On Classification of QCD defects via holography
aa r X i v : . [ h e p - ph ] F e b ITEP-TH-03/09
On Classification of QCD defects via holography
Alexander S. Gorsky, Valentin I. Zakharov
Institute of Theoretical and Experimental Physics,B. Cheremushkinskaya ul. 25, 117259 Moscow, Russia
Ariel R. Zhitnitsky
Department of Physics and Astronomy,University of British Columbia, Vancouver, BC V6T1Z1,Canada
We discuss classification of defects of various codimensions within a holographicmodel of pure Yang-Mills theories or gauge theories with fundamental matter. Wefocus on their role below and above the phase transition point as well as their weightsin the partition function. The general result is that objects which are stable andheavy in one phase are becoming very light (tensionless) in the other phase. Weargue that the θ dependence of the partition function drastically changes at the phasetransition point, and therefore it correlates with stability properties of configurations.Some possible applications for study the QCD vacuum properties above and belowphase transition are also discussed. PACS numbers:
I. INTRODUCTION
The gauge/string duality proved to be effective in description of various aspects of gaugetheories at weak and strong coupling. In the N=4 SYM one can discuss a precise comparisonof the gauge theory results with the sigma model or SUGRA calculation since the relevantgeometry
AdS × S is well established. However the situation in the theory with less amountof SUSY is much more complicated and the explicit background for the pure YM or QCD isnot found yet. The most useful dual model of pure YM at nonzero temperature [1] is basedon a stack of N c D4 branes which are wrapped around a compact coordinate and at large N c provide the geometry of the black hole in AdS . Adding the probe N f D − ¯ D x and the Euclidean time coordinate τ which branes can wrap around.In the previous studies some identifications of the defects have been made. The D0 braneextended along x was identified as the YM instanton [3]. The D2 brane wrapped aroundboth periodic coordinates was identified as the magnetic string [4]. The D6 brane wrappedaround the compact S part of the dual geometry and τ was considered in pure YM [5] anddual QCD [6, 7] where it has interpretation of the domain wall separating two vacua. The D4brane wrapped around the S and extended along τ has the interpretation of the ”baryonicvertex” in pure YM and in the Sakai-Sugimoto model [2]. Some interplay between the Dbranes and the Z N domain walls has been discussed in the holographic picture in [8, 9, 10].There are a few generic facts concerning the branes and their intersections. Let usnumerate few of them relevant for the main text: • p-brane behaves as an instanton on the (p+4)-brane worldvolume [11] • p-brane parallel to the (p+2)-brane gets melted into homogenious field [12] • p-brane transverse to the p+2-brane behaves as the monopole on the (p+2) worldvol-ume [13] • branes in external fields can expand into higher dimensional branes via the Myerseffect [14]Our goal here is to look at a variety of defects using a universal classification scheme.We will be mostly interested in behavior of the corresponding configurations when theconfinement-deconfinement phase transition is crossed and shall emphasize some univer-sal properties of the D-defects. It is very likely that some of the defects to be discussed hereare very important for physics, some of them could be irrelevant. Therefore, we anticipatethat some important/interesting defects will be further discussed and studied in great de-tail in future. It is not the goal of the present paper to go into a deep detail analysis ofeach particular configuration. Instead, our goal is the classification of the D-defects, withemphasis on the change of their properties across the phase transition..To be more specific, our basic tool is the dual description of the deconfinement phasetransition as the Hawking-Page phase transition [1], in which case the two metrics withthe same asymptotic get interchanged. The wrapping around x is stable above the phasetransition T > T c while it is unstable below the critical temperature T < T c , see definitionsand details below. And vice-verse, the wrapping around τ is stable at small temperatures T < T c and unstable at high temperatures, T > T c .Since the nonperturbative physics is sensitive to the θ -term we shall also discuss the θ dependence from the dual perspective. We argue that the behavior of the D defectsis strongly correlated with the θ -dependence when the phase transition is crossed at T c .Such a drastic change in θ dependence at the T c has already been noticed in the literature[3, 4, 15, 16, 17]. We note also that some drastic changes in θ behavior are also supportedby the numerical lattice results [18] -[22], see also a review article [23], which unambiguouslysuggest that the topological fluctuations (related to θ behavior) are strongly suppressed indeconfined phase, and this suppression becomes more severe with increasing N c . Here weshall present some additional examples which support this picture.The paper is organized as follows. In Section II we describe our model based on the N c D4 branes with one compact worldvolume coordinate. In our main Section III we classify D-defects treated in the probe approximation when they do not deform the dual geometry. InSection IV with discuss some composite objects which combine different types of D branes.Finally , in Section V we introduce matter fields in our system which treated as the probes, N f ≪ N c . Section VI is our conclusion. II. DESCRIPTION OF THE MODEL
A natural starting point to discuss the dual holographic geometry is provided by a set of N c D4 branes wrapped around a compact dimension [1]. We shall consider the pure gaugesector first and then add flavor D8 branes along the lines of the Sakai-Sugimoto model [2].We shall assume the large N c limit and consider the supergravity approximation. In this FIG. 1: Metric at
T < T c (left figure) and T > T c (right figure). approximation the geometry looks as M = R , × D × S and the corresponding metricreads as ds = ( uR ) / ( − dt + δ ij dx i dx j + f ( u ) dx ) + ( uR ) − / ( du f ( u ) + u d Ω ) e Φ = ( uR ) , F = 3 N c ǫ π , f ( u ) = 1 − ( u Λ u ) (1)where R = ( πg s N c ) / and R = π ( R u Λ ) / . The coupling constant of Yang-Mills theory isrelated to the radius of the compact dimension R as follows g Y M = 8 π g s l s R At zero temperature theory is in the confinement phase and in the ( u, x ) coordinates wehave the geometry of a cigar with the tip at u = u Λ . The D4 branes are located along our(3+1) geometry and are extended along the internal x coordinate. The key point is that inthe non-zero temperature case there are two backgrounds with similar asymptotic topologyof R × S τ × S × S , where τ is the Wick-rotated time coordinate τ = it , τ ∝ τ + β . Onebackground corresponds to the analytic continuation of the metric described above whilethe second background corresponds to interchange of τ and x , that is the warped factor isattached to the τ coordinate and the cigar geometry emerges in the ( τ, u ) plane instead of( x , u ) plane which now exhibits the cylinder geometry, see Figure 1. It was shown in [1] bycalculation of the free energies that above T c the latter background dominates.That is above phase transition wrapping around the internal x circle is topologicallystable, while the wrapping around the Euclidean time coordinate is unstable. This is oppositeto the stability pattern of the two wrappings below the phase transition.Another issue which we shall be interested in concerns the θ -dependence of the worldvol-ume theories on the different probe branes. It can be traced from the Chern-Simons(CS)terms involving the interaction with the RR one-form C δL = Z C ∧ e F (2)where F is the gauge two-form. Taking into account that θ = Z dx C (3)one immediately recognizes that the θ dependence of the worldvolume theories on the de-fects correlates with the wrapping around x coordinate. Moreover it is clear that the θ dependence of a single defect is poorly defined in the confined phase since the wrappingaround x is topologically unstable and one could discuss the θ dependence of a kind of acondensate of the defects.To model QCD one adds the N f D − ¯ D x coordinate. Thereare a few qualitative phenomena to be mentioned. First, the chiral symmetry breaking isdescribed geometrically in terms of the connectness of the D − ¯ D S and they are the instantons in the D8 brane worldvolume theory, Note thatin the holographic QCD there should be some care concerning the gauge invariance of theRR field. The gauge invariant field strength of the RR field due to the bulk anomaly getsshifted by the η ′ meson and the correct invariant identification of the θ -term reads as Z D F ,inv = θ + p N f f π η ′ (4)where the integration over the u, x disc is implied.In the thermal gauge theory a natural order parameter is the vacuum expectation valueof the Polyakov loop < W ( β ) > = < T rP exp ( Z dτ A ) > (5)which is vanishing at T < T c while < W ( β ) > = 0 at T > T c . This implies that Z N symmetryis unbroken at T < T c and broken at T > T c . It is possible to discuss another R-typesymmetry of the rotation of x coordinate which is assumed to be broken nonperturbativelyat zero temperature to the discrete one similar to SUSY case. Therefore one could expectthe total discrete symmetry to be Z N × Z N where the order parameter for the second factoris < W ( R ) > = < T rP exp ( Z dx A ) > (6)It can serve as the order parameter analogous to the Polyakov loop since it has a nontrivialvev at small temperatures and vanishes in the deconfinement phase. Let us emphasize thatthe total discrete group mentioned above differs from the same product discussed in [8]. Inthat paper the second factor corresponds to the S-dual magnetic center group whose orderparameter is identified with the T’Hooft loop. III. ZOO OF THE D- DEFECTSA. D0 branes
D0 instantons.
The simplest defects to be discussed are D0 branes. It was argued in [3] that instantons arerepresented by the Euclidean D0 branes wrapped around x . In that case it was argued thatthe instanton is well defined above T c as it corresponds to the the geometry of the cylinder.On the other hand, any finite number of instantons are ill-defined below T c because of theD0-brane instability.The θ dependence of the D0 action is captured by the CS term on its worldline. Thechange of the instanton role at the transition point corresponds to the change from theWitten-Veneziano to t’Hooft mechanisms of the solution to the U (1) problem. In QCD-likebrane setup D0 branes wrapped around x intersect with the flavor branes and induce thesources on the flavor brane worldvolumes.This picture can also be readily understood in the quantum-field theory terms since anestimation for T c in the Λ QCD units can be given [16]. Indeed, the wrapping around x corresponds to the well defined small instanton and one can use the standard instantoncalculus to estimate the critical temperature T c and the θ behavior above T c : V inst ( θ ) ∼ cos θ · e − αN ( T − TcTc ) , ≫ (cid:18) T − T c T c (cid:19) ≫ /N,χ ( T ) ∼ ∂ V inst ( θ ) ∂θ ∼ e − αN ( T − TcTc ) → , α ∼ , N ≫ . (7)Such a behavior implies that the dilute gas approximation at large N c is justified evenin close vicinity of T c as long as T − T c T c ≫ N c . Such a sharp behavior of the topologicalsusceptibility χ ( T ) is supported by numerical lattice results [18] -[22] which unambiguouslysuggest that the topological fluctuations are strongly suppressed in the deconfined phase, andthis suppression becomes more severe with increasing N c . These general features observedin the lattice simulations have very simple explanation within QFT framework as eq. (7)shows, as well as in holographic model of QCD [3, 16].Finally, let us address the following question: what happens with our D0 instantons inthe deconfined phase, immediately at T > T c ? We know that at sufficiently large tempera-tures ( T − T c ) /T c ≫ /N the configuration becomes a stable instanton in 4d with the size ρ ∼ ( πT ) − . The density of the instantons is exponentially suppressed ∼ cos θ · e − αN ( T − TcTc ),magnetic charges of the constituents (if exist, see section IV) are completely screened suchthat it makes no sense to speak about individual constituents in this regime. However, forfinite N there is a window of temperatures 0 < ( T − T c ) /T c ≤ /N when the magneticdegrees of freedom are not completely screened yet. This window which shrinks to a pointat N = ∞ is obviously beyond analytical control. However, these magnetic degrees offreedom could be be extremely important in the window 0 < ( T − T c ) /T c ≤ /N . It istempting to assume that these magnetic degrees of freedom is a trace of fractional instantonconstituents which likely to exist in confined phase, see discussions in section IV. D0 - particle
The orientation of the D0 brane worldline along the Euclidean time τ corresponds toits realization as a KK particle without the θ dependence. However the Polyakov loop T rP exp( R dτ A ( τ )) may develop. This configuration at T < T c is a stable scalar glue-likeconfiguration which must have very different properties in comparison with all standardglueballs when the temperature approaches T c from below, ( T c − T ) →
0. Above the criticaltemperature (deconfinement phase) KK modes tend to condense near the tip of the cigarbecause of the instability of the wrapping around τ . In the deconfinement phase KK modesbehave as the instanton -like configuration (with no θ dependence) in the effective 3D gaugetheory. This instability may have enormous consequences for physics since an arbitrary largenumber of such states can be produced in vicinity of T ≃ T c . We shall not elaborate on thisissue in the present work. B. D2 branes
D2 string.
The magnetic string is the probe D2 brane wrapped around S parameterized by x andits tension is therefore proportional to the effective radius R ( u ) [4]. At small temperaturesthis wrapping is topologically unstable and the D2 brane tends to shrink to the tip whereits tension vanishes. This is the large- N c counterpart of the effect of dissolving of p -branesinside p + 2-branes [12]. We see that in this way one immediately reproduces the observedproperty of tensionlessness of the magnetic string in the confining phase which howeverbecomes tensionful above the critical temperature T c of the deconfinement phase transition.The θ -term in the magnetic string worldsheet Lagrangian is induced by CS term L CS = Z d x C ∧ F (8)that is configurations with the flux on the worldsheet amount to the nontrivial 4d topologicalcharge. It was also argued that the magnetic strings amount to the negative contribution tothe total energy of plasma at T > T c [4]. It could explain the negative-sign contribution ofthe lattice magnetic strings into the energy of plasma [26]. Because of the instability of the”thermal” cigar magnetic string becomes effectively particle-like object in the Euclidean 3D[4], in agreement with the lattice studies [27]. D2 domain walls.
Turn now to the discussion of the D2 domain walls. Let us emphasize from the very beginningthat to consider the stable infinite domain walls the degenerate vacua should exist. On theother hand we expect the single stable vacuum in the pure YM case and QCD. That is thearguments concerning the domain walls should be interesting in two aspects. First, thereare metastable vacua whose energy density differs from the density of the true one by theterms O (1 /N ) that is they are almost stable at large N. One can also consider the domainwall balls when the configuration is stabilized by the domain wall tension.There are two types of domain walls in R built from D2 branes. Consider first a D2brane localized in the x coordinate. It corresponds to a domain wall in 4D and has no θ dependence. The theory on the domain wall involves the periodic real scalar field whichcorresponds to the position of the D2 brane on the x circle as well as a scalar correspondingto its radial coordinate. Its tension is small in the deconfinement phase and it behaves asthe string in the 3d effective description. It has no θ dependence.The second type of D2 S-domain walls extended in x involves a nontrivial θ dependentterm. Because of the unbroken electric Z N symmetry in the confinement phase such domainwalls are expected to exist in N-tuples in this phase symmetrically on the τ circle. Theworldsheet theory in the deconfinement phase involves only one real scalar correspondingto its radial coordinate. These D2 domain walls may play an important dynamical rolesupporting the constituents with fractional topological charges, see Section IV. Space filling D2 brane .
One could also consider the D2 branes localized both in τ and x directions. Such space-filling D2 branes most probably are expected to exist in N-tuples in both phases because ineach phase the D2 brane has one unbroken Z N symmetry. Hence the effective gauge theoryshould have SU(N) gauge theory in both phases. These branes could play an importantrole in the effective 3D description of the deconfinement phase. Their worldvolume theorynaturally involves one complex and one real periodic scalars. D2- ¯ D pair . One could also discuss the D2 brane extended along the radial coordinate. Such a config-uration is an analogue of the D8 brane in Sakai-Sugimoto model which is extended alongthe radial coordinate and has U-shape form in the chirally broken phase. In the holographicQCD case one actually has D − ¯ D τ it behaves as string while inthe opposite case as the domain wall. In both cases the tension of the object is finite. Itis tempting to speculate that some kind of the chiral symmetry breaking happens in puregauge theory (without flavor fermions) being localized at lower dimensional defects ratherthan in the entire space.It is also tempting to speculate that some of the D2 branes discussed in this subsec-tion could be mapped into the low-dimensional, chirality-related structures observed on the0latices. C. D4 branes
D4 particle
There are several possible embeddings of D4 branes. One possibility concerns D4 wrappedaround S and extended along the Euclidean time τ . The familiar example of such wrappingin the QCD like geometry [2] has the interpretation of baryon if matter fields in the form ofD8 branes are present in the system. The key point here is that due to the CS term Z d xC ∧ F ∼ N c (9)the “electric charge” N c is induced on the D4 brane that is one has to add N c open strings.It is a static topologically stable configuration. In the QCD-like case these open strings endon the D8 brane yielding the baryonic state.In the pure YM case there are no flavor branes that is one has to add additional low-dimensional branes to compensate charge and make a gauge invariant object. The mostsimple way to achieve this goal is to add N c D0 branes yielding the D0-D4 open strings.Hence we get the D4 particle which does not feel the θ -term and is well-defined below thecritical temperature T < T c .We can combine the “ D4 particle” with “D4 anti-particle” to form a gauge invariantobject, see FIG 2. The mass of this object scales as ∼ N c and is much heavier thanthe usual glueballs. In fact, one can construct gauge invariant objects with any evennumber of vertexes such that the total charge vanishes. It is a new family of glueballs withmass ∼ N c V , where V is the number of D4 and anti-D4 particles which form a desiredconfiguration. It is amusing that such kind of structure in QCD had been previouslydiscussed [28] motivated by the discovery of the carbonic Fullerenes C and C in 1985,which are nano-scale objects [29]. The QCD objects, similar to the carbonic Fullerenes withfemto-meter scale were named Buckyballs. It has been also demonstrated that the “magic”numbers for Buckyballs are V = 8 , , ,
120 which correspond to the most symmetric, andlikely, most stable configurations [28]. The properties of this configuration are not sensitiveto θ . The possibility to discover such kind of configurations at RHIC was discussed in [28].1 FIG. 2: Buckyballs with N = 3 and V = 8 , , ,
120 from [28]. The construction combines “ D4particle” with “D4 anti-particle” to form a gauge invariant object with zero baryon charge. Themass of this object scales as ∼ N c . D4 instanton.
If we consider wrapped D4 brane extended along x we get a new-object, ”D4 instanton”which is to be distinguished from the “D4 particle” discussed above. The origin of this termis due to its similarity in the 4D Euclidean space-time to the canonical instanton (or caloronat T = 0). This object tends to condense below the phase transition and is well definedabove the transition point, similar to the instanton. It carries a nontrivial θ dependence andthe corresponding contribution to the action from the single ”D4 instanton” looks as follows δS ∝ θ Z T rF ∧ F ∝ C (10)Due to the background flux dC through S one has N c Z dx C (11)term in the action implying that the D4-instanton worldvolume is populated by N D0instantons which provide the topological charge N in D4 worldvolume theory. Due tothe unbroken electric Z N in the confinement phase one could expect the N-tuples ofD4-instantons to exist. The nature of large factor N c in both equations (11) and (9) is oneand the same, namely the background flux dC through S which is proportional to N c .2However, the physical interpretation for these two cases is quite different: in the first caseit is the mass of the particle which is ∼ N c , while in the second case it is the action of N c different instantons accompanied by 4 N c zero modes each. It is tempting to identify thisD4- instanton with the configuration consisting of N c different calorons with maximallynontrivial holonomies . As is known, exactly the configuration consisting N c differentcalorons provides an infrared finite contribution to the partition function [30]. D4- ¯ D U-shaped pair.
One can also consider the U-shaped D4 brane extended along the radial coordinate. If itwraps S it is an instanton-like object which however does not carry any θ dependence.Since there is in fact a connected D − ¯ D S . Such object is a space-time filling brane which provides a kind ofhomogeneous ”chiral symmetry breaking” in pure YM theory. D. D6 branes
D6 string .
Let us turn to D6 branes. Consider first the pure YM case at large N c . If D6 is wrappedaround S × x it behaves as the string in the space-time whose tension is defined by thescale of S . Due to the wrapping around S the string carries a CS term generated on itsworldsheet from the CS term Z d xC ∧ F ∧ F (12)which reads as N c R d xA ∧ F . In the confinement phase there is a nontrivial holonomy along x represented by W ( R ), see (6), hence the effective 2d θ term on the D6 string is induced.Note that the induced θ term on t he D6 string is proportional to N c , somewhat similar tothe situation discussed in [31] for the magnetic strings in N = 1 ( ∗ ) theory.Due to the wrapping around x it carries intrinsic θ dependent term Z d xC ∧ F ∧ F ∧ F (13) Instanton at T = 0 becomes a caloron with a generically nontrivial holonomy[25]. θ dependent contribution from this term the topological Chern number c ( F )should be nontrivial. It follows from the D2 branes on the D6 worldvolume. Hence we seethat the interesting ”composite” D6-D2 string has θ dependent contributions from the bothcomponents. Such D6 strings according to our standard arguments are individually unstableand large number of them have tendency to condense at small temperatures T < T c . Theobject is well defined at large temperatures, T > T c and has finite tension. The interpretationof such kind of objects is far from obvious, and the role they play in physics is also unclearat the moment.One can also consider the U-shaped D − ¯ D D6 domain wall .
If D6 brane wraps around τ it behaves as the domain wall which is a source of the corre-sponding RR-form. Such a configuration has been interpreted in ref. [5] as the domain wallwhich separates different metastable vacua known to exist in gluodynamics at large N c . Itsworldvolume theory on the domain wall involves the conventional CS term and has no θ dependent term. IV. COMPOSITE DEFECTS
In this Section we present a few examples of composite defects which exhibit interestingfeatures. Some of them may play a crucial role in understanding of the dynamics.
D0-D2 .
First, we want to address the following question: what happens to instantons in the con-fined phase. Naively, one could think that as the metric takes the geometry of a cigar in( x , u ) plane at T < T c the system becomes unstable, and therefore, there is no subject forthe discussions as instantons simply disappear from the system. However, as we discussedbefore, one should speak about effectively zero action for formation of such kind of objects.Therefore, numerous number of these objects can emerge in the system without any sup-pression. In fact, in [16] it has been argued that this is precisely what is happening when4one crosses the phase transition line from above.Now, in order to investigate what kind of objects may emerge when the phase transitionis crossed from above, we add the D2 domain walls (discussed in section IIIB) localizedat some points along x coordinate. In this construction an object with a fractionaltopological charge 1 /N c may emerge. Indeed, one can follow the construction of ref. [32]for SUSY case when N c D2 branes located symmetrically split the instanton into N c constituents stretched between pairs of domain walls. Each constituent has fractionalinstanton number 1 /N c as well as the fractional monopole number and has no reasonto condense. In our system we have precisely appropriate D2 branes which are neededfor this construction. These monopoles are instantons in the 3d gauge theory on theD2 worldvolume theory which involves the scalar corresponding to the position of D2branes on the cigar. As we mention previously in Section IIIA these magnetic monopolesmay play an important role in the region close to the phase transition 0 < | T − T c | /T c ≤ /N c . D6-D4 .
There are several configurations involving composite D6-D4 defects. The first one to bementioned is the combination ”D6 string-D4 instanton”. Since the D4 instanton shares allcoordinates with D6 string it is melted into the flux on the string of constant ”electric” field.Another example of the melting concerns the combination ”D6 domain wall-D4 particle”when the D4-particle delocalizes on the domain wall into the flux. It is interesting to notethat since in the holographic QCD the D4 particle is identified with the baryon upon meltingwe get a domain wall with the baryonic density. The defect is well defined in the confinementphase.Another interesting possibility concerns the combination ”D6 domain wall-D4 instanton”.In this case a fractional D4 instanton can emerge if there are several domain walls localizedat different positions at x and D4 instanton can be stretched between a pair of domainwalls in the x direction yielding a monopole-like objects on the domain-wall worldvolume. D2-D4 .
An interesting situation happens if we consider the configuration of D2- string and D4-particle localized in x . In this case the magnetic D2 string can be stretched between twoD-particles and therefore does not wrap the x circle. Hence it has finite tension equal tothe distance δx between two D4-particles and does not condense in the confinement phase.5This configuration carries fractional topological charge and is θ dependent. The D4-particlesacquire magnetic charges where the flux of the magnetic string ends on.In the case of D2-domain wall and D4-instanton opposite situation can happen. TheD4-instanton worldline can be split between two D2-domain walls localized at different x coordinates. Such configuration carries fractional topological charge as well. D2-D6 .
The simplest configuration of such type is ” D6 string- D2 string”. That is we have compositestring object with additional ”charge” since the D2 induces instanton-like charge on the D6-brane worldvolume. Such a composite string is unstable in the confinement phase and welldefined in the deconfinement phase. Another combination ”D6 domain wall- D2 string” canbe stable in the confinement phase if there are several domain walls localized at differentpositions in x . The same can be said about the configuration of ” D2 domain wall-D6string” with several D2 walls. V. DEFECTS IN HOLOGRAPHIC QCD
In this Section we discuss defects in the holographic QCD when matter fields are includedby adding N f U-shaped D8 branes [2] in the probe approximation. This case can be consid-ered as a particular example of the composite defects in pure YM theory. To study defectsin holographic QCD we shall add some additional probe branes of different dimensions. Inwhat follows we mention only a few effects which are specific for theory with the fundamentalmatter.First, we can add U-shaped D6 branes parallel to the D8 branes. From the 4d viewpointthey are unstable strings. Indeed the D6 branes share all worldsheet coordinates with theD8 branes, hence there are tachyonic modes in the spectrum of D8-D6 open strings and theD6 brane melts in the D8 worldvolume yielding a flux of the flavor gauge field. Since theD6 branes get delocalized they do not provide string-like localization of the chiral symmetrybreaking.Another interesting example concerns the D4 instanton in holographic QCD extendedalong x or radial coordinate. To estimate its contribution into the partition function remindthat there is CS term on the D8 brane worldvolume multiplied by N c . Since U-shaped D46instanton shares all coordinates with the D8 branes it induces a nontrivial contribution intothe CS part of the action which reads as follows δS CS = N c Z duA u Z d T rF ∧ F (14)where we assume that U (1) A field A u is space-time independent. It can be interpreted asthe constant mode of η ′ meson since in the SS model it is identified as R duA u ( u, x ). Notethat that the constant mode of η ′ mode can be thought of as the effective θ term hence thetotal contribution reads as θ eff N c k where k- is the instanton number in the flavor gaugetheory. Such U-shaped D4 instanton is well defined at any temperature since it does notwrap around any compact coordinate and one could speak about the point-like contributionto the chiral symmetry breaking.One can also discuss fractional flavor D4 instantons. To this aim the flavor branes haveto be placed at different positions at the dual temporal circle. This can be achieved byswitching on chemical potentials which correspond to nontrivial temporal holonomies ofthe flavor gauge fields. The eigenvalues of the flavor holonomy around the temporal circleprovide the positions of the D7 branes on the dual circle. Hence the worldline of the D5brane on the dual circle can split and we get N f fragments of D5 brane stretched between N f D7 branes which carry the flavor ”magnetic” and fractional instanton charges. The systemcan be described in QFT terms as a set of N f ”monopoles” with the total instanton charge Q inst = 1 and the total monopole charge zero, similar to the canonical caloron with nontrivialtemporal holonomies, see for reviews [33, 34] and references therein. Since we are workingin the probe approximation N f << N c the action on each ”monopole” is proportional to N C and therefore these defects are suppressed at large N c .Another possible phenomenon concerns a peculiar manifestation of the Myers effect inthe holographic QCD. It is known that the following configuration: two parallel D4 branes+ D0 branes localized on the D4 branes + D0-D0 open strings, is unstable and decays intothe so-called dyonic instanton [36]. That is this configuration of instantons decays into thecircular D2 brane stretched between parallel D4 branes with electric and topological chargesas well as angular momentum which stabilizes the system. On the D4 worldvolume one getsa magnetically charged ring.In our context, we can consider the set of D8 and D4 branes representing the baryons inthe space-time [ ? ]. We assume that the worldvolume of the D4 brane is S × τ , that is7both type of branes are localized at the x coordinate. We could assume that the D8 branesare generically localized at different values x k where k + 1 , . . . N f . and the ”baryonic” D4branes are localized on the D8 branes. Consider two baryonic D4 branes and connect themby an open string which carries in the spectrum of excitations the gauge boson of the flavorgroup- ρ meson. Similar to the D4-D0-F1 case we expect that the D4 brane is expandedinto a D6 brane with baryonic density and the F1 string is melted into the isotopic chargeamounted from the initial ρ -meson. Hence finally we could expect that the configurationwith the baryonic charge larger than one gets delocalized into the ”baryonic ring” carryingthe isotopic charge. Note that it is known in the Skyrme model that the B=2 state has atorus like ground-state geometry [37], in agreement with our arguments above. Note thatsuch a configuration does not involve wrapping around x , that is it is well defined at T < T c . VI. CONCLUSIONS
In this note we demonstrated that the holographic description of the pure YM or QCD-liketheories implies existence, at least classically of a plenty of defects of different codimensions.We have tried to argue that their existence is insensitive to details of the metric. However theanalysis is certainly oversimplified and we have not aimed at deriving the defects propertiesin detail. We have seen that various types of strings and domain walls and some more exoticcomposite objects are emerging.A few claims seem to be quite generic. If the defect tends to condense (have small tensionvanishing classically) at small temperatures it becomes tensionful and well defined above thephase transition. On the other hand, all the defects apart from the S-branes tend to looseone Euclidean dimension above the critical temperature. The θ dependence of the defect’sworldvolume actions is present in the ”condensing brane” below the phase transition and canbe defined on such defects above the transition point as well on a single defect. This is ananalogue of the instanton solution of the U(1) problem via the Witten-Veneziano mechanismbelow the transition point and via the t’Hooft mechanism above this point. Our analysisimplies that a similar change of mechanisms happens for defects of different dimensions aswell.The theory above the phase transition almost immediately becomes three-dimensionalbecause of the cigar-type instability. This fits well with the lattice studies.8In particular, in holographic description we identified a few very interesting objects suchas heavy buckyballs (D4 particles) whose masses scale as ∼ N c (see section IIIC) or D4instantons whose action scales as ∼ N c . While such objects have been discussed previouslyin the literature within QFT, their future study using the holographic description mayshed a new light on their nature. Another interesting example deserves to be mentioned isthe holographic description of the objects with fractional topological and magnetic charges(section IV). Such objects have been discussed within QFT in late 70s. Future study ofthese objects may provide with a key to understand the QCD vacuum structure and thenature of the phase transition.There are many questions we have only touched upon. In particular, it would be highly in-teresting to understand better the role of the second order parameter ”dual” to the Polyakovloop and the duality between the corresponding pairs of domain walls which gets inter-changed at the phase transition. Our analysis suggests that it is probably reasonable todiscuss the chiral symmetry breaking in pure YM theory induced by U-shaped defects. Atfirst glance it might look strange, but ” chiral symmetry” of the gauge boson can be definedsimilar to fermions. The order parameter for such a ”chiral symmetry breaking” could besimilar to the one recently discussed in [38]. We did not discuss the defects involving NS5branes since it is more natural to discuss such issue upon lifting of IIA setup to the M-theory.In that case the corresponding defects involve M2, M5, KK particles and KK monopoles. Acknowledgments
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