Baryon charge from embedding topology and a continuous meson spectrum in a new holographic gauge theory
aa r X i v : . [ h e p - t h ] A p r Baryon charge from embedding topologyand a continuous meson spectrumin a new holographic gauge theory
Mark Van Raamsdonk and Kevin WhyteDepartment of Physics and Astronomy, University of British Columbia6224 Agricultural Road, Vancouver, B.C., V6T 1W9, Canada
Abstract
We study a new holographic gauge theory based on probe D4-branes in thebackground dual to D4-branes on a circle with antiperiodic boundary conditionsfor fermions. Field theory configurations with baryons correspond to smoothembeddings of the probe D4-branes with nontrivial winding around an S in thegeometry. As a consequence, physics of baryons and nuclei can be studied reliablyin this model using the abelian Born-Infeld action. However, surprisingly, we findthat the meson spectrum is not discrete. This is related to a curious result thatthe action governing small fluctuations of the gauge field on the probe brane isthe five-dimensional Maxwell action in Minkowski space despite the non-trivialembedding of the probe brane in the curved background geometry. Introduction
Gauge-theory / gravity duality [1] provides a powerful tool to construct and studystrongly coupled field theory systems. In recent years, the set of field theories con-structed and analyzed in this way has grown to include examples which are qualitativelysimilar to systems of great physical interest, including QCD (see e.g. [2, 3, 4, 5, 6, 7]),superconductors , superfluids, quantum Hall systems, and cold atom systems (see [8, 9]for recent reviews of applications to condensed matter systems). While it may be toooptimistic to expect that we will be able to find gravitational systems that are exactlydual to real QCD or specific real-world condensed matter systems, these model sys-tems can provide significant qualitative insight into generic phenomena that arise instrongly coupled systems similar to the real-world examples. There are already exam-ples (e.g. the very low viscosity to entropy ratio for the quark-gluon plasma producedin heavy ion collisions) where the insight gained from holographic models offers thebest theoretical understanding of an experimentally measured phenomenon (see e.g.[10, 11, 12]).With such potential for new theoretical insight into physically interesting systems, itseems fruitful to explore a wide variety of holographically constructed field theories. Indoing so, we may uncover new qualitative phenomena in strongly coupled field theoriesthat could help explain real-world physical phenomena or, more generally, lead toan improved understanding of quantum field theory at strong coupling. In addition,amassing a large number of detailed examples will help reveal which features of thesesystems are generic (and thus more likely to apply to other systems for which we maynot have a precise gravity dual), and which are peculiar to specific constructions.Motivated by these considerations, we study in this paper a new holographic fieldtheory closely related to the Sakai-Sugimoto model [7] of holographic QCD. Specifically,our theory has the same adjoint sector, but a different fundamental sector since we useprobe D4-branes instead of probe D8-branes. Thus, our model is based on a braneconstruction where both the “color branes” (which give rise to the adjoint sector) andthe “flavor branes” are D4-branes. The relative orientation for the two sets of branesis: 0 1 2 3 4 5 6 7 8 9 N c D × × × × × N f D × × × × × as shown in figure 1. At weak coupling, this system has a (complex scalar) tachyon inits low-energy spectrum coming from the D4-D4’ strings. However, by separating thebranes in a transverse direction (e.g. the 6 direction), we can arrange for this tachyonto become massless or massive. The lightest D4-D4’ fermions have string scale massesin this case. For the original theory with no transverse separation, the geometrical SO (5) symmetry present in the adjoint sector of the theory is broken to SO (4), whilethe theory defined with a transverse separation between the two sets of branes has thissymmetry broken to SO (3).In order to obtain a decoupled field theory, we want to take a low-energy decoupling1 D4fN D4cX601234 01235
Figure 1: Brane construction for the holographic field theory.limit in this brane setup. We would like to do this in such a way that the lightestmodes of the D4-D4’ strings survive. It is plausible that we can tune the transverseseparation of the two sets of branes as we take the limit to achieve this. In practice, wedo not actually define an explicit decoupling limit starting from a brane configurationin asymptotically flat space. Instead, we do something much simpler (following theSakai-Sugimoto example). We will always consider the limit where N f ≪ N c , sothat the fundamental matter does not affect the physics of the adjoint sector (i.e. thequenched approximation is accurate). Then the addition of fundamental matter can beachieved simply by adding probe D4-branes into the geometry dual to the field theorydescribing the low-energy degrees of freedom of the N c D4-branes. Thus, instead ofstarting with color and flavor branes in asymptotically flat space, we just look for astable configuration of probe D4-branes in the geometry dual to the color branes suchthat the configuration preserves the desired symmetries.Since we are working in a consistent background of string theory, we can say thatthe model we describe is some fully consistent quantum field theory sharing manyqualitative features with QCD. Based on the weak coupling picture, it is tempting tosuggest that the model we describe is one where the fundamental quarks are purelyscalar, since the D4-D4’ fundamental fermions have string scale masses when the branesare tuned to make the lightest scalar massless or massive. Achieving a model withscalar quarks was one of the original motivations for studying this model, since we wereinterested to look at the qualitative similarities and differences between our model andthe Sakai-Sugimoto model (where the fundamental matter is fermionic), and also tosee whether any new qualitative phenomena appear in the physics of strongly coupledfundamental bosons. However, since our actual construction is less direct than anexplicit decoupling limit, we cannot say definitively that the model includes only scalarquarks. 2 utline and Summary
After a review of the basic setup in section 2, we carry out the analysis of probe braneembeddings in section 3, focusing on the case N f = 1. We find a two-parameter familyof D4-brane embeddings. These are labeled by a parameter y that measures how farinto the IR of the geometry the probe brane reaches and a parameter v that controlshow much the brane is tilted into the compact direction of the field theory. Theembeddings in this family correspond to the vacuum solutions for a two-parameterfamily of field theories. In each case, the solution preserves SO (3) ∼ SU (2) globalsymmetry, but for a one parameter family, this symmetry arises via a spontaneousbreaking from SO (4) ∼ SU (2) × SU (2). For this special case, there is a family ofsolutions with the same SO (4)-preserving asymptotics. Each of the solutions preservesonly SO (3), so we would expect an SO (3) vector of massless scalar goldstone bosonsassociated with the broken symmetry. However, we do not find an ordinary discretespectrum of mesons as in other models, but rather a continuous spectrum (as wewould have in a conformal field theory) . This is related to the fact that the actiongoverning small fluctuations of the gauge field on the probe brane is the ordinaryMaxwell action in Minkowski space despite the nontrivial embedding of the probe branein the curved background geometry. We do not have a good interpretation for this fromthe field theory point of view. However, the result crucially depends on the relativenormalization of the Chern-Simons and Born-Infeld terms in the brane action; if thisnormalization is changed at all, we get an ordinary discrete spectrum of mesons. Notethat our results refer only to the small fluctuation analysis (keeping quadratic termsin the action for fluctuations of the probe brane about its equilibrium configurations).It is possible that including the effects of higher-order terms in the Born-Infeld actionmay have interesting effects (in particular, they break the “accidental” five-dimensionalLorentz invariance present in the quadratic action for the gauge field), but this analysisis beyond the scope of the present work.One of the most interesting feature of our model that the baryonic sector can bestudied very reliably, as we discuss in section 5. As in the Sakai-Sugimoto model, baryoncharge arises from D4-branes which wrap an S in the geometry (see section 5 for areview). But in our D4-D4 system, these wrapped D4-branes can smoothly reconnectwith the probe D4-branes of the original embedding. Thus, states in the field theorywith nonzero baryon number correspond to smooth D4-brane embeddings of differenttopology from the vacuum embedding. Baryon charge in the field theory correspondsin the bulk to a topological charge π ( S ) for the embedding (relative to the originalembedding). These smooth embeddings can be studied reliably using the abelian Born-Infeld action, so properties such as baryon mass and nuclear binding energies should beunder complete control in the model. This is in contrast to the Sakai-Sugimoto model, A holographic field theory corresponding to a different class of probe D4-brane embeddings wasstudied in [13] among other examples. Another model with D4-brane probes in a different backgroundwas studied in [14]. A continuous meson spectrum was found also in [17] for a defect theory, but it is not clear whetherthe underlying mechanism is the same. α ′ correctionsto the Born-Infeld action should be important for a completely reliable treatment ofbaryon physics. On the other hand, given the results for the meson sector, our modelis clearly much further from real QCD than the Sakai-Sugimoto model.
In this section, we review the basic construction of our holographic field theory. Webegin by describing the adjoint sector, and then describe the addition of flavor fieldsvia the embedding of probe branes in the dual geometry.
The adjoint sector of our model was originally proposed by Witten [2] as a constructionof non-supersymmetric Yang-Mills theory. It is defined by the low-energy decouplinglimit of N ≡ N c D4-branes wrapped on a circle of length 2 πR with anti-periodicboundary conditions for the fermions. This part of the theory has two dimensionlessparameters, N c and a coupling constant λ = λ D πR , where λ D = g Y M
N .
The dimensionless parameter λ is the effective four-dimensional coupling at the Kaluza-Klein scale. For small λ , this coupling runs to strong coupling at a smaller scaleΛ QCD ∼ R e − cλ where the physics should be exactly that of pure 3+1 dimensional Yang-Mills theory(thanks to fermion masses generated by the antiperiodic boundary conditions and scalarmasses generated at one loop). For large λ , the dual gravity theory becomes weaklycurved, and physics is well described by type IIA supergravity on a background ds = (cid:18) UR (cid:19) ( η µν dx µ dx ν + f ( U ) dx ) + (cid:18) R U (cid:19) ( 1 f ( U ) dU + U d Ω ) e φ = g s (cid:18) UR (cid:19) F = (2 π ) N c ( α ′ ) ω ǫ . (1) There have nevertheless been many studies of baryons and baryon physics in the Sakai-Sugimotomodel making use of various approximations, see e.g. [16] for some of the early work. ω = π is the volume of a unit 4-sphere, ǫ is the volume form on S , and f ( U ) = 1 − (cid:18) U U (cid:19) . (2)The x direction, corresponding to the Kaluza-Klein direction in the field theory, istaken to be periodic, with coordinate periodicity 2 πR , however, it is important to notethat this x circle is contractible in the bulk since the x and U directions form acigar-type geometry.The parameters R and U appearing in the supergravity solution are related tothe string theory parameters by R = πg s N l s U = 4 π R g s N l s while the four-dimensional gauge coupling λ is related to the string theory parametersas λ = 2 π g s N l s R .
In terms of the field theory parameters, the dilaton and string-frame curvature at thetip of the cigar (the IR part of the geometry) are of order λ /N and √ λ , so as usual,supergravity will be a reliable tool for studying the infrared physics when both λ and N are large (in this case, with N >> λ ). The addition of fundamental matter manifests itself through the appearance of N f probe D4-branes in the geometry dual to the adjoint sector. These D4-branes areextended along the x µ directions, and are described by a one-dimensional path in theremaining radial, sphere, and x directions. It is convenient to redefine coordinates sothat the metric in the radial and sphere directions takes the form α ( ρ )( dρ + ρ d Ω ) (3)These coordinates should satisfy dUU p f ( U ) = dρρ From this, we find the map UU = (cid:18) ρρ (cid:19) + 12 (cid:18) ρ ρ (cid:19) ! . where ρ = U − . Locally, the metric (3) is conformally equivalent to R , however weshould note that the space has an infrared end at ρ = ρ where the X circle contractsto a point. Thus, the ball ρ < ρ is not part of the geometry.5he equilibrium brane configurations come in from radial infinity, reach some min-imum value of the radial coordinate, and go back out to radial infinity. These con-figurations asymptote to two specific directions on the S . The boundary conditionsgenerically break the SO (5) symmetry to SO (3), and we expect that the minimumaction embeddings do not break the symmetry further. In other words, we expect thatthe stable configurations will lie in a single plane in the R appearing in (3), so it willsometimes be convenient to use coordinates dρ + ρ d Ω = dr + r dθ + d~x T in terms of which the equilibrium D4-brane configurations will be specified by ~x T = 0and r ( θ ) = ρ ( θ ) (note that ρ = r for ~x T = 0).To write the action for the probe D4-branes, we focus on the case of a single brane,for which we can use the abelian Born-Infeld action S = − µ Z d σe − φ q − det( g ab + ˜ F ab ) (4)together with the Chern-Simons part: S = µ Z X C ∧ e ˜ F (5)where ˜ F = 2 πα ′ F .
We choose static gauge X µ = σ µ for the field theory directions, and describe thenontrivial part of the embedding by functions X ( σ ) , r ( σ ) , θ ( σ ) , X Ti ( σ ), where σ pa-rameterizes the remaining coordinate along the brane. The pull-back metric appearingin the Born-Infeld action is then given explicitly by g µν = G µν + G ∂ µ X ∂ ν X + G rr ∂ µ r∂ ν r + G θθ ∂ µ θ∂ ν θ + G ij ∂ µ X iT ∂ ν X jT g µσ = G ∂ µ X ∂ σ X + G rr ∂ µ r∂ σ r + G θθ ∂ µ θ∂ σ θ + G ij ∂ µ X iT ∂ σ X jT g σσ = G ∂ σ X ∂ σ X + G rr ∂ σ r∂ σ r + G θθ ∂ σ θ∂ σ θ + G ij ∂ σ X iT ∂ σ X jT For now, we are interested in equilibrium brane configurations, which we assume have X iT = 0, so we keep only terms in the action involving X ( σ ), r ( σ ) and θ ( σ ). With thissimplification, the Born-Infeld part of the action becomes S = − µ g s Z dσd x H ( r ( σ )) s r (cid:18) dθdσ (cid:19) + (cid:18) drdσ (cid:19) (6)where H ( r ) = r R (cid:18) ρ r (cid:19)
6e now turn to the Chern-Simons part of the action. Since the background we areconsidering involves a non-zero Ramond-Ramond four-form flux, the potentials C andthe dual C are non-zero. For the configurations that we are considering (which aretranslation-invariant in the field theory directions), the pull-back of C is zero, but wehave a non-zero pull-back for C . We find (see appendix A for a derivation): C = πN ( α ′ ) R ( U − U ) dt ∧ dx ∧ dx ∧ dx ∧ dx , so the Chern-Simons term in the action is S CS = µ Z C = µ πN ( α ′ ) R Z dσd x ( U − U ) ∂X ∂σ For our calculations of the vacuum configurations, it is convenient to fix the remainingreparametrization invariance by choosing σ = θ . If we also define y = rρ x = r ρ R X , the resulting action is S = µ g s ρ R (cid:26) − Z dθh ( y ) p y + ( y ′ ) + g ( y )( x ′ ) + Z dθq ( y ) x ′ (cid:27) where h ( y ) = y (1 + 1 y ) g ( y ) = y ( y − ( y + 1) q ( y ) = ( y − y Since the resulting Lagrangian density does not depend explicitly on θ , we have a θ -independent quantity (analogous to energy for a time-independent Lagrangian density)given by y ′ ∂S∂y ′ + x ′ ∂S∂x ′ − S = hy p y + ( y ′ ) + g ( y )( x ′ ) Since the geometry caps off smoothly at some finite value of y , smooth brane configu-rations must have some minimal value of y for which y ′ = 0. Calling this value y , andcalling the derivative x ′ at this point v , we have h ( y ) y p y + ( y ′ ) + g ( y )( x ′ ) = h ( y ) y p y + g ( y ) v ≡ B x (only on x ′ ), so we have anotherconstant − h ( y ) g ( y ) x ′ p y + ( y ′ ) + g ( y )( x ′ ) + q ( y ) = q ( y ) − h ( y ) g ( y ) v p y + g ( y ) v ≡ C .
From the equations above, we can eliminate x to get: dydθ = ± y s y B ( h ( y ) − ( q ( y ) − C ) g ( y ) ) − . (7)We also find: dxdy = ± y ( q ( y ) − C ) Bg ( y ) q y B ( h ( y ) − ( q ( y ) − C ) g ( y ) ) − θ ( y ) and x ( y ).Integrating, we find θ ( y ) = Z yy d ˜ y y q ˜ y B ( h (˜ y ) − ( q (˜ y ) − C ) g (˜ y ) ) − . (9)where we define θ = 0 to be the angle at which the brane embedding reaches itsminimum value of y . From this expression, it is straightforward to check that for anyvalue y > v , θ approaches a finite value as y goes to infinity.Thus, the brane configurations asymptote to lines of constant θ , as shown in figure2. The relation between y and the maximal value of θ is given by θ ∞ ( y ) = Z ∞ y d ˜ y y q ˜ y B ( h (˜ y ) − ( q (˜ y ) − C ) g (˜ y ) ) − . (10)and plotted in figure 3 for various values of v . For any given v , we find that for large y , the asymptotic angle approaches a limiting value θ Max ( v ), given by θ Max ( v ) = v < . v = 0 π v > v , there is a special value y = y ∗ ( v ) for which the two asymptotic ends of thebrane go towards diametrically opposite points on the sphere. As y approaches 1, θ ∞ increases without bound, corresponding to brane embeddings that wrap multiple timesaround the θ direction. However, for y < y ∗ ( v ) these embeddings are perturbativelyunstable to “unwrapping,” i.e. slipping over the spherical hole in the geometry, as seenin figure 4. The perturbative instability will be demonstrated explicitly in the nextsection. Recall that y=1 represents the IR end of the geometry. y − θ plane) for v = 0. The asymptoticangle between the two ends of the brane is 2 θ ∞ , which ranges from π for the stableembedding which extends to the smallest values of y , down to some value 2 θ Max ≈ . y → ∞ . 9 θ ∞ v = / v = / v =0v = −1 / v = −2 / Figure 3: Asymptotic angle θ ∞ on the sphere vs minimum brane position y in radialdirection, for various values of v . Angle θ is defined to be zero at y = y .For the special case θ ∞ = π/
2, the two ends of the probe brane go to diametricallyopposite points on the S . For these asymptotics, we actually have a family of embed-dings related by the SO (4) rotations that fix these diametrically opposite points onthe sphere as shown in figure 5. This case corresponds to a spontaneous breaking of SO (4) → SO (3) (equivalently SU (2) × SU (2) → SU (2)), and we must therefore havean SO (3) vector of massless goldstone bosons associated with the broken symmetry. Itis these bosons that become tachyonic if we increase θ ∞ beyond π/ v . Thisis very similar to the naive brane picture in figure 1, where a tachyon develops if thetransverse separation between the branes becomes too small.The behavior of the embedding in the X direction can be obtained by integrating(8). We find that for any value of y , the x ( y ) asymptotes to a constant positiveslope dx/dy , so that the brane continues to wrap the x direction as we go out to y = ∞ . Because of the Chern-Simons coupling, the probe brane “prefers” a positiveslope dx/dy ; we see that the asymptotic slope is positive even if slope dx /dθ is negativeat y = y . The behavior of x is shown in figure 6. In this section, we consider small fluctuations about the equilibrium brane configura-tions found in the previous section. We would like to determine which of the embed-dings are perturbatively stable, and for these embeddings, to determine the spectrum10igure 4: Example of multiple embeddings for the same asymptotic sphere angles. Onlyembeddings which do not “wrap” the sphere are stable. The rest are perturbativelyunstable to slipping around the sphere, as shown.11igure 5: Geometrical interpretation of goldstone bosons for the special case θ ∞ = π/ −1.5 −1 −0.5 0 0.5 1 1.5−2−1.5−1−0.500.511.52 θ x y = 2 v =2v =1v =0v =−1v =−2 Figure 6: Behavior of x vs θ for various values of v at y = 2. As a function of y , theslope dx /dy approaches a constant in each case.12f small fluctuations that gives the meson spectrum for the theory.To determine the fluctuation spectrum, we start with the brane action (4) andexpand to quadratic order about a chosen solution, parameterized by ( y , v ). Weconsider all possible bosonic fluctuations, which include fluctuations in the x direction,fluctuations in the three transverse directions along the sphere (which we label by an SO (3) triplet of scalar fields X T ), fluctuations in the r − θ plane, and the gauge fieldfluctuations.In general, for the scalar field modes, the action for small fluctuations about thevacuum solution takes the form S = − C Z d x Z ∞ y dy ( A ( y ) (cid:18) ∂φ∂x µ (cid:19) + 12 ρ R B ( y ) (cid:18) ∂φ∂y (cid:19) + 12 ρ R C ( y ) φ ) (11)while for the gauge fields, we have S = − C Z d x Z ∞ y dy (cid:26) A ( y ) F µν F µν + 12 ρ R B ( y ) F µy F µy (cid:27) It is convenient to define the functions
A, B , and C using the function R ( y ) = (1 + y dθ dy + g ( y ) dx dy ) − . where dθ dy and dx dy refer to the background embedding functions and are given in termsof y , y , and v by the equations (7) and (8). Explicitly, we have R ( y ) = s − B y h ( y ) − ( q ( y ) − C ) h ( y ) g ( y ) . where B ( y , v ) and C ( y , v ) are defined in the previous section. In the special casewhere v = 0, we get: R ( y ) = s − y ( y + 1) y ( y + 1) − ( y − ( y + 1) (cid:18) − y ( y − y ( y − (cid:19) . For the transverse scalar ( X T ) fluctuations, we find A ( y ) = ( y + 1) y R − ( y ) B ( y ) = ( y + 1) y R ( y ) C ( y ) = 12 ( y + 1) y R ( y ) ( (3 y − y dθ dy ) + 6 y ( y − y + 1)( y + 1) dx dy ) − dx dy ( y + 1)( y − y C comes from the Chern-Simons action.For the gauge field fluctuations, we find A ( y ) = 1 y (1 + y ) R − ( y ) B ( y ) = y (1 + y ) R ( y )Finally, the θ and X fluctuations mix with each other, and the fluctuation action forthe combination ( θ, x ) is given as above where now A and B are matrices A ( y ) = R ( y ) ( y +1) y (1 + g ( y ) dx dy ) − dx dy dθ dy ( y − y ( y +1) − dx dy dθ dy ( y − y ( y +1) ( y − y ( y +1) (1 + y dθ dy ) B ( y ) = R ( y ) ( y +1) y (1 + g ( y ) dx dy ) − dx dy dθ dy ( y − ( y +1) y − dx dy dθ dy ( y − ( y +1) y ( y +1) ( y − y (1 + y dθ dy ) . In this special cases v = ±∞ , we have dθ dy = 0, and so these matrices becomediagonal. For the scalar modes, the fluctuation actions above give rise to an equation of motion − A ( y ) ∂ φ∂x µ − ρ R ∂∂y (cid:18) B ( y ) ∂φ∂y (cid:19) + ρ R C ( y ) φ = 0We look for solutions of the form φ ( x, y ) = e ik · x f ( y )where f ( y ) falls off fast enough so that the integral over y in the action converges (i.e.so that φ is a normalizible fluctuation). With this ansatz, the equation reduces to − ρ R ∂∂y (cid:18) B ( y ) ∂f∂y (cid:19) + ( ρ R C ( y ) − λA ( y )) f = 0 (12)where λ = m represents the four-dimensional mass of the fluctuation. In order to solve the gauge field fluctuation equations, it is convenient to choose agauge ∂ µ A µ + A − ∂ y ( BA y ) = 0 . A decouple.For the components in the field theory directions, we have ∂ A ν + A − ∂ y ( B∂ y A ν ) = 0 (13)while for the y component, we have ∂ A y + ∂ y ( A − ∂ y ( BA y )) = 0 . (14)This set of equations has a residual gauge invariance under transformations A µ → A µ + ∂ µ λ A y → A y + ∂ y λ (15)for any λ satsfying ∂ λ + A − ∂ y ( B∂ y λ ) . (16)This allows us to make a further gauge choice A y = 0. The gauge field fluctuationmodes are then captured by solutions to the equation (13). These can be found byseparation of variables, considering solutions of the form A µ = ǫ µ ( k ) e ik · x a ( y )where we require k · ǫ ( k ) = 0by our original gauge condition, and where a ( y ) is a normalizible solution to − k A ( y ) a + ∂ y ( B∂ y a ) . This eigenvalue equation is the same type (12) as we obtain from the scalar equation.
For both gauge and scalar modes, we need to determine the values of λ for whichnormalizible solutions to the equation (12) exist. We can convert this into a simplequantum mechanics problem as follows. First, note that the equation arises from anaction S = Z ∞ y dy ( B ( y ) (cid:18) ∂φ∂y (cid:19) + 12 ( C ( y ) − ˜ λA ( y )) φ ) (17)where we have defined ˜ λ = R λ/ρ . Now, we define a new variable z such that z = Z yy d ˜ yB (˜ y )such that dydz = B ( y ) z ( y ) = 0 .
15n the new variables, the action becomes S = Z z ∞ dz ( (cid:18) ∂φ∂z (cid:19) + 12 B ( y ( z ))( C ( y ( z )) − ˜ λA ( y ( z ))) φ ) (18)where z ∞ = Z ∞ y d ˜ yB (˜ y )This gives rise to the time-independent Schrodinger equation for E = 0, − f ′′ ( z ) + V ( z ) f ( z ) = 0 , so our problem is reduced to determining for which values of ˜ λ the Schrodinger equationwith potential V ( z ) = B ( y ( z ))( C ( y ( z )) − ˜ λA ( y ( z ))) (19)has a bound state with zero energy. We consider first the gauge field fluctuations. Here, we note that C = 0 and A ( y ) B ( y ) =1, so our quantum mechanics potential is simply V A ( z ) = − λ . Also, in this case, we find z ∞ = ∞ . So we do not have any bound states for any λ ,though there are zero-energy solutions f λ ( z ) = e ± i √ λz to the Schrodinger equation for any λ ≥
0. These do not fall off fast enough at large z to be normalizible, but we can superpose solutions with different λ to get normaliziblesolutions. Since the four-momentum in the field theory directions is related to λ by − k = λ , these superpositions will not be eigenstates of four-momentum. Thus thereare no true particle states in the field theory arising from the gauge mode fluctuations.To understand this better, we note that with the new radial variable, the actiongoverning small fluctuations of the gauge field is exactly the Maxwell action in 4 + 1dimensional Minkowski space (despite the nontrivial embedding of the brane in a non-trivial curved space!) S ∝ Z d xdz {− F AB F AB } Configurations with finite energy in the field theory correspond to solutions of these5 D Maxwell equations that fall off sufficiently rapidly for large z and x . These arewavepackets obtained by appropriate superpositions of plane waves A A = ǫ A ( k, k z ) e ik · x + ik z · z Figure 7: Effective potential V ( y ) = C ( y ) B ( y ) for v = 0 and various values of y .Lower graphs have smaller y .So the meson sector of our field theory (at least, the part coming from the gauge fieldfluctuations) behaves more like a conformal field theory with a continuous spectrumthan a massive field theory with particles.It is interesting to note that the behavior we find depends crucially on the rela-tive normalization of the Chern-Simons and Born-Infeld terms in the probe D4-braneaction. If we change the relative coefficient even slightly, the function R ( y ) changesits asymptotic behavior. In the effective quantum mechanics problem, the effectivepotential is still V A ( z ) = − λ but now z ∞ is finite. Since the fluctuation must vanishat z = ± z ∞ , the quantum mechanics problem now has a discrete spectrum (that ofan infinite square well), and we would have a discrete spectrum of mesons in the dualfield theory. For the transverse scalar fluctuations, the effective potential (19) has a lambda inde-pendent part and a term proportional to λ , plotted in figure 7 and 8 respectively. For the λ -independent part of the potential V = BC , we see that for large enough y there will be no bound states, only a continuous spectrum with E ≥
0. For these valuesof y , there will be a zero energy (albeit non-normalizible) solution to the Schrodingerequation for all λ ≥ λ . Thus, the situation is similar to that for thegauge field modes. Note that we are plotting the potentials in this section as a function of y rather than as a functionof z , thus, the actual effective potential in the effective quantum mechanical problems will be relatedto the ones show by a reparametrization of the horizontal axis. Figure 8: Effective potential V ( y ) = − A ( y ) B ( y ) for v = 0 and various values of y .Lower graphs have smaller y .For y small enough, the potential V will have one or more bound states with E <
0. For the full potential V + λV , these bound state energies increase as wedecrease λ below zero, and so we will have a bound state with zero energy for one ormore negative values of λ . Thus, the field theory will be unstable for y below somecritical value y ∗ ( v ) at which the potential V develops a normalizible bound state. Weanticipated this instability in the previous section as the tendency for certain braneconfigurations to “slip” over the sphere and lower their action. Based on that intuition,we expect that the critical value y ∗ ( v ) will be the same value for which the asymptoticbehavior of the brane configuration has θ ∞ = π/ X and θ fluctuations. In general, the spectrum of fluctuations in the X and θ directions is more complicatedto obtain, since the equations are coupled, but we can analyze the fluctuations in thesimple case where v = ±∞ . For v = −∞ , the effective potentials for the X and θ fluctuations are shown in figures 9 and 10.Again, we multiply each of these by λ and ask for which values of λ the effectivequantum mechanics problem has a zero energy eigenvalue. The result is again that anyvalue λ ≥
0, but no negative values of λ will work. Thus, at least for some values ofparameters, we have shown that the model is tachyon-free. Since we do have modesfor which the quadratic action is arbitrarily small, higher order terms (e.g. quartic inthe fluctuations) may be important, but an analysis of these lies beyond the scope ofthe present work. 18 Figure 9: Effective potential V ( y ) = − A ( y ) B ( y ) for v = −∞ and y = 1. –300–250–200–150–100–500V 0.5 1 1.5 2 2.5 3y Figure 10: Effective potential V θ ( y ) = − A ( y ) B ( y ) for v = −∞ and y = 1.19 Baryons
By the gauge-theory / gravity dictionary, gauge fields in the bulk can be associated withconserved currents in the boundary theory. As in the Sakai-Sugimoto model, we havea gauge field living on the probe branes associated with our fundamental matter. Thiscorresponds to the conserved baryon number (more precisely, quark number) currentin the dual field theory. Specifically, the boundary value of the electrostatic potential A (equivalently, the non-normalizible mode) corresponds to a chemical potential forbaryon charge, while the electric flux at the boundary (equivalently, the normaliziblemode of A ) corresponds to the expectation value of baryon charge.In order to have a state in the field theory with baryon charge, we need a sourcefor electric flux on the probe brane. The simplest such source is a fundamental stringendpoint (recall that the string action has a boundary term R A where A is the gaugefield on the brane). We can think of such an endpoint as corresponding to a singlefundamental quark. If a string has both of its ends on the probe brane, we have twoendpoints, but with opposite orientations, so this corresponds to a mesonic state witha quark and anti-quark. In order to have a baryon state, we must have N stringswith the same orientation ending on the probe brane. For a finite energy state, thesestrings must begin at some other source in the bulk. In our background, such a source isprovided by D4-branes wrapped on S [15]. These necessarily have N string endpoints,since the background D4-brane flux gives rise to N units of charge on the spherical D4-branes, so we need N units of the opposite charge (coming from the string endpoints)to satisfy the Gauss law constraint.Thus, a finite energy configuration with a single unit of baryon charge is given bya D4-brane wrapped on S together with N fundamental strings stretched betweenthis D4-brane and the probe D4-branes. A special feature of our model is that thesewrapped D4-branes can smoothly combine with the probe D4-brane (after shrinking thestrings to zero size) to give a configuration with lower energy. In the final configuration,there are no explicit fundamental strings; we simply have a configuration of the probebrane that now wraps the S . In the final configuration, the source for the electricfield on the brane is the bulk flux of the Ramond-Ramond four-form, via the coupling R a ∧ F .Mathematically, the baryon charge in the field theory corresponds to an elementof π ( S ) = Z associated with the embedding. To see this, note that the probe braneembeddings correspond to mappings to the bulk space from R , topologically equivalentto a ball if we add the sphere at infinity. Given any probe brane embedding E with thesame asymptotic behavior as the vacuum embedding E , we can define a map from atopological S to the bulk spacetime by splitting the S into two balls along an S andusing the maps E and E to define the maps from the two balls. By considering onlythe sphere directions in the bulk, we can project this down to a mapping S → S , andsuch mappings may be associated with elements of the homotopy group π ( S ) ∼ Z .This integer gives the baryon number of the configuration in the field theory.In order to find the actual bulk embedding corresponding to a single baryon, it isnecessary to find the probe brane embedding with a single unit of winding on the S n baryons, we want to find theminimal energy brane embedding with n units of winding on the S . In general, anumerical analysis will be required, but it should be possible to obtain precise resultsfor the masses of small nuclei. Acknowledgements
We would like to thank Andreas Karch, Moshe Rozali, Eric Zhitnitsky, and especiallyOfer Aharony for helpful discussions and comments. This work has been supported inpart by the Natural Sciences and Engineering Research Council of Canada, the AlfredP. Sloan Foundation, and the Canada Research Chairs programme.
A Ramond-Ramond forms
The results of our analysis depend crucially on the relative normalization of the Born-Infeld and Chern-Simons parts of the brane action. Since there is some variationin the literature for the normalization of the Ramond-Ramond fields in the Wittenbackground, we include here a derivation of the correct result (consistent with a subsetof previous papers). We use the fact that with the correct Ramond-Ramond flux, aD4-brane wrapped on S should have induced N units of charge, so that a configurationwith N string endpoints (of the correct orientation) on the wrapped D4-brane shouldhave zero integrated charge and thus satisfy the Gauss Law constraint for the compactsurface. The brane action is µ Z (2 πα ′ ) A ∧ F while the action for each string endpoint is simply Z A p where A p is the pullback of the gauge field to the worldline of the string endpoint. Weknow that the four form is constant on the sphere, i.e. we have F = Cǫ where ǫ isthe volume form on the sphere whose integral is 8 π / i is ρ = µ (2 πα ′ ) Cǫ − X i δ (Ω − Ω i )where the delta functions are defined as four-forms localized at the indicated point thatintegrate to 1. Integrating the density over the sphere and setting the result to zerogives: µ (2 πα ′ ) Cω = N µ = (2 π ) − ( α ′ ) − / , we conclude that the Ramond-Ramondfour-form is F = 3 πN ( α ′ ) ǫ From this, we can calculate that the dual six-form in the background is given by (thesign here is related to a choice of convention for the direction of the flux)( F ) U = − G G G G G G UU √− G ( F ) θ θ θ θ = U F πN ( α ′ ) and using F = dC , we obtain the five-form C = U − U F πN ( α ′ ) dt ∧ dx ∧ dx ∧ dx ∧ dx . where we have fixed the constant of integration so that the form is nonsingular at U = U . References [1] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, “LargeN field theories, string theory and gravity,” Phys. Rept. , 183 (2000)[arXiv:hep-th/9905111].[2] E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement ingauge theories,” Adv. Theor. Math. Phys. , 505 (1998) [arXiv:hep-th/9803131].[3] A. Karch and E. Katz, “Adding flavor to AdS/CFT,” JHEP , 043 (2002)[arXiv:hep-th/0205236].[4] I. R. Klebanov and M. J. Strassler, “Supergravity and a confining gauge theory:Duality cascades and chiSB-resolution of naked singularities,” JHEP , 052(2000) [arXiv:hep-th/0007191].[5] J. Babington, J. Erdmenger, N. J. Evans, Z. Guralnik and I. Kirsch, “Chiralsymmetry breaking and pions in non-supersymmetric gauge / gravity duals,”Phys. Rev. D , 066007 (2004) [arXiv:hep-th/0306018].[6] M. Kruczenski, D. Mateos, R. C. Myers and D. J. Winters, “To-wards a holographic dual of large-N(c) QCD,” JHEP , 041 (2004)[arXiv:hep-th/0311270].[7] T. Sakai and S. Sugimoto, “Low energy hadron physics in holographic QCD,”Prog. Theor. Phys. , 843 (2005) [arXiv:hep-th/0412141].228] S. A. Hartnoll, “Lectures on holographic methods for condensed matter physics,”Class. Quant. Grav. , 224002 (2009) [arXiv:0903.3246 [hep-th]].[9] C. P. Herzog, “Lectures on Holographic Superfluidity and Superconductivity,” J.Phys. A , 343001 (2009) [arXiv:0904.1975 [hep-th]].[10] P. Kovtun, D. T. Son and A. O. Starinets, “Viscosity in strongly interactingquantum field theories from black hole physics,” Phys. Rev. Lett. , 111601(2005) [arXiv:hep-th/0405231].[11] D. T. Son and A. O. Starinets, “Viscosity, Black Holes, and Quantum FieldTheory,” arXiv:0704.0240 [hep-th].[12] S. S. Gubser and A. Karch, arXiv:0901.0935 [hep-th].[13] D. Gepner and S. S. Pal, arXiv:hep-th/0608229.[14] R. Casero, A. Paredes and J. Sonnenschein, “Fundamental matter, meson spec-troscopy and non-critical string / gauge duality,” JHEP , 127 (2006)[arXiv:hep-th/0510110].[15] E. Witten, “Baryons and branes in anti de Sitter space,” JHEP , 006 (1998)[arXiv:hep-th/9805112].[16] H. Hata, T. Sakai, S. Sugimoto and S. Yamato, “Baryons from instantons inholographic QCD,” arXiv:hep-th/0701280.D. K. Hong, M. Rho, H. U. Yee and P. Yi, “Dynamics of Baryons from StringTheory and Vector Dominance,” arXiv:0705.2632 [hep-th].D. K. Hong, M. Rho, H. U. Yee and P. Yi, “Chiral dynamics of baryons fromstring theory,” arXiv:hep-th/0701276.K. Nawa, H. Suganuma and T. Kojo, “Brane-induced Skyrmions: Baryons inholographic QCD,” arXiv:hep-th/0701007.K. Nawa, H. Suganuma and T. Kojo, “Baryons in Holographic QCD,” Phys. Rev.D , 086003 (2007) [arXiv:hep-th/0612187].[17] D. Arean, A. V. Ramallo and D. Rodriguez-Gomez, “Mesons and Higgs branchin defect theories,” Phys. Lett. B641