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WIS/08/10-JULY-DPPA

Massive type IIA string theory

cannot be strongly coupled

Ofer Aharony1, Daniel Jafferis2, Alessandro Tomasiello3,4and Alberto Zaffaroni3

1Department of Particle Physics and Astrophysics

The Weizmann Institute of Science, Rehovot 76100, Israel

2School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA

3Dipartimento di Fisica, Universit` a di Milano–Bicocca, I-20126 Milano, Italy

and

INFN, sezione di Milano–Bicocca, I-20126 Milano, Italy

4Albert Einstein Minerva Center, Weizmann Institute of Science, Rehovot 76100, Israel

Abstract

Understanding the strong coupling limit of massive type IIA string theory is a longstand-

ing problem. We argue that perhaps this problem does not exist; namely, there may be

no strongly coupled solutions of the massive theory. We show explicitly that massive type

IIA string theory can never be strongly coupled in a weakly curved region of space-time.

We illustrate our general claim with two classes of massive solutions in AdS4×CP3: one,

previously known, with N = 1 supersymmetry, and a new one with N = 2 supersym-

metry. Both solutions are dual to d = 3 Chern–Simons–matter theories. In both these

massive examples, as the rank N of the gauge group is increased, the dilaton initially

increases in the same way as in the corresponding massless case; before it can reach the

M–theory regime, however, it enters a second regime, in which the dilaton decreases even

as N increases. In the N = 2 case, we find supersymmetry–preserving gauge–invariant

monopole operators whose mass is independent of N. This predicts the existence of branes

which stay light even when the dilaton decreases. We show that, on the gravity side, these

states originate from D2–D0 bound states wrapping the vanishing two–cycle of a conifold

singularity that develops at large N.

arXiv:1007.2451v1 [hep-th] 14 Jul 2010

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1Introduction and summary of results

One of the most striking aspects of string theory is its uniqueness, realized by the fa-

mous “web of dualities” that interconnect its various perturbative realizations. A famous

thread in this web connects weakly coupled, perturbative type IIA string theory with

its strong coupling limit, M theory (which reduces at low energies to eleven–dimensional

supergravity).

It has been known for a while, however, that this duality does not work when the

Romans mass parameter F0 [1], which can be thought of as a space-filling Ramond-

Ramond (RR) 10-form flux, is switched on. There is no candidate parameter in eleven–

dimensional supergravity to match with F0, unlike for all the other fluxes; nor is there

any massive deformation of the eleven–dimensional theory [2–4]. And, from the type IIA

point of view, the D0-branes which give rise to the momentum modes in the eleventh

dimension at strong coupling do not exist in the massive theory (as there is a tadpole

for their worldvolume gauge field). This would then appear to be an imperfection in our

understanding of string duality: it would be one string theory whose strong coupling limit

is not known.

In this paper, we will argue that this strong coupling limit may not exist, and we will

show this explicitly at least at the level of weakly curved solutions. In general these are the

only solutions we have any control over, unless we have a large amount of supersymmetry;

one can separately consider cases with a large amount of supersymmetry, and none of them

seem to lead to strong coupling either. (The type I’ theory of [5] contains in some of its

vacua strongly coupled regions of massive type IIA string theory, but these regions have

a varying dilaton and their size is never larger than the string scale.) Thus, we claim that

there is no reason to believe that any strongly coupled solutions exist (with the exception

of solutions with small strongly coupled regions), and we conjecture that there are none.

This is consistent with the fact that no suggestion for an alternative description of the

massive theory at strong coupling is known.

In section 2 we provide a simple argument that the string coupling gs in massive

type IIA string theory must be small, if the curvature is small. Generically, we find that

gs∼

equations of motion and flux quantization.

< ls/R, where R is a local radius of curvature. The argument just uses the supergravity

This result is in striking contrast with what happens in the massless case. In the

ten–dimensional massless vacuum, for example, the dilaton is a free parameter, and in

particular it can be made large, resulting in the M–theory phase mentioned earlier. The

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massive theory has no such vacua.

It is of interest to consider examples with AdS4 factors, where we can take advan-

tage of a dual field theory interpretation via the AdS/CFT correspondence, which also

provides a non-perturbative definition for the corresponding string theory backgrounds.

In particular, it is natural to consider solutions like the N = 6 supersymmetric solution

AdS4× CP3of the massless type IIA string theory [6–8]. In this solution, the dilaton is

determined by the internal flux integers k ∝?

large dilaton with small curvature. In this limit, the solution is better described as the

AdS4× S7/ZkM–theory background. The dual field theory has been identified in [9] as

the N = 6 superconformal Chern-Simons-matter theory with gauge group U(N)×U(N)

and Chern-Simons couplings k and −k.

Massive type IIA solutions are also known on AdS4×CP3, and it is natural to compare

their behavior to the massless case. For example, some solutions with N = 1 supersym-

metry are known explicitly [10,11]; they contain the N = 6 solution as a particular case.

The field theory duals are Chern–Simons–matter theories whose levels do not sum up to

zero. Even though F0is quantized as n0/(2πls), one might think that introducing the

smallest quantum of it, say n0= 1, should have little effect on the solutions, if the other

flux integers k and N are already very large. It would seem, then, difficult to understand

how a massless solution with large dilaton can suddenly turn into a massive solution with

small dilaton when n0is turned on.

CP1F2and N ∝?

CP3F6, gs∼ N1/4/k5/4,

whereas the curvature radius R/ls ∼ N1/4/k1/4. In particular, for N ? k5one has a

As we will see in section 3, in general this “small deformation” intuition is flawed.

When trying to express the dilaton in terms of the flux parameters, in the massive case

one ends up with expressions in which F0multiplies other, large flux parameters. Hence

F0 can have a large effect on the behavior of the solutions even if it is the smallest

allowed quantum. As it turns out, as we increase N, the dilaton does start growing

as gs ∼ N1/4/k5/4, as in the massless case. But, before it can become large, gsenters

a second phase, where it starts decreasing with N. Specifically, for N larger than the

“critical value” k3/n2

0

0, we have gs∼ N−1/6n−5/6

. Both behaviors are visible in figure 2.

Notice that what happens for these N = 1 solutions is not entirely a consequence of

the general argument in section 2. One could have found, for example, that for large N

the radius of curvature became small in string units. In such a situation, our supergravity

argument would not have been able to rule out a large dilaton; even worse, it would

actually generically predict it to be large. It is interesting to ask whether there are

situations where that happens. Of course, one would not trust such strongly–curved,

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strongly–coupled solutions, since we have no control over them; but, if they existed, they

would suggest that perhaps strongly coupled solutions do exist and need to be understood.

To look for such a different behavior, we turn to a second class of massive solutions,

still on AdS4× CP3, but this time with N = 2 supersymmetry. Such gravity solutions

were predicted to exist via AdS/CFT [12], and found as first–order perturbations in F0

of the N = 6 solution in [9]. The field theory duals are again Chern–Simons–matter

theories whose levels do not sum up to zero. In section 4 we point out that these theories

have certain gauge–invariant monopole operators, whose mass (which is protected by

supersymmetry) is independent of the rank N. This suggests the existence of wrapped

branes that remain light in the large N limit. This cannot happen for backgrounds which

are both weakly–coupled and weakly–curved.

To see what happens at large N, in section 5 we find these N = 2 gravity solutions,

generalizing the construction in [13]. We reduce the equations of motion and supersymme-

try equations to a system of three ODEs for three functions, which we study numerically.

As in section 3, we then study the behavior of gsas a function of the flux integers. We

find exactly the same phenomenon as in section 3: gsfollows initially the same growth

observed for the N = 6 solutions, and departs from that behavior before it can get large.

The existence of the light states found in section 4 is not a consequence of strong coupling,

but is instead explained by the fact that the internal space develops a conifold singularity

where branes can wrap a small cycle. We compute numerically the mass of D2–D0 bound

states wrapping the vanishing cycle, and we reproduce very accurately the mass predicted

in section 4 from AdS/CFT.

Hence, in both examples we examined, the curvature stays bounded almost every-

where, and the dilaton does not become strongly coupled. Our argument in section 2

does not rule out the possibility of solutions with large curvature and large dilaton, and

it would be nice to find a way to rule them out. In general, such solutions would not be

trustworthy, but in some situations one might understand them via chains of dualities.

For example, in some cases it might be possible to T–dualize to a massless solution with

small curvature, which in turn might be liftable to M–theory, along the lines of [14]. The

behavior found in the two examples analyzed in this paper may not be universal, and we

expect the AdS/CFT correspondence to be very helpful in any further progress.

One motivation for understanding the strong coupling limit of massive type IIA string

theory is the Sakai–Sugimoto model [15] of holographic QCD, which has Nf D8–branes

separating a region of space with F0 = 0 from a region with F0 = Nf/(2πls). The

solution of this model is known in the IR, where it is weakly coupled and weakly curved

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and the D8–branes may be treated as probes; but it is not clear what happens in the

UV, where, before putting in the D8–branes, the coupling became large (see [16] for an

analysis of the leading order back-reaction of the D8–branes in this model). Our analysis

rules out the possibility that the region of massive type IIA string theory between the

D8–branes becomes strongly coupled while remaining weakly curved in the UV. It would

be interesting to understand whether there is a sensible UV completion of this model,

and, if so, what it looks like.

2A general bound on the dilaton

In this section, we will find a bound for the dilaton for type IIA solutions with non-zero

0-form flux F0?= 0, assuming that the ten–dimensional curvature is small.

The argument is simply based on the equations of motion of type IIA supergravity.

Note that due to supersymmetry, these equations are actually exact (at two-derivative

order) and can be trusted even when the coupling constant becomes large. The Einstein

equations of motion in the string frame take the form

?

e−2φ

RMN+ 2∇M∇Nφ −1

4HMPQHNPQ

?

=

?

k=0,2,4

TFk

MN,(2.1)

where

TFk

MN=

1

2(k − 1)!FMM2...MkFNM2...Mk−

1

4k!FM1...MkFM1...MkgMN. (2.2)

The equations (2.1) are valid at every point in spacetime, away from possible branes or

orientifolds. On such objects, we would need to include further localized terms, but they

will not be needed in what follows. In fact, all we need is a certain linear combination:

let us multiply (2.1) by e0Me0N, where e are the inverse vielbeine; 0 is a frame index in

the time direction. We can now use frame indices to massage T00on the right hand side:

?

=

2(k − 1)!F0A2...AkF0A2...Ak+

2TFk

00=

1

(k − 1)!F0A2...AkF0A2...Ak− η00

1

1

2(k − 1)!F0A2...AkF0A2...Ak+

2k!FA1...AkFA1...Ak≡1

1

2k!FA1...AkFA1...Ak

?

1

2(F2

0,k−1+ F2

⊥,k) .

(2.3)

We have defined the decomposition Fk= e0∧F0,k−1+F⊥,k. (In particular, F⊥,0is simply

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F0.) Applying this to (2.1), we get

?

e−2φ

e0Me0N

?

RMN+ 2∇M∇Nφ −1

4HMPQHNPQ

??

=1

4(

?

k=2,4

F2

0,k−1+

?

k=0,2,4

F2

⊥,k) .

(2.4)

Again, this is satisfied at every spacetime point (away from possible sources): there is no

integral in (2.4). RMN needs to be small in the supergravity approximation. In fact, all

the remaining terms in the parenthesis on the left-hand side need to be small too: they

are all two–derivative NS–NS terms. If any of them is large in string units, we cannot

trust the two–derivative action any more; hence that parenthesis needs to be ? l−2

On the other hand, when F0?= 0, the right-hand side of (2.4) is at least of order one in

string units. To see this, recall that RR fluxes are quantized, in appropriate sense. The

Fkare actually not closed under d, but under (d − H∧). However, the fluxes

˜Fk= e−B(F0+ F2+ F4+ F6)

s.

(2.5)

are closed; these satisfy then the quantization law

?

Ca

˜Fk= nk(2πls)k−1, (2.6)

where nkare integers and Caare closed cycles. In particular, F0= n0/(2πls). Since the

right-hand side of (2.4) is a sum of positive terms, we get that it is > 1/l2

order one factors).

s(up to irrelevant

Let us now put these remarks together. Since the parenthesis on the left-hand side is

? 1/l2

s, and the right-hand side is > 1/l2

s, we have

eφ? 1 . (2.7)

For generic solutions, the parenthesis on the left hand side of (2.4) will be of order

1/R2, where R is a local radius of curvature. In that case, we can estimate, then,

eφ∼

<ls

R,

(2.8)

which of course agrees with (2.7).

When F0= 0, the conclusion (2.7) is not valid because all the remaining terms on the

right hand side can be made small, in spite of flux quantization. For example, assume all

the components of the metric are of the same order 1/R2everywhere, and that H = 0.

Then, the integral of F is an integer nk, but the value of F2

kat a point will be of order

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(nk/Rk)2(in string units). At large R, this can be made arbitrarily small. This is what

happens in most type IIA flux compactifications with F0= 0; the dilaton can then be

made large, and the limit φ → ∞ reveals a new phase of string theory, approximated by

eleven–dimensional supergravity.

To summarize, we have shown that F0?= 0 implies that the dilaton is small (2.7), as

long as the two–derivative action (the supergravity approximation) is valid.

3The N = 1 solutions

In this section, we will see how the general arguments of section 2 are implemented in the

N = 1 vacua of [10].

3.1The N = 1 solutions

We recall here briefly the main features of the N = 1 solutions in [10] on AdS4× CP3.

The metric is simply a product:

ds2

N=1= ds2

AdS4+ ds2

CP3,N=1. (3.1)

Topologically, CP3is an S2fibration over S4. We use this fact to write the internal metric

as

ds2

8(dxi+ ?ijkAjxk)2+

where xiare such that?3

radius, related to the AdS radius by

CP3,N=1= R2?1

i=1(xi)2= 1, Aiare the components of an SU(2) connection on

S4(with p1= 1), and ds2

1

2σds2

S4

?

, (3.2)

S4 is the round metric on S4(with radius one). R is an overall

RAdS≡ L =R

2

?

5

(2σ + 1). (3.3)

The parameter σ in (3.2) is in the interval [2/5,2]; this implies, in particular, that L/R is

of order 1 for these N = 1 solutions. For σ = 2, (3.2) is the usual Fubini–Study metric,

whose isometry group is SU(4) ? SO(6). For σ ?= 2, the isometry group is simply the

SO(5) that rotates the base S4.

The metric (3.1) depends on the two parameters L and σ. A third parameter in the

supergravity solution is the string coupling gs. Yet another parameter comes from the

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B field. For 2/5 < σ < 2, supersymmetry requires the NS-NS 3-form H to be non–zero

(see [10, Eq. (2.2)]). One can solve that constraint by writing

?(2 − σ)(σ − 2/5)

σ + 2

where β is a closed two–form [10, Eq. (4.5)]. Because of gauge invariance B∼= B + dλ1,

the space of such β is nothing but the second de Rham cohomology of the internal space,

H2(CP3) = R. So we have one such parameter, which we can take to be the integral of

β over the generating two–cycle in H2,

B = −

J + β (3.4)

b ≡

1

(2πls)2

?

CP1β ,(3.5)

where we normalized b so that large gauge transformations shift it by an integer.

To summarize, the N = 1 supergravity solutions depend on the four parameters

(L,σ,gs,b).

3.2Inverting the flux quantization equations

We now apply the flux quantization conditions (2.6). It is convenient to separate the

contribution from the zero–mode β:

˜Fk= e−β˜Fk|β=0;

k(2πls)k−1, which can be computed explicitly [10]. We have

nb

6

b 1

(3.6)

we then define?˜Fk|β=0≡ nb

1

lgsf0(σ)

l

gsf2(σ)

l3

gsf4(σ)

l5

gsf6(σ)

=

nb

0

nb

2

nb

4

≡

10 0 0

b10 0

1

2b2

b1 0

1

6b3

1

2b2

n0

n2

n4

n6

,(3.7)

where

l = L/(2πls) , (3.8)

and

f0(σ) =5

4

?

(2 − σ)(5σ − 2)

(2σ + 1)

√2σ + 1 ,

,f4(σ) = −25π2

3 · 52

(σ − 1)(2σ + 1)5/2

σ2(σ + 2)2

?

(2 − σ)(5σ − 2) ,

f2(σ) =8π

√5

(σ − 1)

(σ + 2)

f6(σ) = −27π3

3 · 57/2

(σ2− 12σ − 4)(2σ + 1)7/2

σ2(σ + 2)2

.

(3.9)

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Equation (3.7) is [12, Eq. (4.26)], which in this paper we chose to reexpress in terms

of l (the AdS radius in string units) rather than r (the internal size in string units), to

harmonize notation with section 5.

We want to invert these formulas and get expressions for the parameters (l,gs,σ,b)

in terms of the flux integers ni, as explicitly as possible. If one assumes b = 0, this

is easy [10]; with b ?= 0, it is a bit more complicated. A good strategy is to consider

combinations of the flux integers that do not change under changes of the b field: in

addition to n0, two other combinations are

(nb

2)2−2n0nb

We then find

4= n2

2−2n0n4,(nb

2)3+3n2

0nb

6−3n0nb

2nb

4= n3

2+3n2

0n6−3n0n2n4. (3.10)

n2

2− 2n0n4= (f2

2− 2f0f4)

?l

gs

?2

=16

15π2

?l

gs

?2(σ − 1)(4σ2− 1)

?l

σ2

,

?3(−6 + 17σ − 6σ2)(2σ + 1)3/2

(3.11)

n3

2+ 3n2

0n6− 3n0n2n4= (f3

2+ 3f2

0f6− 3f0f2f4)

gs

?3

=8π3

53/2

?l

gs

σ2

(3.12)

.

We see that (3.11) and (3.12) give two independent expressions for l/gs; this implies

(n2

(n3

2− 2n0n4)3

2+ 3n2

0n6− 3n0n2n4)2=

64(σ − 1)3(2σ − 1)3

27σ2(−6 + 17σ − 6σ2)2≡ ρ(σ) . (3.13)

This determines σ implicitly in terms of the fluxes. The function ρ(σ) (which we plot in

figure 1) diverges at σ =17−√145

12

∼ .41, and has zeros at σ =1

have multiplicity three, and hence they are also extrema and inflection points. Moreover,

it has a minimum at σ ∼ .65; and it goes to 1 for both σ = 2 and σ =2

We can now combine the equation for n0 in (3.7), which determines gsl, with the

expression for l/gsin either (3.11) or (3.12). We prefer using the latter, since it turns out

to contain functions of σ which are of order one on most of the parameter space:

2and σ = 1. These zeros

5.

l =

53/4

23/2√π

(2 − σ)1/4(5σ − 2)1/4σ1/3

(2σ + 1)1/2(−6 + 17σ − 6σ2)1/6

(2 − σ)1/4(5σ − 2)1/4(−6 + 17σ − 6σ2)1/6

σ1/3n

The function in the expression for l diverges at σ =17−√145

and 2, whereas the function in the expression for gsvanishes for σ =2

?n3

2

n3

0

+ 3n6

n0

− 3n2n4

n2

0

?1/6

,(3.14)

gs= 51/4

?π

2

1

2

0(n3

2+ 3n2

0n6− 3n0n2n4)1/6

.(3.15)

12

∼ .41 and vanishes for σ =2

5,17−√145

5

12

and 2.

Finally, the second row of equation (3.7) determines b in terms of n2, n0 and the

remaining fields l, gs and σ. One could eliminate l and gs from that expression using

(3.14) and (3.15), but we will not bother to do so.

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2

5

12

Σ

0.5

1

Ρ

1

2

1

Σ

?0.001

0.005

0.01

Ρ

Figure 1: A plot of the function ρ(σ) in (3.13).

3.3A phase transition

We will start by taking for simplicity

n4= 0 ,(3.16)

and we will call

n2≡ k ,n6≡ N(3.17)

as in [9].

In this case, (3.13) reads

ρ(σ) =

?

1 + 3Nn2

0

k3

?−2

.(3.18)

From the graph in figure 1, we see that the behavior of the solution depends crucially on

the ratio

N ?k3

Nn2

k3 . If for example

0

n2

0

, (3.19)

we have ρ(σ) ∼ 1. Looking at figure 1, we see that a possible solution is σ = 2. Around

this point, ρ goes linearly; so, if we write σ = 2−δσ, we have δσ ∼

(3.15) we then have

Nn2

k3 . From (3.14) and

0

l ∼ δσ1/4

?k

n0

?1/2

=N1/4

k1/4,gs∼ δσ1/4(kn0)−1/2=N1/4

k5/4. (3.20)

This is the same behavior as in the N = 6 solution [9].

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If, on the other hand,

N ?k3

n2

0

, (3.21)

we have ρ(σ) ∼ 0. The possible solutions are σ ? 1 or σ ?1

in the expressions for l and gsin (3.14) and (3.15) are then both of order one. We have

2. The σ–dependent functions

l ∼N1/6

n1/6

0

,gs∼

1

N1/6n5/6

0

. (3.22)

Notice that this behavior occurs for example in the nearly K¨ ahler solutions of [17].

For those vacua, we have l5/gs= n6and 1/(lgs) = n0, which gives the same behavior as

in (3.22). Notice also that σ = 1 corresponds indeed to a nearly K¨ ahler metric.

If one were to find a Chern–Simons dual to a vacuum whose only relevant fluxes are

n6and n0, such as the nearly K¨ ahler solutions, it would be natural to identify n6with

a rank N and n0with a Chern–Simons coupling˜k (because F0induces a Chern–Simons

coupling on D2–branes). In such a dual,

coupling. We see then that l and gsN in (3.22) are both functions of this˜λ, as expected.

From (3.22) one can calculate the finite temperature free energy to be βF ∼ V2T2 N2

V2T2N5/3n1/3

at strong coupling βF ∼ V2T2N3/2k1/2.

In figure 2 we show a graph of gsas a function of N; we see both behaviors (3.20) and

(3.22).

n6

n0=

N

˜k≡˜λ would then be the new ’t Hooft

˜λ1/3∼

0 , which grows with a higher power of N than in the massless case, for which

1 100

104

106

108

1010

n6

0.1

0.2

gs

Figure 2: The behavior of gsas a function of N = n6, for n2= k = 100, n4= 0 and

n0= 1. We see both the growth in the first phase (3.20), for n6? n3

decay in the second phase (3.22), for n6? n3

2/n2

0= 106, and the

2/n2

0= 106.

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Our analysis above was limited for simplicity to the case n4= 0, but it is easy to argue

that also for other values of n4, gscannot become large. Equation (3.15) tells us that

gs= f(σ)/n1/2

and m ≡ n3

from above in the massive theory by the maximal value of |f(σ)|. If m = 0, then (3.12)

implies that (−6+17σ −6σ2) also vanishes, and we can then use (3.13) to rewrite (3.15)

in the form gs=˜f(σ)/n1/2

0

˜ m1/4, where˜f(σ) is again bounded in the relevant range and

˜ m ≡ n2

maximal value of |˜f(σ)|, but this must be true since m and ˜ m cannot vanish at the same

time (as is clear from (3.11) and (3.12)). Thus, for any integer fluxes with n0?= 0, gsis

bounded from above by a number of order one.

0 m1/6, where f(σ) is bounded from above in the relevant range of values,

2+ 3n2

0n6− 3n0n2n4is an integer. Thus, if m ?= 0, then gsis clearly bounded

2−2n0n4is another integer. Thus, if ˜ m ?= 0 then gsis bounded from above by the

3.4 Probes

We will now see that the “phase transition” between (3.20) and (3.22) has a sharp con-

sequence on the behavior of the probe branes in the geometry. We will consider branes

which are particles in AdS4and that wrap different cycles in the internal space CP3.

Not all such wrapped branes are consistent. In the N = 6 case, where F0= 0 and

?

tadpole for the world–sheet gauge field A, because of the coupling

1

2πls

CP1F2 = n2 ?= 0, the action for a D2-brane particle wrapping the internal CP1has a

?

R×CP1A ∧ F2= n2

?

R

A

(3.23)

(the R factor in the D2-brane worldvolume being time). D0-branes, in contrast, have no

such problem. In the field theory, they correspond [9] to gauge–invariant operators made

of monopole operators and bifundamentals.

For the solutions with both F0? n0?= 0 and?

have a tadpole. If one considers a bound state of nD2D2 branes and nD0D0 branes, the

tadpole for A is

(nD2n2+ nD0n0)

CP1F2? n2?= 0, both D2’s and D0’s

?

R

A .(3.24)

For relatively prime n0and n2, the minimal choice that makes this vanish is nD2= n0

and nD0 = −n2. These branes also correspond to a mix of monopole operators and

bifundamentals; we will discuss analogous configurations in more detail in section 5.4.

Consider now the case n0= 1, and n2= k ? 1. Here we should consider a bound state

of one D2 brane and k D0 branes. In the context of AdS/CFT, all masses are naturally

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measured in units of the AdS mass scale mAdS≡

is of order L. The masses of a D2 and of a D0 particle would then be (setting the string

scale to one)

mD2L ∼L2/gs

1/Lgs

1

RAdS=

1

L; recall also from (3.3) that R

=L3

,mD0L ∼1/gs

1/L=L

gs

. (3.25)

Thus, the bound states we are considering here (the particles that have no world–sheet

tadpole) have a mass of order

mD2−kD0=L

gs

√k2+ L4. (3.26)

Which of the two terms dominates? it turns out that the answer depends on which of

the two phases, (3.20) or (3.22), we are considering. In both phases the ratio of the two

masses is a function of

N

k3.

A simple computation gives that, in the first phase (3.20), the D2’s mass is ∼

whereas the k D0 branes have mass k ×k. The D0’s dominate the mass, which then goes

like k2?

In the second phase, the D2’s mass is ∼ N2/3, whereas the k D0’s mass goes like

k × N1/3. Hence the D2 dominates. The mass then goes like N2/3

Another type of branes that have no tadpole problems are D4 branes. In the field

theory, these correspond [9] to baryon operators. In AdS units, these have a mass of

order

(3.22), which looks reasonable for a baryon.

√Nk,

1 +N

k3.

?

1 +?N

k3

?−2/3.

L4/gs

1/L=L5

gs. Interestingly, this turns out to be of order N in both phases (3.20) and

3.5Field theory interpretation

The field theories dual to the vacua analyzed in this section were proposed in [12]. Because

of the low amount of supersymmetry, we do not expect to be able to make here any useful

check of this duality. However, we can use our gravity results to make some predictions

about those field theories, under some assumptions.

First of all, let us recall briefly the N = 1 field theories defined in [12]. They are similar

to the N = 6 theory of [9,18], in that they also have a gauge group U(N1)×U(N2). The

matter content can be organized in (complexified) N = 1 superfields XI, I = 1,...,4;

they transform in the (¯ N1,N2) representation of the gauge group. The biggest difference is

that the Chern–Simons couplings for the two gauge groups are now unrelated: we will call

them k1and −k2. For k1?= k2, it is no longer possible to achieve N = 6 supersymmetry,

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and there are several choices as to the amount of flavor symmetry and supersymmetry that

one can preserve. In this section, we consider a choice that leads to N = 1 supersymmetry

and SO(5) flavor symmetry; in the following sections we will consider a different choice,

that leads to N = 2 and SO(4) flavor symmetry.

This theory can be written in terms of N = 1 superfields; the superpotential then reads

WN=1= Tr[c1X†

terms are manifestly invariant under Sp(2)=SO(5), as promised. When k1= k2≡ k, the

theory has N = 6 supersymmetry when the parameters are c1= −c2= 2π/k, c3= −4π/k.

For k1?= k2, this choice is no longer possible, as we already mentioned. In spite of there

being only N = 1 supersymmetry, however, it was argued in [12] that there still exists

a choice of cithat makes the theory superconformal, as long as k1− k2is small enough

with respect to the individual ki.

IXIX†

JXJ+c2X†

IXJX†

JXI+c3ωIKωJLX†

IXJX†

KXL]. Notice that all the

If we define the ’t Hooft couplings

λ1=N

k1

,λ2=

N

−k2

,λ±= λ1± λ2, (3.27)

the N = 6 theory would correspond to λ+= 0. The argument in [12] then says that there

is a CFT in this space of theories if λ+? λ−, although at strong coupling it is difficult

to quantify just how much smaller it has to be.

Let us now try to translate in terms of these field theories the “phase transition” we

saw in section 3.3. To do so, we can use the dictionary (5.35) between the field theory

ranks and levels on one side, and flux integers on the other. This dictionary is also valid

for N = 1 theories [12]. The phase transition in section 3.3 happens for N ∼ n3

k1∓ k2

N

2/n2

0. Since

=1

λ1±1

λ2=

±4λ±

λ2

+− λ2

−

, (3.28)

when λ+? λ−we have n0/N ∼ λ+/λ2

at

−, n2/N ∼ 1/λ−. So the phase transition happens

λ−∼ λ2

+.(3.29)

In particular, the “ABJM phase” (3.20) corresponds to λ+??λ−; the “nearly–K¨ ahler”

regime where

described by the field theories described in [12] and reviewed in this section. However,

given the low amount of supersymmetry, this can only be a conjecture at this point.

phase (3.22) corresponds to λ+??λ−. At strong coupling, then, there is an intermediate

?λ−? λ+? λ−, where it is possible that the second phase (3.22) is also

Rather than trying to test further this correspondence, we will now turn our attention

to N = 2 theories, on which there is much better control.

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4 Monopoles in N = 2 Chern–Simons–matter theo-

ries

In this section, we will recall some general facts about monopole operators in Chern–

Simons–matter theories, and we will apply them to a particular quiver theory, similar to

the ABJM theory; its gravity dual will be examined in section 5.

4.1Construction of monopole operators in general

Consider a d = 3 gauge theory with gauge group?m

in the IR, then these must be dimension 2 operators in the IR. There may or may not be

operators charged under the corresponding U(1)mflavor symmetry; if they exist we will

call them monopole operators.

i=1U(Ni). Then there are m currents,

ji= ∗ Tr(Fi), which are conserved by the Bianchi identity. If the theory flows to a CFT

In a conformal field theory, it is convenient to use radial quantization and consider

the theory on R × S2. Let us apply the state–operator correspondence to a monopole

operator, with charge vector ni. This results in a state in the theory on an S2, such that

?

Fi= diag(w1

S2Tr(Fi) = 2πni. We will denote the diagonal values of Fiby wa

i:

i,...,wNi

i)volS2 ; (4.1)

taking the trace over the gauge group U(Ni), we have that the magnetic charges are

ni=

?

a

wa

i. (4.2)

We are interested in d = 3 N = 2 Chern–Simons–matter theories. We can take them to

be weakly interacting at short distances by adding Yang–Mills terms as regulators [19–22].

In such a regulated theory, there are BPS classical configurations with the gauge field as

in (4.1) and non–trivial values for the scalar fields. The BPS equations on R1,2include the

Bogomolnyi equations Fi= ∗Dσi, where σiis the adjoint scalar in the vector multiplet.

In R × S2, this equation is different because the metric needs to be rescaled, and the

fields need to be transformed accordingly; the equations then read Fi= σivolS2. Notice in

particular that the σiare constant. There are also other BPS equations, which involve the

other scalars in the theory (for explicit computations for N = 3 theories, see [20, §3.2],

and in N = 2 language, [23]).

After adding the regulating Yang-Mills term, g2

N = 2 vector multiplet should be treated classically, while the chiral matter fields should

Y Mbecomes small in the UV, so the

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