Z dtA t . (4.65)Therefore, for non-zero mass we have to add a number of F1’s that is proportional to theproduct of the mass with the magnetic flux: q = F N / P the relevant coupling is S CS = 13! (2 π ) T Z F A ∧ dA ∧ dA , (4.66)therefore for a non-vanishing magnetic flux we need q = F N / P the relevantcoupling is S CS = 14! (2 π ) T Z F A ∧ dA ∧ dA ∧ dA (4.67)and the number of F1’s that must be added for non-zero mass is q = F N /
48, which isagain F times the number of D0-branes dissolved in the D6-brane.We have summarized in Table 1 the number of fundamental strings that are requiredin order to cancel the tadpoles originating from all the background fluxes for each type ofwrapped brane. D p -brane Number of F1’sD0 F D2 k + F N D4 k N + F N D6 N + k N + F N Table 1: Number of F1’s that must end on each D p -brane in the presence of mass (andmagnetic flux).Note that although F << k, N , N can be sufficiently large so as to make F N ≈ k .This is certainly the case for the values found in (3.51), (3.55), (3.59). In the next sectionwe study the dynamics of the particle-like brane configurations with these F1’s attached.22
10 15 20 25 30 F L - - - - - - E bin (cid:144) qT F1 Ρ Figure 7: Binding energy per string of the di-monopole, in units of T F ρ , as a function of F L k . The dynamics of the various brane configurations discussed in section 3 is modified in thepresence of a non-zero mass due to the fact that the number of F1’s attached to the branedepends on the mass as shown in Table 1.Let us consider first the di-monopole, or D0-brane. Following the analysis in section 3we have that Q = T /g s and q = F . Therefore, β = r − (cid:16) kF L (cid:17) (4.68)and the bound (3.47) leads to F ≥ kL (4.69)Therefore, the configuration is stable if the mass is sufficiently large. Note that this boundis perfectly compatible, in the regime of validity of the supergravity description, with thefact that F << k, N . We have plotted in Figure 7 the behavior of the binding energy perstring as a function of the mass. Here we see that the configuration is maximally stablewhen F → ∞ , for which β max = 1, and reduces to F free quarks plus a D0 at the bound,when β = 0.The D2-brane with flux turns out to be more stable in the presence of mass. In thiscase β = s − π (1 + F N k ) (cid:16) π N L (cid:17) (4.70)This function has a maximum at N = F L π k . Since the binding energy per string decreasesmonotonically with β this is the value of the magnetic flux for which the binding energy(per string) is minimum. 23igure 8: Binding energy per string of the ’t Hooft monopole, in units of T F ρ , as afunction of the magnetic flux and the mass.The values of the magnetic flux for which the configuration can form a bound statedepend on the mass. When F satisfies the bound (4.69), required by the stability of theD0-brane, the configuration is stable for all values of the magnetic flux. On the otherhand, when F < kL there is a maximum value for the magnetic flux beyond which theconfiguration is no longer stable, and reduces to k + F N free quarks. As in previous sectionsthis is the value for which β = 0, which in this case is: N max = kL π (4 k − F L ) (cid:16) πF L + q F L + 4 k (4 π − (cid:17) (4.71)This behavior of the binding energy per string as a function of N and F can be shown inFigure 8.The D4-brane with flux in the massive background has β = s − L π (4 N + F k N ) (cid:16) π N L (cid:17) (4.72)This function has a maximum at N = F L π k (1 + q π k F L ). For this value the configura-tion is maximally stable. On the other hand β = 0 is reached when N = L π for F = kL ,and N = kL k − F L (1 ± q − k − F L π k ) for F = kL . For these values the configuration is nolonger stable and reduces to k N + F N free quarks. In summary the values of the magnetic24igure 9: Binding energy per string for the di-baryon, in units of T F ρ , as a function of themagnetic flux and the mass (right: F L k = { , . , . , . , . } for (1)-(5) respectively).flux for which the configuration can form a bound state must satisfy: N ≥ L π for F = 2 kL , (4.73) N ≥ kL F L − k (cid:16)r F L − k π k − (cid:17) for F > kL , (4.74)and 2 kL k − F L (cid:16) − r − k − F L π k (cid:17) ≤ N ≤ kL k − F L (cid:16) r − k − F L π k (cid:17) (4.75)for F < k/L . Note that in all cases there is a minimum value required for the magneticflux, consistently with the fact that also in the massive case a configuration with a D4-braneand fundamental strings attached does not exist for vanishing magnetic flux.The behavior of the binding energy per string as a function of N and F is shownin Figure 9 (left). Figure 9 (right) exhibits the value of the magnetic flux for which theconfiguration is maximally stable for different values of the mass.Finally, the D6-brane with flux has β = vuut − π (cid:16) π N L (1 + F N k ) (cid:17) (cid:16) π N L (cid:17) (4.76)This function reaches its maximum value when N = 0 for arbitrary mass. On the otherhand, β = 0 is reached for finite N when F < kL . Beyond this value of N the configurationis no longer stable and reduces to N + k N + F N free quarks plus a wrapped D6-brane.25igure 10: Binding energy per string for the baryon vertex, in units of T F ρ , as a func-tion of the magnetic flux and the mass (right: F L k = { , . , . , . , . } for (1)-(5)respectively).The behavior of the binding energy per string as a function of N and F is shown inFigure 10 (left). Figure 10 (right) exhibits more clearly the behavior of the binding energywith the magnetic flux for various values of F . In this paper we have analyzed various configurations of particle-like branes in ABJM,focusing on the study of their dynamics. This study has revealed that new and moregeneral monopole, di-baryon and baryon vertex configurations can be constructed if theparticle-like branes carry lower dimensional brane charges.We have seen that a new di-baryon configuration with external quarks can be con-structed out of the D4-brane wrapped on the P ⊂ P . In the presence of a non-trivialmagnetic flux F = N J , with J the K¨ahler form of the P , this brane develops a tad-pole that has to be cancelled with k N / k N / − q − π ≤ N L ≤ q − π . Dynamically, the upper bound arises because if theenergy of the D4-brane with flux is too large the F-strings cannot form a bound statewith it. For this value the strings become radial, and the configuration reduces to free k N max / k N min / N / k . We have seen that the configuration formed by the boundstate D2-D0 plus the k F-strings is stable for N /L ≤ q − π , reducing to k free quarksplus a D2-brane with L q − π D0-brane charge when the upper bound is reached. TheD6-brane in turn is the analogue of the baryon vertex in
AdS × S [6]. The generalizationof the later to include a non-vanishing magnetic flux was studied in [19]. In that referenceit was found that the magnetic flux had to be bounded from above in order to find a stableconfiguration, like for the D2 and D6 branes considered in this paper. For the D6-branethe number of F-strings depends as well on the magnetic flux, but this fact does not modifysubstantially its dynamics.As we have mentioned, all the configurations that we have considered reduce to freequarks when the magnetic flux reaches the highest possible value (also the lowest for theD4-brane). For this value the brane can be located at an arbitrary position in AdS , withthe free radial strings stretching from there to ∞ . This is different from the free quarkconfiguration of [20], where the D5-brane is located at ρ = 0, where it has zero-energy,and the radial strings stretch from 0 to ∞ . For the maximum (and minimum, if applicable)value of the magnetic flux the D-brane is located at an arbitrary ρ , where it has someenergy which is compensated by the lower energy of the strings stretching between ρ and ∞ . In the presence of magnetic flux the location of the D p -brane has therefore become amoduli of the system.We have already stressed that in the probe brane approximation used in this paper allsupersymmetries are broken. However, in analogy with the baryon vertex construction in AdS × S we expect that, at least when the charged branes are supersymmetric, somesupersymmetries will be preserved when the strings join the brane at the same point. Inthis case we would have to consider the full DBI problem and look for spiky solutions[44, 45]. The description of the baryonic brane in AdS × S in terms of a single D5-branedeveloping a spike was done in [46]. This configuration is BPS, and this is reflected in the27act that its binding energy is zero. An attempt to find similar spiky solutions in AdS × P has been made recently in [18], with rather negative results even for the D6-brane with zeromagnetic flux, which should be analogous to [46]. We hope that some spiky configurationscan still be found in this background by relaxing some of the ansatze taken in [18]. Wewill report on these issues in a future publication.It is significant that for all the configurations that we have discussed the binding energyis non-analytic in the ’t Hooft coupling, with this non-analyticity being of the precise formof a square-root branch cut, like in AdS × S . This hints at some kind of universal behaviorbased on the conformal symmetry of the gauge theory.An important question that remains open is what are the field theory realizations ofthe D p -branes with charge that we have considered. Since we do not expect that the D2and D6 brane configurations are supersymmetric it is hard to have an intuition about theinterpretation of the new charges in the field theory side. It is interesting to note that thenumber of extra fundamental strings required to cancel the worldvolume tadpole is thatrequired to cancel the tadpole on the dissolved D2 branes. This might suggest that thedual operators are doped versions of the original ones with an operator representing theD2 branes. It is hard to be more precise, in particular due to the expected lack of SUSY.However, for the D4-brane with D0-charge one can expect that a supersymmetric spikysolution exists, in which case it makes sense to try to interpret the bounds on the magneticflux in the gauge theory dual. In field theory language the bound (3.47) would read: N + N k ≤ π n f √ λ , (5.77)where n f is the number of external quarks, which is a function of the magnetic flux: n f = k N /
2, and λ is the ’t Hooft coupling. Therefore, at strong ’t Hooft coupling weexpect a bound on the baryon charge of (generalized) di-baryon configurations with n f external quarks. This should be related in some way to the stringy exclusion principleof [21], although we have not been able to find a direct interpretation. Note that for allbranes the bound on the magnetic flux exhibits the same non-analytic behavior with λ asthe binding energy, which seems to indicate that the bounds should have its origin in theconformal symmetry of the gauge theory.Finally, the role of the B field, and its potential relation to the Abelian part of thegauge symmetry, remains to be understood. We postpone these investigations for furtherwork. Acknowledgements
We would like to thank Joan Sim´on for useful discussions and the anonymous referee forpointing out some important numerical errors in a previous version of this paper. D.R-G.28ould like to thank N. Benishti and J. Sparks for collaborations on related topics and manyenlightening conversations about physics and mathematics. N.G. wants to thank the CERNTH-division, where part of his work was done, for the kind hospitality. The work of N.G.was supported by a FPU-MICINN Fellowship from the Spanish Ministry of Education andthe European Social Fund. This work has been partially suported by the research grantsMICINN-09-FPA2009-07122, MEC-DGI-CSD2007-00042 and FICYT-IB09-069.