Shamit Kachru, John Pearson and Herman Verlinde- Brane/Flux Annihilation and the String Dual of a Non-Supersymmetric Field Theory

8/3/2019 Shamit Kachru, John Pearson and Herman Verlinde- Brane/Flux Annihilation and the String Dual of a Non-Supersym

1/29

SLAC-PUB-9083

Brane/Flux Annihilation and the String Dualof a Non-Supersymmetric Field Theory

Work supported by Department of Energy contract DEAC0376SF00515.

Shamit Kachru, John Pearson and Herman Verlinde

Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309

January 2002SU-ITP-01/50

PUPT-2019hep-th/0112197


2/29

a r X i v : h e p - t

h / 0 1 1 2 1 9 7 v

3 2 6 D e c

2 0 0 1

SU-ITP-01/50, SLAC-PUB-9083PUPT-2019

hep-th/0112197

Brane/Flux Annihilation and the String Dual

of a Non-Supersymmetric Field Theory

Shamit Kachru 2, John Pearson 1 and Herman Verlinde 1

1 Department of Physics, Princeton University, Princeton, NJ 08544

2 Department of Physics and SLAC, Stanford University, Stanford, CA 94305/94309

Abstract

We consider the dynamics of p anti-D3 branes inside the Klebanov-Strassler geometry, thedeformed conifold with M units of RR 3-form ux around the S 3. We nd that for p


3/29

1 Introduction

The search for calculable and nontrivial nonsupersymmetric string vacua remains a problemof basic interest in string theory. In the context of AdS/CFT duality [ 1], this is related to

the construction of gravitational anti-holographic descriptions for nonsupersymmetric 4dquantum gauge theories. Here, we describe the construction of a simple nonsupersymmetricstring background which has both a holographic description in terms of 4d gauge theory andrealization in the context of warped string compactications. The model possesses severalattractive features, among them the existence of certain discrete parameters which allow itto be rendered classically stable and arbitrarily long-lived and the ability to set the regimeof nonsupersymmetric dynamics at energies exponentially far below the Planck scale.

One of the recent advances in the anti-holographic description of gauge theories is thediscovery of several smooth gravitational duals for N = 1 supersymmetric pure Yang-Millstheory [2, 3, 4]. One such dual, as discussed by Klebanov and Strassler (KS), starts with the

theory of N D3 branes probing M fractional wrapped D5 branes in a conifold geometry.In the nal geometry, the branes have disappeared; they are replaced by RR and NS three-form uxes through two intersecting three-cycles in the deformed conifold [ 2]. The crosseduxes carry N units of F 5 -form charge, in keeping with the starting point. The holographicdescription of the geometry involves a 4d SU (N + M ) SU (N ) gauge theory undergoingan RG cascade, in which repeated Seiberg duality transformations relate weakly-coupleddescriptions of the physics. In the case N = KM for some integral K , the IR eld theoryat the end of the cascade is a pure N = 1 supersymmetric SU (M ) Yang-Mills theory. TheKS model can be embedded into a compact geometry [ 5] and used to realize warping as ameans of generating large hierarchies as in the Randall-Sundrum scenario [ 6] along the lines

advocated in [ 7] (for other discussions along these lines, see e.g. [8]). Roughly speaking, thepure Yang-Mills theory becomes the theory on the infrared brane in the RS description,while the string compactication manifold provides an effective Planck brane.

In our model, along with the conifold geometry with crossed three-form uxes, we in-troduce an additional ingredient which breaks the supersymmetry of the solution: We placesome number p


4/29

our basic physical expectations. We also discuss the two different vacuum branches of thesupersymmetric KS system, since this structure will be of importance to us later. Then,in sections 3 and 4, we study the basic dynamics of the anti-D3 branes in the conifoldand ux background. We will do this from two perspectives. In 3, we will look at the uxinduced potential for anti-D3 branes and present evidence that it induces them to expandinto an NS 5-brane giant graviton conguration. In 4 we will then examine this systemdirectly from the viewpoint of the NS 5-brane action with induced anti-D3 charge. Bycombining both perspectives and establishing a precise correspondence between them, weobtain a consistent picture where the anti-branes blow up into a vebrane wrapping an S 2

of some xed size on the small (A-cycle) S 3 inside the conifold. For small enough p/M ,this describes a metastable false vacuum. In 4.3 we present the analysis of the EuclideanNS 5 brane bubble wall which mediates the quantum decay of our theory to a nearbysupersymmetric vacuum with slightly different uxes and M p D3 branes. This is the truevacuum, and is one of the models discussed in [ 2]. In 5 we present our best guess for thecorrect holographic description of our non-supersymmetric theory in terms of a long-livedfalse vacuum in the KS eld theory. We also discuss the type IIA dual description. Finally,in section 6, we give a short discussion of the model when considered as a warped stringcompactication and describe some features of the 4d effective supergravity theory.

As an interesting aside, we note that our study also sheds light on the geometric transition[4] between string backgrounds with D3-branes sources and smooth KS-geometries with only3-form ux [2]. Contrary to expectation, we will see that this transition can be understoodin terms of the dynamics of the D3-branes themselves, which undergo via the formation of afuzzy NS 5-brane a continuous transmutation into pure ux. This transition takes placeinside supersymmetric domain walls interpolating between the two phases, the BPS domain

wall solution described in 4.2.2 Preliminaries

In this section, we review some of the relevant aspects of the KS geometry and its holo-graphically dual gauge theory description. We describe our non-supersymmetric model andsome of our physical expectations in 2.2.

2.1. The Klebanov-Strassler geometry from F-theory

In most of our investigation, we will consider the non-compact limit of the Klebanov-Strassler geometry. For purposes of presentation and to facilitate later discussions of ourmodel in the context of warped string compactications, we will start from the point of viewof F-theory with non-zero 3-form uxes H 3 and F 3 . Such models have been investigatedfrom an M-theory and F-theory perspective in, e.g.,[ 9, 10, 5, 11]. To simplify notation, weassume as in [5] that we are working in a limit where the F-theory compactication on the

2


5/29

fourfold X can be viewed as an orientifold of IIB string theory on a Calabi-Yau threefold Y .Besides the 3-form ux, the geometry will typically involve the insertion of N 3 D3-branesand/or N 3 anti-D3 branes. The net D3 charge

Q3 = N 3 N 3 (1)is xed via the global tadpole condition

(X )24

= Q3 +1

2210T 3 Y H 3 F 3 . (2)Here (X ) is the Euler characteristic of the CY fourfold that species the F-theory com-pactication, and T 3 is the D3-brane tension.

We are interested in expanding about a singular point in moduli space where Y hasdeveloped a conifold singularity. The KS solution corresponds to the placing of M units of F 3 ux through the A-cycle of the conifold and K units of H 3 through the dual B-cycle:

142 A F 3 = M ,1

42 B H 3 = K . (3)Assuming no other (crossed) uxes are present, we could in principle choose M and K such that 24 = MK , so that D3 charge conservation is satised without the need of extraD3-brane insertions. We can therefore expect to nd a smooth compactication geometry.Indeed, as discussed in detail in [ 5], these uxes give rise to a non-trivial superpotentialfor the complex structure modulus zA = A that controls the relative size of the A-cycle,stabilizing it at an exponentially small value

zAexp(2K/Mg s). (4)

As a result, the compactication geometry develops a smooth conical region, described bythe warped geometry of the deformed conifold

4

i=1

z2i = 2 , (5)

with zA exponentially small.The exponential hierarchy ( 4) has a natural explanation from the holographic perspective

as follows [2, 5]: The geometry (5) is dual to an SU (N + M ) SU (N ) gauge theory, whichhas a non-trivial -function proportional to gs M . As a result, its renormalization group owtowards the infrared involves a cascade of successive strong coupling transitions at successivescales n with log(n / n +1 )(gs M )

1 . Each strong coupling transition involves a Seiberg

3


6/29

A

K6

anti-D3sp

B

K6

B

D3sM-p

A

B

3H = K-1

V( )

V( )

H = K3

B

Fig 1: The decay, described in 2.2 and 4.3, takes place between an intitial non-supersymmetric situation with p D3 branes near the tip of the conifold, to a nal supersym-metric situation with M p D3-branes. The total D3-charge is preserved via the simultaneous jump in the H 3 ux around the B-cycle by one unit, from K to K 1.

duality transformation which lowers the rank of the larger of the two gauge groups by 2 M .Assuming that N = KM , the maximal number of such dualities is K , leaving in the IRa pure supersymmetric SU (M ) Yang-Mills theory. The total range of scales traversed in

this duality cascade is therefore log( 0/ K ) K/gs M , consistent with the supergravity

prediction ( 4).

2.2. Our Model and Physical Expectations

We choose to study the model of 2.1 in the case that

N 3 = 0 , N 3 = p,24

= KM p (6)

with p


7/29

decay back to a supersymmetric KS vacuum by undergoing brane/anti-brane annihilation.In our conguration, however, there is no obvious way for this to occur.

To gain more insight, let us look at our situation ( 6) from the dual eld theory perspec-tive. It is natural to identify the total D3 charge carried by the compactication manifold,

/ 24, with the total rank N of the SU (N + M ) SU (N ) dual gauge theory. Indeed, if wetook N = KM + p, with p


8/29

up to a net attractive force ( 8). Therefore, even if we began with a more generic initialdistribution, we expect the anti-D3 branes to accumulate quickly near the apex at r = 0.

The metric near the apex reads [ 12] (we work in string units, = 1)

ds2 = a20 dxdx + gs M b2012dr 2 + d23 + r 2d22

a204/ 3

gs M b20 0.93266. (9)

Since we assume that all physics takes place at r = 0, our space-time has the topologyR 4 S 3. The RR eld F 3 = dC 2 has a quantized ux around the S 3

S 3 F 3 = 42M, (10)while in the supersymmetric background dictates 6H 3 = gsF 3, so that

dB6 =1g2s

10 H 3 = 1gs

dV 4F 3 , (11)

with dV 4 = a40 d4x. The dilaton eld is constant, and the self-dual 5-form eld vanishes atthe tip.

2.3. Branches of Moduli Space in the KS system

Here we review one of the relevant features of the KS low energy eld theory, followingsection 7 of [2]. This will be important in understanding our proposal for the holographiceld theory description of the metastable false vacuum, as well as of the tunneling processthat describes its quantum decay.

Consider the RG cascade of the KS gauge theory with N = KM in its penultimate step.In this case, the unbroken gauge group is SU (2M ) SU (M ), and the SU (2M ) factor hasan equal number of avors and colors. There is also a quartic superpotential whose roughform is

W = (AiB j AkB l)ik jl

(12)where the Ai are in the (2 M, M ) representation, the B i are in the conjugate representation,and i = 1 , 2. We know from the analysis of [13] that in N = 1 supersymmetric QCD withN f = N c, the moduli space is quantum mechanically modied. Treating the SU (M ) as aglobal symmetry (i.e. taking its dynamical scale to vanish), this situation reduces to theone studied in [13] (with the added complication of the quartic tree-level superpotential).

6


9/29

Dene the meson elds N ij, and the baryon elds B, BN ij, = Ai, B j, , B= ( A1)M (A2)M , B= ( B1)M (B2)M , (13)

where and are SU (M ) avor indices. In order to reproduce the quantum modiedmoduli space of the SU (2M ) theory, we should add a Lagrange multiplier term to ( 12)

W = N ij, N klik jl + X (det( N ) BB 4M ) , (14)

where the determinant is understood to be that of a 2 M 2M matrix (coming from the i, jand color indices on N ).

There are distinct mesonic and baryonic branches of supersymmetric vacua arisingfrom the superpotential ( 14). On the baryonic branch

X = N = 0 , B= B= i2M . (15)

On this branch, the SU (M ) factor in the gauge group remains unbroken, and one is leftwith pure N = 1 gauge theory in the IR. However, there is also a branch where B= B= 0and the mesons have non-vanishing VEVs.

det( N ) = 4M , B= B= 0 . (16)

Now the gauge group is generically Higgsed, and there is a moduli space of vacua consistingof the theory of M D3 branes probing a deformed conifold geometry.The fact that the mesonic and baryonic branches of moduli space are disconnected is a

result of the tree level superpotential ( 12). In the theory without ( 12), the quantum modulispace is dened by the equation det( N ) BB= 4M and one can smoothly interpolatebetween these branches, via a continuous path of supersymmetric vacua. In the presentcase, on the other hand, eld congurations that interpolate between the two branches takethe form of localized supersymmetric domain walls with non-zero energy density. Thesedomain walls solve BPS equations of the form

zI = gIJ J W () (17)

with I = {N ij , B, X }, and z the coordinate transverse to the domain wall. The walltension is proportional to

| W | = |W (b) W (m )|, (18)

7


10/29

where b and m denote the respective vacuum values ( 15) and ( 16) of the two branches.By considering e.g. the special mesonic vacuum where all meson elds have the sameexpectation value (i.e. independent of their avor index), we deduce

| W

|= 2M 4 . (19)

In 4.2 we will consider the dual supergravity description of the domain wall. In thiscase it will represent the transition from the smooth deformed conifold with only ux (corre-sponding to the baryonic branch) to the situation with one less unit of NS 3-form ux and M D3-branes. In 5 we will use this dual understanding of the supersymmetric domain wall tomotivate a similar holographic description of our non-supersymmetric D3 background andits decay process. Here we will similarly argue that there are two relevant branches: Therst, the analogue of the baryonic branch, will be the (now nonsupersymmetric) metastablevacuum; and the second, the analogue of the mesonic branch, will be the theory with M pD3 branes probing the deformed conifold ( 7).

3. The D 3 brane PerspectiveWe are interested in the dynamics of p D3 branes sitting at the end of the KS throat,

under the inuence of the F 3 and H 3 uxes (3). We will do our analysis within the probeapproximation, taking the KS background as xed, while ignoring the backreaction due tothe D3 branes. The characteristic size of the geometry is set by R gs M , while wecan estimate that the backreaction due to the p anti-branes extends over a region of order

r gs p. Hence the distortion of deformed conifold due to presence of the D3 branesremains small as long as p


11/29

3.1. Dielectric D 3 Branes

The worldvolume action of the p anti-D3 branes, placed at the apex of the deformedconifold and in the S-dual frame, is given by the Born-Infeld action

S BI = 3

gs Tr det( G + 2 gs F ) det( Q) 23 Tr iiB6 (20)Here

Qi j = i j +2igs

[i, k ](Gkj + gs C kj ) . (21)

Because we are working in the S-dual frame, relative to the usual Born-Infeld action, wereplaced B 2 by C 2 and C 6 by B 6 . The two-form F here denotes the non-abelian eldstrength on the D3 brane worldvolume.

The scalar elds parametrize the transverse location X of the D3 branes, via therelation = 2 X . By making these matrix coordinates non-commutative, the anti D3-branes can collectively represent a 5-dimensional brane which can be identied with the NS5-brane. The topology of this fuzzy NS 5 brane is R 4 S 2, where the two-sphere S 2 hasan approximate radius R equal to

R242

pTr(( i)2). (22)

It is instructive to look for the non-commutative solution for p


12/29

and IASD uxes). This logic, or alternatively, direct calculation of the second term in ( 20),tells us that the full potential coming from ( 20) in this ISD ux background will be

gs V eff ()

det( G ) p

i42

3F kjl Tr [k , j ]l

2

g2sTr [i , j ]2 + . . . . (25)

As in [15], this potential has extrema away from the origin = 0. To get some intuition,let us set F kjl = f kjl , where we make the approximation that the A-cycle S 3 is large (whichis good in the limit of large gs M ). The magnitude of f can be read off from the normalizationof the integrated RR ux as in ( 3), which requires

f 2

b30 g3s M . (26)

Demanding that V () i = 0 yields the equation of motion

[[i , j ], j ] ig2s f ijk [ j , k] = 0 . (27)

To solve (27), notice that if one takes constant matrices i satisfying the commutationrelations

[i , j ] = ig2s f ijk k , (28)

then ( 27) is automatically satised. But, up to rescaling the , ( 28) are just the commuta-tion relations which are satised by a p p dimensional matrix representation of the SU (2)generators

[J i, J j ] = 2i ijk J k . (29)

So by setting i = 12 g2s f J i , with J i the generators of any p-dimensional SU (2) represen-tation, we nd solutions of ( 28).

Which solution is energetically preferred? Using the known value of the quadraticCasimir c2 = Tr(( J i)2) in each p-dimensional SU (2) representation, one can see that theenergetically preferred solution is to take the p-dimensional irreducible representation of SU (2), for which one nds

V eff 3gs

p2

6g8s f

4 p( p2 1) (30)3 pgs

1 82 ( p2 1)

3 b120 M 2. (31)

10


13/29

The negative term in ( 30) comes about through a competition between the (positive) quarticterm and the (negative) cubic term in ( 25). The other p-dimensional reducible representa-tions occur as metastable vacua of ( 25), where the D3 branes have blown up to a number of less giant NS 5 branes but can still satisfy V eff < p/g s . These separate 5-branes all want

to cluster together to form the most giant NS 5-brane, with minimal energy equal to ( 30).It is interesting to compare the energy in the giant graviton vacuum ( 30) to the nal

energy in the supersymmetric ground state, V = 0. We see that V eff > 0 implies that p


14/29

2F 2 = 2F 2 C 2 . (34)This action has been obtained by S-duality from that of the D5-brane. Here F 2 = dA isthe two-form eld strength of the world-volume gauge eld, G denotes the induced metricalong the internal S 2, and G encodes the remaining components along the d and R 4

directions. Using the explicit form ( 9) of the metric, we have

ds2induced = b20 gs M dxdx

+ d2 + sin 2 d22 (35)

= ds2 + ds2

where (relative to eqn ( 9)) we have absorbed the factor of a0/R 0 into x . (This means thatfrom now on, all time and distance scales in the R 4 directions are measured in red-shiftedstring units, or in holographic dual terminology, in terms of the dimensional transmutationscale of the low energy gauge theory.) We can evaluate the following integrals over S 2

S 2 det G = 4 b20 gs M sin2 , (36) S 2 C 2 () = 4 M 12 sin(2) , (37)

2 S 2 F 2 = 42 p . (38)This last equation reects the fact that the NS 5-brane carries D3 brane charge p. Combining(36)-(38) gives:

S 2 det( G+ 2 gsF ) = 4 2Mgs V 2() (39)

V 2() =1 b40 sin4 + pM + 12 sin(2) 2. (40)

Adding the 5 B 6 term, obtained from eqn ( 11), gives a total NS 5-brane actionS = d4x det G L() , (41)

L() = A0 V 2() 1 2 12 (2 sin2) , (42)with

A0 =425M

gs=

3M gs

. (43)

12


15/29

0.5 1 1.5 2 2.5 3

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.5 1 1.5 2 2.5 3

0.1

0.2

0.3

0.4

Fig 2: The effective potential V eff () for pM 3%, showing the stable false vacuum, and for pM 8%, with only a marginally stable minimum.

We can use this action to introduce a canonical momentum P conjugate to , and writethe resulting Hamiltonian density Has

H(, P ) = A02

2 sin2 + A20 V 2() 2 + P 2 (44)which generates the time evolution of (t). For the moment, however, we are just interestedin nding whether there exists a static solution corresponding to the giant graviton of 3.

Some useful intuition is obtained by considering the effective potential

V eff () H(, P = 0)= A0 V 2()

12

(2 sin2) (45)

13


16/29

In gure 2 we have plotted this effective potential V eff () for two different values of prelative to M : the top graph corresponds to p/M 3%, and shows a stable false vacuum,and the lower graph corresponds to the special case p/M 8% at which there is only amarginally stable intermediate minimum. For p/M > 8% the slope of the effective potential

is everywhere negative.In both cases we can draw an interesting conclusion. In the regime p/M > 8%, the

nonsupersymmetric conguration of p D3 branes relaxes to the supersymmetric minimumvia a classical process: The anti-branes cluster together to form the maximal size fuzzyNS 5-brane which rolls down the potential until it reduces to M p D3-branes locatedat the north-pole at = . In the regime with p/M (sufficiently) smaller than 8%, thebranes reach a meta-stable state, the fuzzy NS-5 brane located at the location min forwhich V eff (min ) = 0. This conguration is classically stable, but will decay via quantumtunneling. We will study this tunneling process in 4.3. In both cases, the end result of theprocess is M p D3-branes in place of p anti-D3-branes with the H 3 ux around the B-cyclechanged from K to K 1.Now let us check the correspondence with the non-abelian description of 3. For smallvalues of we can expand

V eff () A0p

M 4

33 +

b40M 22 p

4 (46)

which has a minimum at min = 2pb40 M equal to

V eff (min ) 3 pgs 1 82

p2

3 b120 M 2 , (47)

in exact agreement with the value ( 30) found earlier. Moreover, the size R min R0 (withR0 the S 3 radius) of the NS 5 brane giant graviton exactly matches with our earlier result(32).

Another quantitative conrmation of the result ( 45) for the effective potential is that thedifference in vacuum energy between the south and north-pole is equal to twice the tensionof the anti-D3 branes

V eff (0)

V eff () =

2 p3

gs. (48)

This is the expected result. One way to understand this [ 16] is to compare our nonsuper-symmetric model with a hypothetical situation with all p anti-D3 branes replaced by pD3-branes, i.e. branes with opposite charge and tension as D3-branes. This last situationwould preserve supersymmetry and would therefore have zero vacuum energy. To changeit back to our physical situation, however, one needs to add back p D3/anti-D3 pairs, with

14


17/29

zero charge but with total tension 2 p3 /g s . Notice, however, that in order to obtain thetrue total vacuum energy, we need to add to the result ( 45) a term

V tot () = V eff () +p3

gs, (49)

so that the supersymmetric vacuum indeed has V tot = 0. This extra contribution comesfrom a term 3gs / 24 H 3G3 in the string action which, via the global tadpole condition,adds up to p3 /g s .

Finally, let us return to the validity of our description. As mentioned, this is a slightlyproblematic question, since both the S-dual D3-brane variables and the NS 5 world-volumetheory are strongly coupled. It seems reasonable, however, that at least our main qualitativeconclusions, (i) the anti-D3 branes expand to a fuzzy NS 5 brane, ( ii ) for small enough p/M , the NS 5 brane will stabilize at some S 2-radius min proportional to p/M , and (iii )

for large enough p/M the anti-D3/NS 5 conguration will be classically unstable, will re-main unchanged in a more complete treatment. One forseeable quantitative difference, forexample, is that inclusion of the backreaction of the NS 5-brane on the S 3 geometry mighttrigger the classical instability for smaller values of p/M than found above.

4.2 BPS Domain Wall

As discussed in 2.3, one can consider supersymmetric domain walls that interpolate betweenthe supersymmetric mesonic and baryonic vacuum branches in the pure KS gaugetheory with p = 0. As we now show, it is possible to write a special BPS solution to the NS5 brane equations of motion that describes a supersymmetric domain wall between the twophases. Specically, we assume that the mesonic vacuum is such that all meson elds ( N ij )

have the same expectation value. In the supergravity, this is described by the congurationof M D3-branes located at the same point on the S 3, which we take to be the north-pole = .

Before we describe this domain wall solution, we note that, for general p, the total F 2 ux through the S 2 satises

2 S 2 F 2 = 4 p M ( 12 sin(2))= 4 (M p) M (

12

sin(2)) (50)

with = . In other words, we can think of the F background from the south-pole perspective as carrying p units of D3 charge, or from the north-pole perspective ascarrying M p units of D3-charge. Notice that this implies that there must be M p units of F 2 ux placed at the north-pole, that is, M p D3-branes. Hence, in the special case that

15


18/29

p = 0, the NS 5-brane at the north-pole represents M D3 branes, while at the south-pole itcan simply shrink from view without a trace.

The domain wall solution corresponds to an NS 5-brane conguration described by aspatial trajectory (z) (z the coordinate transverse to the wall) interpolating between =

and = 0. Following the same steps as above, we nd that the z-evolution of (z) isgoverned by the Hamilton equations z = H zP and z P = H z with

Hz(, P ) = A02

2 sin2 + A20 V 2() 2 P 2 (51)We look for a trajectory that, for large negative z, starts at rest at the north-pole, i.e., = and P = 0. Therefore, this solution has H= 0. Solving for P we nd

P = b20 sin2 . (52)

We thus obtain the following rst order equation for (z)

z = b20 sin

2 12 sin2

. (53)

This can be integrated to (choosing the location of the domain wall around z 0)

z =

cot

b20 . (54)

The right-hand side covers the half-space from z = to z = 1 /b 0, where the NS 5-branetrajectory has reached the south-pole at = 0. At this point the brane has disappeared,leaving behind the pure ux KS solution. In the dual gauge theory, this is the baryonicvacuum.

It is useful to think of the domain wall as an NS 5 brane wrapped around the S 3 A-cycleof the conifold, deformed near the north-pole due to the presence of the M D3-branes endingon it. From this perspective, it clearly has the property of inducing a jump by one unit inthe H 3 ux around the B-cycle. Indeed, if we consider two such B-cycles B(z+ ) and B (z )located at opposite sides of the domain wall, their difference B (z+ ) B (z ) represents an(otherwise contractible) 3-cycle that surrounds the NS 5-brane once. Since the NS 5-braneacts like a magnetic source for H 3 we have

B (z+ )H 3 B (z )H 3 = 4 2. (55)

16


19/29

The tension of the domain wall is obtained by evaluating the classical action of the abovesolution per unit time and area. We have

S =

dtd2x

det G3 T wall (56)

with

T wall = A0 b0 gs M

0d

b20 sin2

=5 22b30 (gs M )3/ 2

g2s(57)

As expected, this result for the domain wall tension breaks up as the product of the NS5-brane tension 5/g 2s times the volume of the S 3 (with radius R0 = b0gs M ) wrapped byit. From a holographic viewpoint, this should be compared with the formula ( 19) obtainedfrom the gauge theory effective superpotential.

We should mention that the probe approximation used here is no longer strictly validnear the north-pole = , since the M D3-branes represent an appreciable stress-energythat will have a non-negligible effect on the background geometry. Previous experience withsupersymmetric congurations of this kind [ 14], however, suggests that such backreactioneffects do not signicantly alter the results for quantities like the domain wall tension.

4.3 Vacuum Tunneling

We now turn to a description of the decay of the non-supersymmetric conguration with p non-zero. This takes place via nucleation of a bubble of supersymmetric vacuum ( 7)

surrounded by a spherical NS 5 domain wall which expands exponentially as a consequenceof the pressure produced by the drop in the vacuum energy. To obtain the nucleation rate, itis standard practice to look for a corresponding Euclidean solution. The relevant solution forus is an NS-5 brane trajectory (R), where R is the radial coordinate in R 4, connecting thegiant graviton conguration at = min at large R to an instantonic domain wall locatednear some appropriate radius R = R at which the solution reaches the supersymmetricminimum = .

The total action functional for such a trajectory reads

S = B0

R dR R 3 V 2()

1 + ( R )2 +

1

(

p

M +

1

2sin2) (58)

with

B0 = 22b403gsM 3. (59)

As before, it is convenient to write the classical equations of motion for this action in the

17


20/29

0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

0.5 1 1.5 2 2.5 3

0.5

1

1.5

2

2.5

3

Fig 3: The Euclidean NS 5-brane trajectory (R) for two values of p/M : the upper trajectory corresponds to pM 3%, and the lower one to the near critical value

pM 8%.

form of Hamilton equations R = H RP and R P = H R with

HR (, P ) = B0R3

22 sin2 + B 20 R6 V 2() 2 P 2 (60)

In gure 3 we have drawn the resulting Euclidean NS 5-brane trajectories (R) for twovalues of p/M .

In principle, we could extract the nucleation rate from the above formulas for general p/M by evaluating the total action of the (numerically obtained) classical solutions, thoughthis is a laborous prospect. Instead, we can quite easily obtain the leading order decayrate in the limit of small p/M as follows: From gure 3, the classical trajectory for small p naturally divides up into two separate regions: it stays at near the non-supersymmetric

18


21/29

minimum until coming very close to the domain wall radius R where it quickly movestoward the north-pole value = . On these grounds, we divide the total action into acontribution coming from the non-zero vacuum energy (given in ( 49) and (47)) of the non-supersymmetric giant graviton at R > R and a second contribution coming from the

tension of the domain wall. Using the thin wall approximation [ 17] we may writeS = T wall Vol3(R) V tot (min ) Vol4(R) (61)

where

Vol3(R) = 2 2b30(gs M )

3/ 2R3 Vol4(R) =12

2b40(gsM )2 R4. (62)

denote, respectively, the 3-volume of and the 4-volume inside a 3-sphere of radius R asmeasured by the metric ( 36).

It now seems reasonable to assume that, when p


22/29

5 Two Dual Perspectives

In this section we consider two dual perspectives on our model. In particular, we proposea dual holographic description in terms of a nonsupersymmetric, metastable vacuum in the

KS eld theory. Secondly, as an additional motivation for this proposal, we summarize howour model may be described from the type IIA point of view.

5.1 Holographic Dual Field Theory

For simplicity, we rst consider the case that p = 1, which we expect to be related to theSU (2M 1) SU (M 1) KS eld theory. In addition to the N = 1 gauge multiplets,this theory has bifundamental elds, two in the (2 M 1, M 1) representation and twoin the (2M 1, M 1) representation. So, from the perspective of the SU (2M 1), thereare N f = 2( M 1) avors, one less than the number of colors. Hence it is no longerpossible to write the color neutral baryonic combinations Band

B, and, as a result, the onlysupersymmetric vacuum of the system is the mesonic branch ( 16). This is the holographicdual of the stable vacuum with M 1 D3-branes, the situation we expect to land on after thedecay process in the case p = 1. The question now becomes, Where do we nd the other,nonsupersymmetric metastable vacuum corresponding to a single anti-D3 brane probing theconifold?

Though qualitatively important, for large M , the presence of the single anti-D3 braneacts only as a small perturbation of the situation with p = 0. (For example, the effectiveNS 5-brane potential V eff given in (45)-(40) has a perfectly smooth limit for pM 0.) Itseems reasonable to assume, then, that the nonsupersymmetric minimum for p = 1 can be

thought of as closely related to the baryonic vacuum of the supersymmetric theory with p = 0. Indeed, as we have argued in 4.2, the NS 5 domain wall that separates it from thetheory with M 1 D3-branes in the supersymmetric case represents a transition betweenthe baryonic and mesonic branch.

We will now try to use this intuition to obtain a description of the nonsupersymmetricminimum. We introduce, in spite of the fact that N f = N c1 in our case, the two baryonicsuperelds

Ba = ( A1)M 1(A2)M 1, Ba = ( B1)M 1(B2)M 1 , (66)which are no longer colorless but carry a color index a transforming in the fundamentalrepresentation of SU (2M 1). The idea is that the nonsupersymmetric theory correspondsto a false vacuum of the p = 1 KS gauge theory characterized by a non-zero expectationvalue of these color charged baryon elds. Naturally, this will cost energy, but it seemsa reasonable assumption that (for p/M very small) this nonsupersymmetric vacuum maynonetheless be classically stable because it is separated from the supersymmetric minimumvia a potential barrier.

20


23/29

To make this proposal more concrete, let us derive the form of the superpotential of ourmodel with Ba and Ba included. To this end, we start by adding to the SU (2M 1)SU (M 1) gauge theory a single pair of scalar multiplets Aa and B a (with a denoting the 2 M 1color index), which we will then make very massive. The motivation for introducing these

extra elds is that, before we decouple them, they augment the system to have N f = N cso that we can introduce color neutral baryonic elds. Dene the combination ab = Aa Ba .Now write the superpotential

W = N ij, N klik jl + X (det N abBa Bb 4M 2) + mtr (67)

with det N the determinant of the (2 M 1)(2M 1) meson matrix obtained by includingAa and Ba . Here the last term gives rise to a mass m for the extra elds Aa and Ba . Wecan decompose

detN = tr + abA,a

i B,b

j (N 1

)ij

det N. (68)The lagrange multiplier term in ( 67) is the standard one for an N = 1 gauge theory withN f = N c. Notice that the extra eld ab does not appear in the rst term in eqn ( 67); weomit it here to avoid a symmetry breaking expectation value for ab . Instead, we would liketo keep N c equal to 2M 1 after integrating out ab . The supersymmetric ab equations of motion now read

m + X (det N trBB) = 0 (69)from which we can solve for X , and

A,ai B,b

j (N 1)ij det N = Ba Bb. (70)

Inserting the solution for X back into W gives the superpotential (with p = 1)

W = (N ij ) (N k )

ik j + p4M p1

det ij, N tr BB

1p

(71)

with 4M 11 = m4M 2. This is our proposed superpotential of the theory with the bary-onic superelds present.

In case of general p, we can similarly write color charged baryon elds, which transformin the p-th anti-symmetric product of the fundamental. Although we have not done theexplicit analysis in this general case, a natural guess is that the superpotential will take theform (71). As a primitive reasonability check, we note that the supersymmetric equationof motion for the baryon eld, BW = 0, yields the condition that Ba = Ba = 0, so themesonic vacuum remains present as the only supersymmetric vacuum, according with ourexpectations.

21


24/29

In general, without more detailed control over the dynamics, the superpotential on itsown provides at most inconclusive evidence of the possible existence of other, nonsupersym-metric vacua of the theory. Still, if our proposal is right, it should at least give some hint. Ageneral comment: Formally, the equation BW = 0 also admits one other solution, namely

tr B

B . While it is of course dangerous to suggest that this implies the existence of another supersymmetric vacuum, it does indicate that, as a function of the baryon conden-sate, the full potential V = |dW |2 of the theory will have a maximum at some intermediatescale (which one would expect to be near tr BB4M 4 p1 ). It is conceivable, therefore, thatthere exists another minimum at large tr BB.

The strongest evidence for the existence of the nonsupersymmetric baryonic vacuum,however, still comes from the supergravity analysis. The characterization of the nonsuper-symmetric model in terms of p anti-D3 branes inside the conifold geometry suggests that,somewhere in the dual eld theory, there should be a hint of an (unbroken) SU ( p) gaugesymmetry. Indeed, our proposed dual interpretation in terms of a phase with a non-zero

condensate for tr BBnaturally leads to a breaking of the SU (2M p) gauge symmetry toSU ( p). It seems natural to identify the worldvolume theory of the p anti-D3 branes withthe effective low energy description of this SU ( p) sector. In the following subsection, we willnd an independent indication from the type IIA perspective that the nonsupersymmetrictheory is described by an SU ( p) SU (M p) gauge theory.

5.2 Type IIA Brane Congurations

The RG cascade in the KS system can also be understood via a dual type IIA perspective[2]. In the IIA description, one studies a theory on D4 branes suspended between NS 5

branes. Consider an NS 5 and an NS 5 brane, the rst lling out the 012345 directions,and the latter the 012389 directions in spacetime. Suppose them separated only along thex6 direction, compactied on a circle. One can then stretch N D4 branes around the circle,and M D4 branes on one of the two segments. The former correspond to the N D3 branesand the latter to fractional branes. The resulting eld theory has SU (N + M ) SU (N )gauge group and the matter stretching across the NS 5 and NS 5 branes gives precisely thebi-fundamentals which arise in the KS eld theory.

In this description, the forces on the branes are not perfectly balanced the NS branesbend together on the segment with the additional fractional branes. This corresponds tothe fact that the SU (N + M ) gauge theory has N f = 2N avors and becomes strongly

coupled as one ows to the IR. One can move the 5 brane through the NS 5 brane andaround the x6 circle to avoid this intersection; this will have the effect of reducing the gaugegroup to SU (N ) SU (N M ). This is the rst step in the RG cascade, and it is repeateduntil the rank of the gauge groups is low enough that N f < N c in one of the factors and thenon-perturbative dynamics becomes more subtle.

The difference in our setup is the addition of p D3 branes to the conifold. In the dualbrane conguration, these should appear as p D4 branes stretched between the NS 5 and

22


25/29

the NS 5 branes in addition to those already present in the KS setup. There are now (atleast) two obvious options:

The p anti-branes can annihilate with the D4 branes in each segment, leaving an SU (N +M p) SU (N p) gauge theory. This subsequently undergoes the RG cascade as describedabove; assuming N = KM and p


26/29

= K-1H

B3

NS5

NS5

M-p D3

S 3

= KH

B3

( i i )

p D3

Tunneling

( i )

Fig 4: Schematic depiction of the brane/ux annihilation process for (i) subcritical and (ii)supercritical values for p/M .

when we embed our model in a true string compactication with a nite volume as con-structed in [ 5]. In this case, the holographically dual gauge theory will be coupled to 4-dgravity. Looking at the form of the potentials in gure 4, it is then natural to ask whattype of cosmological evolutions are possible in this set-up.

Besides the eld, the string compactication will generally give rise to many otherlight moduli elds. In the basic model of [5], all can be made massive except for the K ahlermodulus u(x) controlling the overall volume of the 6d internal space Y . If we ignore thebackreaction due to the branes and uxes, it is dened by

ds2 = g dxdx + e2ugi j dzidz j , (72)

where ds2 is the 10d string frame metric and gi j the Ricci-at metric on Y . Following thesteps outlined in [ 5], one obtains for the 4d low energy effective action

S =1

224 d4x (g4)1/ 2 R4 6 ( u )2 a20 e 6u V 2()( )2 + a40 e 12u V tot () , (73)24


27/29

with g4 the 4d Einstein frame metric, and V 2 and V tot as in (40) and (45)-(49). The coefficienta0 is the warpfactor at the location of the anti-D3-branes.

The dynamics of this low energy eld theory is dominated by the steep inverse-volumedependence evident in ( 73), which implies that the presence of the extra energy density in

the anti-branes will quickly force the Calabi-Yau manifold to decompactify. In addition, itprevents the model from giving rise to any appreciable ination. To stop this decompact-ication process, or to get an inationary solution, it seems that one would have to nda novel means of stabilizing the K ahler moduli (for a discussion of moduli stabilization inroughly this context, see e.g. [ 5, 19]).

The gravitational effect of brane nucleation processes that induce discrete ux jumps hasrecently been investigated in [ 21], as a possible dynamical mechanism for neutralizing thecosmological constant. In particular the set-up considered in the second reference appearsclosely related to ours.

Acknowledgements

We would like to thank A. Bergman, C. S. Chan, K. Dasgupta, S. Giddings, C. Herzog, I.Klebanov, J. March-Russell, J. Maldacena, S. Mukhi, P. Ouyang, J. Polchinski, M. Schulz,N. Seiberg, M. Strassler, S. Trivedi, E. Verlinde and E. Witten for helpful discussions. Thework of J.P. and H.V. is supported in part by a National Science Foundation grant 98-02484and an NSF Graduate Research Fellowship. The work of S.K. is supported in part by aPackard Fellowship, a Sloan Fellowship, the DOE under contract DE-AC03-76SF00515, and

National Science Foundation grant PHY-0097915.

References

[1] For a review, see: O. Aharony, S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, LargeN Field Theories, String Theory and Gravity, Phys. Rept. 323 , 183 (2000) [hep-th/9905111 ].

[2] I. Klebanov and M. Strassler, Supergravity and a Conning Gauge Theory: DualityCascades and SB Resolution of Naked Singularities, JHEP 0008 , 052 (2000) [hep-

th/0007191 ].

[3] J. Maldacena and C. Nunez, Towards the Large N Limit of Pure N = 1 SuperYang-Mills, Phys. Rev. Lett. 86 , 588 (2001) [hep-th/0008001 ].

[4] C. Vafa, Superstrings and Topological Strings at Large N , J. Math. Phys. 42 , 2798(2001) [hep-th/0008142 ].

25
http://arxiv.org/abs/hep-th/9905111http://arxiv.org/abs/hep-th/9905111http://arxiv.org/abs/hep-th/0007191http://arxiv.org/abs/hep-th/0007191http://arxiv.org/abs/hep-th/0008001http://arxiv.org/abs/hep-th/0008142http://arxiv.org/abs/hep-th/0008142http://arxiv.org/abs/hep-th/0008001http://arxiv.org/abs/hep-th/0007191http://arxiv.org/abs/hep-th/0007191http://arxiv.org/abs/hep-th/9905111http://arxiv.org/abs/hep-th/9905111


28/29

[5] S. Giddings, S. Kachru and J. Polchinski, Hierarchies from Fluxes in String Compact-ications, [hep-th/0105097 ].

[6] L. Randall and R. Sundrum, A Large Mass Hierarchy from a Small Extra Dimension,Phys. Rev. Lett. 83 , 3370 (1999) [hep-th/9905221 ].

[7] H. Verlinde, Holography and Compactication, Nucl. Phys. B580 , 264 (2000) [hep-th/9906182 ];C. Chan, P. Paul and H. Verlinde, A Note on Warped String Compactication, Nucl.Phys. B581 , 156 (2000) [hep-th/0003236 ].

[8] P. Mayr, On Supersymmetry Breaking in String Theory and its Realization in BraneWorlds, Nucl. Phys. B593 , 99 (2001) [hep-th/0003198 ];P. Mayr, Stringy World Branes and Exponential Hierarchies, JHEP 0011 , 013 (2000)[hep-th/0006204 ].

[9] K. Becker and M. Becker, M-theory on Eight Manifolds, Nucl. Phys. B477 , 155(1996) [hep-th/9605053 ];K. Becker and M. Becker, Supersymmetry Breaking, M-theory and Fluxes, JHEP0107 , 038 (2001) [hep-th/0107044 ].

[10] S. Gukov, C. Vafa and E. Witten, CFTs from Calabi-Yau Fourfolds, Nucl. Phys.B584 , 69 (2000) [hep-th/9906070 ];K. Dasgupta, G. Rajesh, and S. Sethi, M-theory, Orientifolds and G-ux, JHEP9908 , 023 (1999) [hep-th/9908088 ];B. Greene, K. Schalm and G. Shiu, Warped Compactications in M and F-theory,

Nucl. Phys. B584 , 480 (2000) [hep-th/0004103 ].[11] G. Curio, A. Klemm, D. Lust and S. Theisen, On the Vacuum Structure of Type II

String Compactications on Calabi-Yau Spaces with H Fluxes, Nucl. Phys. B609 , 3(2001) [hep-th/0012213 ];G. Curio, A. Klemm, B. Kors and D. Lust, Fluxes in Heterotic and Type II StringCompactications, [ hep-th/0106155 ].

[12] C. Herzog, I. Klebanov and P. Ouyang, Remarks on the Warped Deformed Conifold,[hep-th/0108101 ].

[13] N. Seiberg, Exact Results on the Space of Vacua of Four-Dimensional SUSY GaugeTheories, Phys. Rev. D49 , 6857 (1994) [hep-th/9402044 ].

[14] J. Polchinski and M. Strassler, The String Dual of a Conning Four-DimensionalGauge Theory, [ hep-th/0003136 ].

[15] R. Myers, Dielectric Branes, JHEP 9912 , 022 (1999) [hep-th/9910053] .

26
http://arxiv.org/abs/hep-th/0105097http://arxiv.org/abs/hep-th/9905221http://arxiv.org/abs/hep-th/9906182http://arxiv.org/abs/hep-th/9906182http://arxiv.org/abs/hep-th/0003236http://arxiv.org/abs/hep-th/0003198http://arxiv.org/abs/hep-th/0006204http://arxiv.org/abs/hep-th/9605053http://arxiv.org/abs/hep-th/0107044http://arxiv.org/abs/hep-th/9906070http://arxiv.org/abs/hep-th/9908088http://arxiv.org/abs/hep-th/0004103http://arxiv.org/abs/hep-th/0012213http://arxiv.org/abs/hep-th/0106155http://arxiv.org/abs/hep-th/0108101http://arxiv.org/abs/hep-th/9402044http://arxiv.org/abs/hep-th/0003136http://arxiv.org/abs/hep-th/9910053http://arxiv.org/abs/hep-th/9910053http://arxiv.org/abs/hep-th/0003136http://arxiv.org/abs/hep-th/9402044http://arxiv.org/abs/hep-th/0108101http://arxiv.org/abs/hep-th/0106155http://arxiv.org/abs/hep-th/0012213http://arxiv.org/abs/hep-th/0004103http://arxiv.org/abs/hep-th/9908088http://arxiv.org/abs/hep-th/9906070http://arxiv.org/abs/hep-th/0107044http://arxiv.org/abs/hep-th/9605053http://arxiv.org/abs/hep-th/0006204http://arxiv.org/abs/hep-th/0003198http://arxiv.org/abs/hep-th/0003236http://arxiv.org/abs/hep-th/9906182http://arxiv.org/abs/hep-th/9906182http://arxiv.org/abs/hep-th/9905221http://arxiv.org/abs/hep-th/0105097


29/29

[16] J. Maldacena and H. Nastase, The Supergravity Dual of a Theory with DynamicalSupersymmetry Breaking, JHEP 0109 , 024 (2001) [hep-th/0105049 ].

[17] S. Coleman, The Fate of the False Vacuum I. Semiclassical Theory, Phys. Rev. D15 ,2929 (1977).

[18] S. Mukhi and N. Suryanarayana, A Stable Non-BPS Conguration from IntersectiongBranes and Anti-branes, JHEP 0006 , 001 (2000) [hep-th/0003219 ].

[19] E. Silverstein, (A)dS Backgrounds from Asymmetric Orientifolds, [ hep-th/0106209 ].

[20] F. Cachazo, K. Intriligator and C. Vafa, A Large N Duality via a Geometric Transi-tion, Nucl. Phys. B603 , 3 (2001) [hep-th/0103067 ];F. Cachazo, B. Fiol, K. Intriligator, S. Katz and C. Vafa, A Geometric Unication of Dualities, [ hep-th/0110028 ].

[21] R. Bousso and J. Polchinski, Quantization of Four-form Fluxes and Dynamical Neu-tralization of the Cosmological ConstantJHEP 0006 006 (2000) [hep-th/0004134 ];J.L. Feng, J. March-Russell, S. Sethi and F. Wilczek, Saltatory Relaxation of theCosmological Constant, Nucl.Phys. B602 307 (2001) [hep-th/0005276 ]
http://arxiv.org/abs/hep-th/0105049http://arxiv.org/abs/hep-th/0003219http://arxiv.org/abs/hep-th/0106209http://arxiv.org/abs/hep-th/0103067http://arxiv.org/abs/hep-th/0110028http://arxiv.org/abs/hep-th/0004134http://arxiv.org/abs/hep-th/0005276http://arxiv.org/abs/hep-th/0005276http://arxiv.org/abs/hep-th/0004134http://arxiv.org/abs/hep-th/0110028http://arxiv.org/abs/hep-th/0103067http://arxiv.org/abs/hep-th/0106209http://arxiv.org/abs/hep-th/0003219http://arxiv.org/abs/hep-th/0105049

Documents

Shamit Kachru, John Pearson and Herman Verlinde- Brane/Flux Annihilation and the String Dual of a Non-Supersymmetric Field Theory