###### Abstract

D4-D8 and D3-D7 systems are studied and a possible holographic dual of large N QCD (SU(N) gauge fields and fundamental quarks) is sought. A candidate system is found, for which however no explicit solution is available. Susy is broken by having a condensing to a D5. The mechanism for supersymmetry breaking is then used to try to construct a Standard Model embedding. One can either obtain too few low energy fields or too many. The construction requires TeV scale string theory.

hep-th/0305069

BROWN-HET-1356

On Dp-Dp+4 systems, QCD dual and phenomenology

Horatiu Nastase

Brown University

Providence, RI, 02912, USA

## 1 Introduction

One of the reasons why people are interested in gravity-field theory dualities is the hope that we can describe QCD via gravity. The original paper [1] treated the duality between =4 SU(N) SYM at large N and string theory, followed shortly thereafter by the paper of Witten [2] describing how to get a description of the pure glue theory (SU(N) Yang-Mills) via gravity. Other developments afterwards involve breaking supersymmetry e.g. [3] and/or conformal invariance e.g. [4, 5, 6, 7], and also introducing bifundamental fields via D7 branes or D5 branes e.g. [8, 9, 10], but the question of having a theory without conformal invariance and supersymmetry and with dynamical massless quarks has eluded attempts to solve it. For a review of the developments until 1999 see [11]. In this paper I will address this question, and find that while the decoupled D3-D7 theory describes a supersymmetric version of QCD, there is a modification of this system which describes the (large N) QCD, but unfortunately it is not possible to write down explicitly (not even implicitly).

The observation which allows us to do this is that one can write down the solution for a system, and that if this system condenses to a D6, one can write that solution down as well. I use the embedding of massive 10d IIA theory in M theory defined in [12] and extended in [13] to give a UV definition of the massive IIA solutions. It is then found that one has to T dualize and go to a D3-D7 system, smeared over an overall transverse coordinate. The holographic dual to QCD is then a system of branes, which however doesn’t have a known supergravity solution. The decoupled solution can be written explicitly (up to a one dimensional integral). The solution for the can be found up to an integro-differential equation for a function , but for the even the ansatz cannot be written.

An obvious question then is can one lift this D brane construction for the holographic dual of QCD to a Standard Model embedding? I study this question in the context of D-brane-world GUT models and find that one needs to have TeV-scale string theory. In the context of an SU(5) susy GUT we can obtain massles states corresponding to the 5 fermions, Higsses and gauge fields, but no fermions (which contain the fields in the of ). By adding orientifolds, one is able to obtain the required fields, but much more on top of that, and the corresponding masses seem to be wrong anyway. In any case, that system is very hard to analyze.

The paper is organized as follows. In section 2 I study D4-D8 solutions, in 2.1. previous solutions and in 2.2 solutions which will be used thereafter. In section 3 I try to define the D4-D8 system and its M theory embedding using the construction in [12, 13]. Section 4 is devoted to motivating the supergravity- field theory correspondence for the D4-D8 system and identifying the gauged supergravity describing it. In section 5.1 I describe the proposed set-up for the holographic dual of QCD, analyze the brane-antibrane condensation process and then in section 5.2 study the susy breaking using the system. In section 6 I try to embed the QCD description into a Standard Model description using D-brane worldvolumes. After a study of supersymmetric lagrangians for D3-D7-O(7) systems in 6.1, I try to build a model. In 6.2 systems without orientifolds are studied and section 6.3 introduces orientifolds. I finish in section 7 with discussion and conclusions. The appendix reviews GUTs for our purposes.

## 2 D4-D8 systems and QCD

From the perspective of a braneworld scenario or holographic duality, if one wants to realize a QCD-like system one has to introduce fundamental quarks, and the most obvious way is to look at Dp-Dp+4 systems. Since it is not clear how a D5-D9 system would be useful for either holography or braneworld phenomenology, we are left with D3-D7 systems and D4-D8. Let’s start out with D4-D8 systems for their simplicity and then notice that one needs to go back to D3-D7, but keep the advantages of the D4-D8.

The D4 brane theory dimensionally reduced to 4d is =4 SYM, which has fermions in the adjoint representationof . That theory is conformal, however in 5d the D4 theory has a dimensionful coupling constant and is getting strongly coupled in the UV, therefore the theory is not well defined. We will get back later to the question of defining the theory. By introducing D8s we have quarks and scalars in the bifundamental representation , forming an =2 hypermultiplet. In the holographic context, since

(1) |

in the decoupling limit , keeping fixed means , so we are left with fundamental hypermultiplets.

The D8 has the string metric and dilaton ( is the -dimensional Minkowski metric)

(2) |

where is an arbitrary constant of integration or (by the usual rescaling for p-branes)

(3) |

where is defined as the coupling constant at the position of the D8 brane. Here , so the mass is piecewise constant, and jumps at the positions of the D8 branes. The in the mass corresponds to D8 branes vs. anti-D8 branes. One can obviously redefine x such that the harmonic function appears just as a conformal factor for flat space.

### 2.1 D4-D8 in the literature

The solution for a D4-D8 system can be written as

(4) |

and here also corresponds to the D8 vs. anti-D8. In the literature, people have considered a “D4 inside D8” solution [14] and a “partially localized D4-D8 system [15]. The “D4 inside D8” is just a D4 delocalized over the transverse coordinates to the D8 and is given by (in Einstein frame)

(5) |

Here is the density on the unit of z direction. It was obtained by lifting the D4 in 9d solution via KK reduction on the D8 domain wall (that paper introduced the notion of dimensional reduction on a domain wall; by contrast one can always dimensioanlly reduce on a coordinate paralel to the domain wall), with ansatz

(6) |

The partially localized solution of Youm [15]reads

(7) |

and corresponds to the case

(8) |

which is a form where now (compact coordinate, in , since ) can be interpreted as being the transverse coordinate to the D8 due to the lucky coincidence that there is no r dependence in the transverse metric. Note that just for the D8 solution it would not be true, but since we have a D4 harmonic function with the correct r dependence, the total r dependence cancels. Because of that cancellation, (8) can be interpreted as the metric in the presence of an O(8) (and at infinite coupling; presumably there is a finite coupling O(8) one could add to the D4-D8 system, and it should limit to this). One needs to add O(8) planes at the fixed planes to cancel the brane charge (16 branes to cancel -16 units of charge from two O8’s). The authors of [16] found that the corresponding dual - theory is a fixed point with global global symmetry, derived from a gauge theory theory at infinite bare coupling. So, although the D4 theory is not conformal, by adding the D8 and O(8), the theory flows to a nontrivial conformal fixed point.

### 2.2 D4-D8 solutions

However, finding a localized solution is not so difficult after all. The point is that there are so called partially localized intersections, where brane 1 with harmonic function lives on , and brane 2 with harmonic function lives on , with overall transverse space . They are written in terms of harmonic functions and in the usual way, except that now and satisfy the equations (e.g. [15], [17])

(9) |

In other words, we delocalize one brane (say brane 2) over the worldvolume coordinates of the other brane (1), and then is harmonic (obeys the laplace equation) in the background of brane 2. In the case of a Dp-Dp+4 system, this condition is automatically satisfied, and then obeys the equation

(10) |

where , with some numerical constant, and then

(11) |

If we put , then the resulting equation

(12) |

is solved by

(13) |

The constant is fixed by matching with the normalization of the function source, and one gets

(14) |

and so

(15) |

with

(16) |

Let us now define the decoupling limit. As I mentioned, we want to keep the D4 SYM coupling fixed, , so that

(17) |

(I have rescaled as usual and so and the number of D8’s is . Then by rescaling also and the integration variable , we get in the limit

(18) |

and the decoupled D4-D8 system is

(19) |

A similar analysis for the case of the D2-D6 system was done in [18] and for a D1-NS5 system it was done in [13]. Notice that the near core D8 can be always trusted, independent of the number of D8 branes (the curvature in string units is always small, curvature scalar , so one needs actually , which can be satisfied for ). The number of D4’s however, has to be very large, as usual.

Let us now look at various ways of breaking supersymmetry. The most used is the method of Witten [19], of putting the system at finite temperature. This corresponds to compactifying on a supersymmetry breaking circle. The fermions aquire masses of the order of the compactification scale at tree level, and the scalars at quantum level, by fermion loops.

For the holographic dual, Witten’s solution had the AdS black hole as a starting point. Then scale the mass M to infinity, together with r to infinity and t to 0 in a particular way. The resulting solution has only one parameter (the radius of AdS).

But equivalently [20] one can just take the near horizon limit of the nonextremal solution. Although this solution has apparently two parameters (the AdS radius and the nonextremality parameter M, or temperature T), calculations in this background will not depend on T alone. Indeed there exists a rescaling (with no parameters going to infinity!) which takes the D3 nonextremal near-horizon solution to Witten’s metric, namely (R=AdS radius). In particular, for the Wilson loop [21] calculation of potential in [22], this means that E(L,R,T)=E’(LT,R) (TR) or equivalently EL=f(LT,R). The bottom line is that when one computes either potential or glueball masses, one can use either Witten’s type of construction, or a near horizon nonextremal solution, in which case by scaling of the coordinates one gets the desired nonsusy theory. Both ways were used in glueball calculations [23, 24, 25, 26, 27], but when one starts with a nonconformal theory before the compactification, there is no analog of the AdS black hole solution (since there is no AdS background), so one has to use the scaling in the nonextremal solution. A similar case, of glueballs in the N=1 nonconformal cascade theory of Klebanov and Strassler [5], was treated in [28].

So let us try to put the D4-D8 system at finite T by making it nonextremal. It is easy to do so for the “D4 inside D8” solution. Just lift the solution for D4 in 9d at finite temperature on the D8 with the ansatz (6) and get

(20) |

where

(21) |

But this is again the delocalization over z of the full nonextremal D4-D8 solution, which however now is hard to find.

If one could find the finite temperature localized D4-D8 solution, it would still not be so useful, since the corresponding field theory will be the same as for pure D4 branes at finite temperature: pure 4d Yang-Mills theory. Maybe though by comparing the two descriptions one would be able to find out the spurious effects of the construction (by seeing if there are quantities which do change).

We want however to keep the fundamental fermions after susy breaking. When compactifying the D4-D8 field theory, we would like therefore to put antiperiodic boundary conditions for the (4,4) fermions, so that they become massive, and periodic boundary conditions for the (4,8) fermions, so that they remain massless. One would have to check whether such boundary conditions are consistent with the interactions, and whether unitarity is preserved in such a system.

But let’s see whether one can find a holographic dual to such a system. By the general argument in [29], if one dimensionally reduces on a euclidian black hole spacetime, all the fermions will be antiperiodic around the KK coordinate, so they will get a mass. The argument is that there is only one spin structure available to the spinors around the KK coordinate. In general, the possible phases around it are dictated by the invariances of the lagrangian. At large distances from the black hole, the space is topologically flat (times the KK circle), so that all phases are allowed. Near the horizon however, the spacetime is flat space times the transverse sphere and admits a unique spin structure, which becomes the antiperiodic one at infinity. The same argument can be extended to nonextremal branes, for instance nonextremal branes, as in [19]. The spacetime is flat at infinity and has a transverse sphere near the horizon. Only the antiperiodic spin structure is valid over the whole spacetime.

So in the case of the euclidian black hole or nonextremal D4, fermions defined over the whole space become massive. They couple to fermionic operators on the D4. Therefore the D4 fermions are antiperiodic and become massive.

But we also see a way out. If there are fermions which are defined only over a part of the holographically dual spacetime, they can be mapped to fermionic operators remaining in the spectrum, whereas the ones defined over the whole spacetime are mapped to operators dissappearing from the spectrum.

In the context of compactification, there doesn’t seem to be a solution of this type, but there is a domain wall type solution (“alternative to compactification”) which has the necessary properties.

The nice thing about D8 branes is that one can also write down a solution (unlike for other branes), in the particular case where in between the two branes we have flat space. This can be achieved by writing

(22) | |||||

If one would have for it would be a D8-D8, but now it is a , and if the D8 can be trusted -curvature small in string units means

(23) |

and small implies no quantum corrections- the can be trusted as well. All we did was change the sign of the mass on one side, which changes the sign in the Killing spinor equation, therefore the Killing spinor on one side is not valid on the other. So there is no globally defined fermion in this background.

One can still write down a localized D4 inside the D8, even in the presence of the . Piecewise, the equation is still (12), with c and m derived from (22), and then we just have to match the solutions over the branes.

In the case , we have a on top of each other. From the string theory point of view, that gives a gravitational solution, but can also (depending on the K theory class of the system) give a lower dimensional brane, e.g. a D6. From the gravity point of view, the holographic dual is the same as for the D4-D8, just that now on both sides of the brane.

I will postpone the discussion of a specific set-up for later, but let us note that whereas there is no globally defined fermion, there are fermions defined on the D8 (and at x=0 we still have supersymmetry)-or rather on the z=0 slice of the holographic dual (19), corresponding to the D8.

So in the case of the solution, fermions defined over the whole spacetime couple to field theory operators which will dissappear from the spectrum. These will be operators with no D8 charges. On the other hand, fermions defined only on the D8 will couple to fermionic operators in the field theory which remain in the spectrum. These are operators charged under the D8 global symmetry.

Note that the fact that there are fermions which are defined only on a subset of the holographic dual is not a new concept. The =2 superconformal theory of D3-D7-O(7) described in [30] has bulk modes coupling to operators with no charges and vector modes defined on an orientifold fixed plane inside coupling, to operators with (vector) charges. It is in fact very similar to this system: without the O(7) and after a T duality it becomes the D4-D8, so the coupling of the operators with charges to vector modes defined on D8 (or rather the z=0 slice of (19)) is an established fact. The only new observation is that therefore the uncharged fermionic operators dissappear from the spectrum (get high anomalous dimensions), whereas the charged fermionic operators don’t.

## 3 Defining the D4-D8 system

As I mentioned, the D4 field theory (and the D8 field theory actually, but that is now “frozen”) is not well defined in the UV, so one must allow for a UV completion. In the UV, the effective D4 coupling is large, and the theory must be described by M theory, therefore the UV completion of the (nonrenormalizable) D4 theory is given by the M5 brane field theory. But what about the D4-D8 case?

Let us start with seeing how to embed the D8 in M theory. There have been attempts to embed the D8 directly in M theory, as an “M9” domain wall. One of these is solution in [31], where the 11th direction is an isometry direction for the metric. But the D8 is a solution to Romans’ massive supergravity [32], and a fully covariant M9 would be also a solution to an 11d supergravity with a mass parameter (cosmological constant).

It is unkown how to lift the massive IIA theory and its D8 background solution directly into M theory. The point is that M theory doesn’t seem to admit a cosmological constant, and if one wanted to lift massive 10d supergravity directly, one would get a cosmological constant in 11d. The only possible way around this is if the 10d mass arises via a Scherk-Schwarz generalized reduction on a circle. But for that, one would need a global symmetry in 11d, and the action doesn’t have such a symmetry. The equations of motion have however a scaling symmetry, which was exploited in [33] to reduce to a massive 10d sugra. However, that is a different massive supergravity in 10d (one that admits, in particular, the de Sitter space as a background), and moreover, it amounts in 11d to a compactification on the euclidian radial direction. That massive supergravity can also be obtained as a usual reduction of a modified M theory, as in [34].

Instead, the most conservative embedding in M theory was done by Hull [12], who was able to embed the massive supergravity and the D8 background in M theory by introducing two extra T dualities, one of which was a “massive T duality” as defined in [35]. M theory on a of zero area is type IIB and IIB can be compactified on via Scherk-Schwarz. After a “massive T duality,” we get massive IIA.

The endpoint is that massive IIA supergravity is equivalent to M theory on the space B(A,R), in the limit , with metric

(24) |

and all the radii going to zero, and the x’s have periodicity 1, . In the limit, we should keep the massive IIA quantities fixed, so

(25) |

So how does the Hull duality help us in defining the D8 and the D4-D8 field theories? The correct description of the D8 brane field theory is not clear, but the D8 goes over to a gravitational background in the (Hull) dual M theory, so the field theory on that soliton in M theory will be the correct description.

In the D4-D8 case, the D8 field theory decouples, but one is left with (4,8) fields, which can’t be lifted to usual M theory, so one needs to look for the Hull dual. The field theory description is given by the lift of the D4-D8 system to M theory. One is T dualizing twice to get to M theory, so it matters where are located the dualized coordinates. There are 3 choices: T dualize along two transverse coordinates, two paralel coordinates, or along one each. The latter situation is the most useful, since then after two T dualities, one is still describing a D4 brane, albeit with two small radii. The D8 background has now become a 6-brane smeared over two transverse directions. Then the lift to M theory gives as usual a M5 brane, in the background dual to the D8 (i.e. a 7+1-dimensional worldvolume, described in (33)). The decoupling of the D8 field theory corresponds to decoupling of the degrees of freedom localized at the M theory 7+1d background, but one still has degrees of freedom coming from M2’s stretched between the M5 and the “M7” (corresponding to (4,8) strings). Unfortunately, it is unclear how to describe these degrees of freedom, but at least it is possible in principle. In the next subsection I will analyze in more detail the embedding in M theory via Hull duality.

### 3.1 Embedding in M theory

Let us now derive the embedding of supergravity solutions of massive IIA into 11d supergravity solutions (if the solutions are BPS, the embeddings are still valid, even if the space is singular, and one can’t use quantum perturbation theory).

Dimensionally reducing M theory to massless IIA on one has

(26) |

Now one has to perform a T duality on to get to IIB and then a massive T duality on to get to massive IIA.

The full set of T duality rules giving the (hatted) massless IIA fields in terms of the IIB ones are

(27) |

The inverse T duality rules, this time with the added complication of them being massive, are (this time the hatted quantities are IIB and unhatted massive IIA)

(28) |

Applying the above T duality rules going from IIA to IIB on and then to massive IIA on one gets for the 11d metric (keeping only the fields relevant for our discussion)

(29) | |||||

and the corresponding massive IIA metric is then

(30) |

while the dilaton is ( is the massive IIA dilaton, is the IIB dilaton and the massless IIA dilaton)

(31) |

When one applies this prescription to the D8 solution of massive type IIA

(32) |

one indeed finds the 11d gravitational metric

(33) |

with

(34) |

Hull [12] has also found this solution as the correct 11d gravitational background corresponding to the D8 background. We should note here that the massive IIA sugra does not admit flat space as a solution, the background with maximal supersymmetry is the D8.

Now when one lifts a solution of massive IIA to M theory, it matters where one chooses to make the two T dualities, i.e. where one puts and . The best choice is of course to arrange and such as to get the same type of solution after the two T dualities.

In the particular case of the D4-D8 solution, the best choice is to have one direction paralell to the D4, one perpendicular. Then after two T dualities, one still has the D4 solution, and it will lift to an M5.

Let us however first treat the case where both T dualities are paralel to the D4. We will reach a D2 which lifts to an M2 in the gravitational background. Indeed, one gets

(35) |

which implies

(36) | |||||

Restricting the D4-D8 to the “D4 inside D8” solution corresponds as before just to dropping the x dependence of .

When one T duality is paralel and one perpendicular, the same solution as above, but with a transverse coordinate, lifts to

(37) | |||||

which corresponds to an M5 in the gravitational background (33). Note that in both cases there is also a nontrivial field.

In [13], a procedure was developped for getting a Matrix model [36, 37, 38] corresponding to the massive IIA supergravity, and was applied to the D8 background. I will apply it now to the D4-D8 system.

Since massive 10d IIA string theory is equivalent to M theory on the singular background (33), one defines Matrix theory in that background and compactifies it. After T dualities in all the , one gets a Matrix model of D3 branes. As an intermediate step necessary to decouple gravity from the D3 brane theory, following Sen [39] and Seiberg [40], an theory was introduced, such that

(38) |

are held fixed in the limit, and the limit is imposed afterwards.

The metric (24) for B(A,R) is invariant under the isometries (we have put for simplicity)

(39) |

with Killing vectors , and . One also notes that . Since and don’t commute, it matters in which order one makes the T dualities. We choose to do , then , then .

After one has :

(40) |

After one has

(41) |

is generated by the vector . Making this so that one can apply the T duality rules means the coordinate transformation

(42) |

The metric in the new coordinates is (after dropping primes on coordinates)

(43) |

After the third -duality, we have

(44) |

So let us apply this procedure for the D4-D8 solution. One goes to an theory to decouple string theory, compactifies on a lightcone coordinate, and then T dualizes on all 3 ’s.

We will drop the bars from all quantities (in the end, nothing will depend on the theory anyway). Let’s start with the background corresponding to an M2. The IIA metric after dimensional reduction on the lightlike coordinate will be (we choose that coordinate to be perpendicular to the M2, thus getting a D2 brane)

(45) |

After the T duality on it will become a D3 brane ending on a NS5 brane in the direction. The metric is