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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.
2 A 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.
3 The 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].
3.1 The 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).
3.2 Inverting 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
b1 0 0
1
2b2
b 1 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
11
Page 13
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,
12
Page 14
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.
13
Page 15
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
14
Page 16
not. This justifies not solving Gauss’s law in describing the “pure” monopole operator,
which then behaves as if it were a local, non-gauge invariant chiral field [20].
The scalar σiis set by the BPS equations to be equal to the inverse Chern–Simons
level times the moment maps of the matter fields,
σi= 2πDi
ki
, (4.3)
and to have a spin 0 operator, one should satisfy the constraint [23]
σiXij= Xijσj
(4.4)
for any bi–fundamental field Xij connecting the i-th and j-th gauge group [24–26]. In
addition, the fields Xijneed to satisfy the F-term equations of the N = 2 theory. We see
from (4.3) that some matter fields, which should be neutral under the background U(1),
are necessarily non vanishing in the background; hence, the possibility of satisfying all
equations gives a nontrivial constraint on the possible BPS monopole operators.
In general, monopole operators T creating such configurations at a point will not be
gauge invariant. However, they will behave exactly like local fields. Hence, they can
be combined with other local operators O, to write gauge–invariant expressions of the
schematic form
Tr(TO) ,
where the indices are contracted as appropriate for the representations in which the op-
erators transform.
(4.5)
Let us determine how the monopole transforms under the gauge group. This is easy to
find for the Abelian factors of the theory. There is an obvious contribution to the electric
charges of a monopole operator, from the Chern–Simons term?ki
nikiunder the ithelectric U(1) Abelian factor.
4π
?Tr(Ai∧ Fi). This
says that a monopole with magnetic charges niwill behave like a particle with charges
This result gives a constraint on the possible gauge invariant operators Tr(TO) we can
obtain. If all matter is in bifundamental and adjoint representations, no gauge invariant
operators can be formed from monopoles that are charged under the overall U(1), since
no matter field transforms under it. Since we just computed the electric charge under the
ithU(1) to be kini, the charge under the overall U(1) is?kini. Thus, if we are to form
?
15
any gauge invariant operators of the form (4.5), we need to require
m
i=1
kini= 0 .(4.6)
Page 17
This result will be useful in the theories with gauge group U(N1)×U(N2), which we will
discuss in section 4.2.
Let us now ask how the monopole will transform under the full non-Abelian group.
One method to determine this is the following. One fixes a particular configuration on
the sphere (breaking the gauge symmetry), one computes the charge under the whole
Cartan subalgebra of the gauge group, and then one integrates over the gauge orbits.
Thus monopole operators can be labelled by U(1) subgroups of the gauge groups. For the
U(Ni) factor of the gauge group, the charges under this Cartan subalgebra are the wa
with a = 1,...,Ni, that we saw in (4.1).
i,
Therefore, a monopole associated with magnetic flux wa
weight vector [27, Sec. 4.2]
iis in the representation with
(kiw1
i,...,kiwNi
i) (4.7)
of the ithgauge group. (Our notation here is that the weight vector denotes the number
of boxes in each row of a Young diagram of the representation; thus (k,0,...,0) is, for
example, the completely symmetric representation.)
Quantization in a background of the type discussed in this section can result in anoma-
lous contributions to the charges and energy of the state. For the non–chiral theories we
we will consider in section 4.2, there is no such a correction to the gauge charges1. How-
ever, as we will discuss in section 4.2, the dimension of the monopole operator is given by
the energy of the state on the sphere, which often includes a non–zero Casimir energy.
4.2 Dimensions and charges of the monopoles
We will apply in this section the results of section 4.1 to the Chern–Simons–matter theories
with N = 2 and N = 3 given in [12]. In particular, we will compute the dimensions of
particular monopoles, which will be useful later, when comparing to the gravity solutions
of section 5.
Let us first recall some details about the field theories of interest. They are similar to
the N = 6 theory of [9,18], in that they also have a gauge group U(N1) × U(N2), and
N = 2 “chiral” superfields Ai,Bi, i = 1,2; the Aitransform in the (¯ N1,N2) representation,
1If the matter content is chiral, as for the theories in [13], there will be an additional one–loop correction
to the gauge charges. One way to understand this effect is that the state on the sphere has a constant
value for the scalar in the vector multiplet, which gives a mass to any matter fields charged under that
U(1) subgroup in which the magnetic flux lives. Integrating them out at one–loop can shift the effective
Chern–Simons level in that background if the theory has chiral matter.
16
Page 18
whereas the Bitransform in the (N1,¯ N2). Just as in section 3.5, the crucial difference
between the N = 6 theory and the N = 2 theory is that the Chern–Simons couplings for
the two gauge groups are now unrelated; we again call them k1and −k2. The theories we
want to consider in this section are defined by the superpotential
W = Tr(c1(AiBi)2+ c2(BiAi)2) .(4.8)
For generic ci, the theory has N = 2 supersymmetry and SU(2) flavor symmetry. For
ci=
still SU(2). For c1= −c2= c, the supersymmetry stays N = 2, but W can be rewritten
as
W = cTr(?ij?klAiBkAjBl) ,
1
ki, supersymmetry turns out to be enhanced to N = 3, while the flavor symmetry is
(4.9)
which shows that the flavor symmetry is enhanced to SU(2)×SU(2). This N = 2 theory
is dual to the gravity solution discussed in the next section.
We will now apply to these theories the discussion of section 4.1 about monopole
operators. The following computation is a straightforward generalization of that done
in [19,20] for the N = 3 theory. The results for the N = 3 and N = 2 theories appear to
be identical, since the flavor symmetry guarantees that the matter fields have the same
dimensions as in the more supersymmetric theory.
As we saw in section 4.1, there are non–trivial conditions on the scalars for the
monopole to be BPS. For the theories we are considering, the conditions read
AA†− B†B =k1
B B†− A†A = −k2
σ1Ai=Aiσ2,
2πσ1,
2πσ2,
σ2Bi=Biσ1,
(4.10)
together with the F-term constraints coming from (4.8). In view of (4.1) and Fi= σivol2,
such equations relate the magnetic fluxes wa
can consider are defined by magnetic charges wa
w1= (1,...,1,0,...) (with n11’s) and w2= (1,...,1,0,...) (with n21’s). The non-zero
elements of the fields Aiand Biare n1×n2and n2×n1rectangular matrices, respectively,
which are required to satisfy the first two lines in (4.10) and the F-term constraints.
The problem of finding appropriate vacuum expectation values for the matter fields is
equivalent to finding the BPS moduli space of the generalized U(n1) × U(n2) Klebanov–
Witten theory with superpotential (4.8) and Fayet-Iliopoulos (FI) parameters turned on.
Many of these moduli spaces are non–empty.
1with the wa
1which are all either 0 or 1: namely,
2. The simplest monopoles we
17
Page 19
Using now (4.6), we see that this monopole operator can be coupled to elementary
fields in a gauge invariant way only if
n1k1= n2k2.(4.11)
The matter content is non–chiral, so there are no anomalous contributions to the gauge
charges of the monopole. There is, however, a one–loop correction to the dimension of
the operator, which is given by [20–22]
∆ = −1
2
?
2 × 1(n1(N1− n1) + n2(N2− n2)) − 4 ×1
= (n1− n2)2− (N1− N2)(n1− n2) ,
fermions
?
|q|R
= −1
2 2(n1(N2− n2) + n2(N1− n1))
?
(4.12)
where R is the R-charge of the fermion and q the charge under the U(1) subgroup specified
by the vectors w1= (1,..n1..,1,0,...) and w2= (1,..n2..,1,0,...). The various contribu-
tions arise as follows. The four bi-fundamental fermions have R-charge −1/22. Each
bi-fundamental fermion is a matrix with N1N2entries; the n1(N2− n2) + n2(N1− n1)
off-diagonal entries have charge ±1 under the magnetic U(1) subgroup, while the remain-
ing entries are neutral. The two adjoint gauginos have R-charge +1. They are square
matrices with N2
charge ±1, while the other are neutral. We used the fact that in this theory, both for
N = 3 and N = 2, the R–charges in the UV and IR are identical.
In the N = 6 theory, k1= k2and it follows from (4.11) that n1= n2. The simplest
monopole has just w1= w2= (1,0,...,0). We need to turn on Aiand Bifields that solve
the U(1) Klebanov-Witten theory with a FI term. According to (4.7), such a monopole
transforms in the k-fold symmetric representation of U(N1) and in the conjugate k-fold
symmetric representation of U(N2). The monopole can combine with k fields Aito form
a gauge-invariant operator (we can analogously form a gauge-invariant operator with the
conjugate monopole and k fields Bi).
ientries; the 2ni(Ni− ni) off-diagonal entries of the i-th fermion have
In the N = 3 and N = 2 theories, we cannot have n1= n2, but we can now take
n1= k2and n2= k1and rectangular matrices Aiand Bithat solve (4.10). In general, the
2Since the superpotential (4.8) must have R-charge +2 and there is a discrete symmetry between
Aiand Biwe have that R(Ai) = R(Bi) = 1/2; the R-charge of the fermionic partners is R(A) − 1 by
supersymmetry.
18
Page 20
matrices AiBjwill not be diagonalizable. Recalling equation (4.7), the monopole opera-
tor is in a representation of the gauge group with weight vectors (k1,..k2..,k1,0,...) and
(k2,..k1..,k2,0,...). A gauge invariant combination must include k1k2matter bifundamen-
tals (if k1and k2are not relatively prime, some operator with smaller dimension could
exist). The total dimension of the gauge-invariant operator, dressed with k1k2elementary
fields, is then
∆ =k1k2
2
+ (k2− k1)2− (k2− k1)(N1− N2) .(4.13)
Note that we have determined the vacuum expectation values of the matter fields
needed to “support” the flux to form a BPS state on the sphere using the classical moduli
space. This is justified since the Higgs branch does not receive quantum corrections.
More precisely, the ring of chiral operators is the ring of algebraic functions on the moduli
space. There is a natural map [24–26] from the moduli space of Chern–Simons–matter
theories to the moduli space of the four–dimensional Yang–Mills theory with the same field
content. That moduli space cannot receive quantum corrections (aside from wavefunction
renormalization which fixes the coefficients of the superpotential), and only the S1bundle
over that space, associated to the dual gauge fields, is quantum corrected. This precisely
corresponds to 1-loop corrections to the charges and dimensions of monopole operators,
which are, however, constructed in the UV weakly coupled Yang–Mills–Chern–Simons–
matter theory.
Let us summarize the results of this section. The monopole operators that create k2
units of flux for the first gauge group and k1 for the second have k1k2 bifundamental
indices, and hence we can contract them with k1k2bifundamental fields to construct a
gauge–invariant operator. Such operators have dimension given by (4.13). In particular,
they stay light when N1= N2≡ N → ∞. Since in general monopole operators correspond
to D–branes, this seems to indicate a limit where D–branes become light, which usually
signals some sort of breakdown of the perturbative description. We will see in section 5.4
precisely how this happens.
5 The N = 2 solution
We now turn to writing and studying the N = 2 solution predicted to exist in [12], and
found in [28] at first order in F0. This solution will be the gravity dual of the field theory
defined by the superpotential (4.9), and it will serve as another illustration of the general
result of section 2.
19
Page 21
We will start in section 5.1 by reducing the equations of motion and the supersym-
metry conditions to a system of three equations for three functions of one variable. This
procedure closely parallels [13], where an analogous solution for the gravity dual of the
Chern-Simons theory based on the C3/Z3quiver was found. In section 5.2, we will impose
flux quantization, and derive expressions for the supergravity parameters in terms of the
flux integers; in section 5.3 we find, just like in section 3.3, a “phase transition” that
prevents the dilaton from growing arbitrarily large. Finally, in section 5.4 we find light
D–brane states dual to the monopole operators discussed in section 4.2.
5.1 The N = 2 solutions
The ten dimensional metric we will consider is a warped product of AdS4with a compact
six-dimensional internal metric with the topology of CP3:
ds2
10= e2Ads2
AdS4+ ds2
6.(5.1)
As discussed in [28,29], there is a foliation of CP3in copies of T1,1, which is in turn a
S1fibration over S2× S2. The usual Fubini–Study metric can be written as
ds2
44
where Ai, i = 1,2, are one-form connections, with curvatures
6=cos2(t)
ds2
S2
1+sin2(t)
ds2
S2
2+ dt2+1
16sin2(2t)(da + A2− A1)2, (5.2)
dAi= Ji,(5.3)
where Jiare the volume forms of the two spheres S2
interval [0,π/2]; all the functions in our solution (including A in (5.1)) will depend on this
coordinate alone. At one end of the interval [0,π/2], one S2shrinks; at the other end, the
other S2shrinks. To make this metric regular, we take the periodicity of a to be 4π.
i. The coordinate t parametrizes the
The Fubini–Study metric is appropriate for the N = 6 solution, which has F0= 0.
Once we switch F0on, as we saw in section 4.2, AdS/CFT predicts the existence of an
N = 2 solution with isometry group SU(2)×SU(2)×U(1) (the first two factors being the
flavor symmetry which is manifest in (4.9), and the third being the R–symmetry). The
internal metric for such a deformed solution is then given by3
ds2
6=e2B1(t)
4
ds2
S2
1+e2B2(t)
4
ds2
S2
2+1
8?2(t)dt2+1
64Γ2(t)(da + A2− A1)2.(5.4)
3One could have reparameterized the coordinate t so as to set one of the functions in (5.4) to a
constant, for example ?, as in (5.2). We have chosen, however, to fix this reparameterization freedom
in another way: by choosing the pure spinors (A.8) to be as similar as possible to those for the N = 6
solution, see in particular (A.11).
20
Page 22
Were the functions e2Binon-vanishing, we would have a metric on the total space of an
S2bundle over S2× S2. To maintain the topology of CP3, we require that e2B2vanishes
at t = 0 and e2B1vanishes at t = π/2. With an abuse of language, we will refer to t = 0
as the North pole and t = π/2 as the South pole, although there is no real S2fiber. To
have a regular metric, ?(t) and Γ(t) must behave appropriately at the poles.
It is convenient to define the combinations
wi= 4e2Bi−2A
(5.5)
which control the relative sizes of the two S2’s. As discussed in Appendix A, the super-
symmetry equations reduce to three coupled first order ordinary differential equations for
w1, w2and a third function ψ which enters in the spinors:
Ct,ψ(w1+ w2) + 2cos2(2t)w1w2
Ct,ψ(w1+ w2)cos2(2ψ) + 2w1w2
4w1
sin(4t)Ct,ψ(w1+ w2)cos2(2ψ) + 2w1w2
4w2
sin(4t)Ct,ψ(w1+ w2)cos2(2ψ) + 2w1w2
ψ?=sin(4ψ)
sin(4t)
,
w?
1=
Ct,ψ(w1w2− 2w2− 2sin2(2ψ)w1)
, (5.6)
w?
2=
Ct,ψ(w1w2− 2w1− 2sin2(2ψ)w2)
,
where
Ct,ψ≡ cos2(2t)cos2(2ψ) − 1. (5.7)
All other functions in the metric and the dilaton are algebraically determined in terms
of w1,w2,ψ:
√2eA(cot(ψ) − tan(ψ))csc2(2t) sin(2ψ) − cos(2ψ) cot(2t)ψ?
Γ = 4eAsin(2t) + cos(2t)cot(2t)sin2(2ψ)
2?1 + cot2(2t) sin2(2ψ)
F0csc(4t) sec(2ψ) tan(2ψ)
?
Here, c is an integration constant, that so far is arbitrary. The fluxes are determined as
well, and have the general form
? =
2?1 + cot2(2t) sin2(2ψ)
(5.8)
(5.9)
e4A= −4c
(5.10)
e3A−Φ= csec(2ψ) 1 + cot2(2t) sin2(2ψ) . (5.11)
F2= k2(t)e2B1J1+ g2(t)e2B2J2+˜k2(t)i
2z ∧ ¯ z ,
2z ∧ ¯ z ∧ J1+ ˜ g4(t)e2B2i
F4= k4(t)e2B1+2B2J1∧ J2+˜k4(t)e2B1i
F6= k6(t)e2B1+2B2
16
2z ∧ ¯ z ∧ J2,
i
2z ∧ ¯ z ∧ J1∧ J2,
(5.12)
21
Page 23
where
˜ gican be found in (A.12). The fluxes satisfy the Bianchi identities, which require that
i
2z ∧ ¯ z =
?Γ
16√2dt∧(da+A2−A1). The full expressions for the coefficients ki,˜ki, gi,
˜F ≡ e−B(F0+ F2+ F4+ F6) (5.13)
is closed. This dictates in particular that F0is constant.
We can now study the regularity of the differential equation near its special points,
t = 0 and t = π/2, by finding a power series solution of the equations. The general
solution will depend on three arbitrary constants. However, we are after solutions with
particular topology, where w2vanishes at t = 0 (the “North pole”) and w1vanishes at
t = π/2 (the “South pole”). Near t = 0, we obtain
ψ = ψ1t −2
w1= w0+ (4 + 4ψ2
w2= (4 + 4ψ2
3(4ψ1+ 5ψ3
1)t3+ O(t5),
1− 2w0+ 2w0ψ2
1)t2+ O(t4) .
1)t2+ O(t4),
(5.14)
In our solution, w0and ψ1are not independent: imposing that w1vanishes at t = π/2
determines w0in terms of ψ1. The power series expansion in˜t ≡ π/2 − t near t = π/2 is
identical, with the role of w1and w2exchanged:
ψ =˜ψ1˜t −2
w1= (4 + 4˜ψ2
w2= ˜ w0+ (4 + 4˜ψ2
3(4˜ψ1+ 5˜ψ3
1)˜t2+ O(˜t4),
1− 2 ˜ w0+ 2 ˜ w0˜ψ2
1)˜t3+ O(˜t5),
1)˜t2+ O(˜t4) .
(5.15)
The constants ˜ w0and˜ψ1should also be determined by ψ1; we can then think of ψ1as
the only parameter in the internal metric. To find a solution with the required topology,
we note that the equations (5.6) are symmetric under the operation t →π
w1 ↔ w2, and we look for solutions which are left invariant by this symmetry. This
determines˜ψ1= ψ1and ˜ w0= w0, and it allows us to restrict the study of the equations
to the “north hemisphere” t ∈ [0,π/4]. The only thing left to impose is that the solution
is differentiable at t = π/4. This is what determines w0as a function of ψ1, which we
plot in figure 3. w0(ψ1) is monotonicaly decreasing; our numerical analysis shows that it
vanishes at a point very well approximated by ψ1=√3.
2− t,ψ → −ψ,
The perturbative expansion of the solutions near the “poles” t = 0 and t = π/2 allows
to check the regularity of the six-dimensional metric. In fact, the only special points in
the metric are the poles, where a copy of S2degenerates. Using the previous expansion,
22
Page 24
1
2
1
3
2
3
Ψ1
1
2
3
4
w0
Figure 3: A plot of w0as a function of ψ1. It vanishes linearly around the point ψ1=√3.
it is straightforward to check that, at both poles, the shrinking S2combines with (t,a) to
give a piece of the metric proportional to
dt2+1
4t2?ds2
Si + (da ± Ai)2?
. (5.16)
Thanks to the fact that the periodicity of a is 4π, this is the flat metric of R4. For all
ψ1∈ [0,√3) the metric is then regular. For ψ1=
pole and the metric develops two conifold singularities. ψ1 =
limiting point in our family of solutions.
√3, both spheres degenerate at each
√3 is thus the natural
We can examine now the number of parameters in the solution. As discussed above,
the differential equations provide just one parameter, ψ1, the value of the derivative of ψ
at the North pole t = 0. It is convenient to define two more parameters by
gs≡ eφ|t=0,2L = eA|t=0. (5.17)
Both φ and A vary over the internal manifold, but numerical study reveals that they only
do so by order one functions. So gsand L can be thought of as the order of magnitude
of the dilaton and AdS radius in our solutions4. We can now reexpress the integration
constant c by evaluating (5.11) at t = 0:
c =8L3
gs
1
?1 + ψ2
1
. (5.18)
4The normalization has been chosen so that in the metric, at t = 0, L2multiplies an Anti-de Sitter
space of unit radius, and the relation between the mass of a particle at t = 0 and the conformal dimension
of the dual operator is mL = ∆(∆−4). This normalization is related to the fact that, in our conventions,
ds2
AdS4has cosmological constant Λ = −3|µ|2and, as discussed in appendix A, we chose µ = 2.
23
Page 25
The F0flux is then determined by evaluating (5.10) at t = 0:
F0= −
1
Lgs
ψ1
?1 + ψ2
1
. (5.19)
Finally, a fourth parameter comes from the B field. As in section 3, there is a zero–
mode ambiguity coming from the presence of a non–trivial cohomology in our internal
manifold. To see this, let us call B0a choice of B–field such that H = dB0solves the
equations of motion. For example, we can choose B0such that
˜F2= F2− B0F0= 0 .(5.20)
H = dB0is guaranteed to solve the equations of motion, since equation (5.20) implies
that dF2= HF0, which we have already solved. However, this will also be true for any B
of the form
B = B0+ β ,(5.21)
for any β which is closed. We can apply to this β the same considerations as in section
3.1: 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. And, just as in (3.5), we define the integral of β over the generating two–cycle
in H2: b ≡
not generate confusion, as the contexts are different.
1
(2πls)2
?
CP1β. The fact that we use the same notation as in section 3 should
Summarizing, our solutions are parameterized by the four numbers (L,ψ1,gs,b). The
situation is very similar to the N = 1 solutions we studied in section 3, with σ replaced
by ψ1.
5.2Inverting the flux quantization equations
This section will follow closely the corresponding treatment for the N = 1 solutions in
section 3.2. The equations are formally very similar:
nb
6
1
lgsf0(ψ1)
0
l3
gsf4(ψ1)
l5
gsf6(ψ1)
=
nb
0
nb
2
nb
4
≡
100 0
b1 0 0
1
2b2
b 1 0
1
6b3
1
2b2
b 1
n0
n2
n4
n6
, (5.22)
where l = L/(2πls), as in (3.8). The vector on the left hand side is given by the integrals
1
(2πls)k−1
?
Ck˜Fk, where Ckis the single k–cycle in CP3(k = 0,2,4,6), and˜Fkis defined
24
Page 26
using the particular B0in (5.20); this also explains why the second entry of the vector is
zero (this is simply our choice for the definition of b). We could have made such a choice
for the N = 1 solution as well; we did not do so because for SU(3) structure solutions
there is a different and particularly natural choice of B–field.
Using equation (5.19) we can write
f0(ψ1) = −
ψ1
?1 + ψ2
1
. (5.23)
We know the other functions fk(ψ1) only numerically. We obtain 2f4(ψ1) by integrating
˜F4 over the diagonal S2times the “fiber” (t,a), which is a representative of twice the
fundamental four-cycle. The plots of fk(ψ1) are given in figure 4. Our numerical analysis
indicates the following asymptotics at the two extrema ψ1= 0, ψ1=√3:
f4∼ ψ−1
f4∼ (√3 − ψ1) ,
1
,f6∼ ψ−2
f6→ const
1
for ψ1→ 0 ;
for ψ1→
(5.24)
√3 .(5.25)
1
2
1
3
2
3
Ψ1
1
2
1
f4
1
2
1
3
2
3
Ψ1
0.02
0.04
0.06
f6
Figure 4: Plots of f4(ψ1) and f6(ψ1). Their asymptotic behavior near 0 and√3 is given
in (5.24), (5.25). Notice in particular that f6(√3) is small but non–zero.
We can now proceed in the same fashion as in the N = 1 case to determine ψ1from
the flux parameters. Namely, we write the combination
(n2
2− 2n0n4)3
2+ 3n2
(n3
0n6− 3n0n2n4)2= −
8f4(ψ1)3
9f6(ψ1)2f0(ψ1)≡ ρ(ψ1) , (5.26)
25
Page 27
which allows us to determine ψ1in terms of fluxes. l and gsare then given by
?n2
g6
s
= 3f5
l6
=
f0(ψ1)
3f6(ψ1)n0
?3?
1 + 3n2
0n6
n3
2
?
− 3n0n4
1 + 3n2
n2
2
?
, (5.27)
0(ψ1)f6(ψ1)(n2n0)−3
0n6
n3
2
− 3n0n4
n2
2
?−1
. (5.28)
A crucial role is played by the function ρ(ψ1) which we plot in Figure 5. It decreases
monotonically from 1 at ψ1= 0 to zero at ψ1=√3. Its asymptotic behavior at ψ1= 0
and ψ1=√3 is:
ρ ∼ (√3 − ψ1)3for ψ1→
ρ ∼ 1 − ˜ cψ2
1for ψ1→ 0 ,
√3 ,(5.29)
for some constant ˜ c. This is in agreement with (5.24), (5.25). The fact that ρ vanishes at
the same point, ψ1=√3, where the solution develops a singularity, is strongly supported
by our numerical analysis, and will be crucial in reproducing the field theory results.
1
2
1
3
2
3
Ψ1
1
2
1
Ρ
Figure 5: A plot of the function ρ(ψ1) in (5.26).
5.3A phase transition
As in the N = 1 case, consider for simplicity the case n4= 0 and call, as usual, n6= N
and n2= k. Equation (5.26) becomes
?
As in the N = 1 case, there are two interesting regimes. For N ? k3/n2
we are near the undeformed solution. From (5.29), we see that 1 − ρ(ψ1) ∼ ψ2
ρ(ψ1) =1 + 3n2
0N
k3
?−2
.(5.30)
0, ψ1→ 0 and
1; hence we
26
Page 28
can identify in this regime
ψ1∼
?n2
0N
k3
?1/2
. (5.31)
Moreover, we see from (5.23) that f0(ψ1) ∼ ψ1; using also (5.24), we easily compute from
(5.27)
l ∼N1/4
which is indeed the behavior of the N = 6 solution [9].
For N ? k3/n2
√3. From (5.29) and (5.30), we see that
k1/4,gs∼N1/4
k5/4,(5.32)
0, the function ρ(ψ1) should approach zero, and this happens for ψ1→
δψ1≡
?√3 − ψ1
?
∼
?
k3
Nn2
0
?2/3
. (5.33)
From (5.25) we then conclude
l ∼
N1/6
|n0|1/6,gs∼
1
N1/6|n0|5/6, (5.34)
which is the same behavior as in (3.22). Again, as in the N = 1 case, we can also argue
generally that gsremains bounded for any integer values of the fluxes, with n0?= 0.
At first sight, this seems puzzling. At the end of section 4.2, we noticed that this
gravity solution is expected to develop light D–branes in the limit N1= N2= N → ∞.
As argued in [18], N1= N2precisely when n4= 0 (see also (5.35)). But there do not
seem to be any light D–branes in a limit where gsis small and the internal manifold is
large, since a D–brane mass scales as Lk/gs, with k ≥ 0.
As we remarked after equation (5.16), however, in the limit ψ →√3 (which is relevant
for large N) the internal manifold develops two conifold–like singularities, since the two-
cycle is now shrinking to zero at the “poles”. As we will now see, the new light states are
obtained from D-branes wrapping the vanishing cycle for that singularity.
5.4 Probes
We now want to compare this gravity solution with the field theory we saw in section 4.2;
specifically, the one defined by the superpotential in (4.9), which has the right symmetries
to be the dual of the gravity solution we found in section 5.1.
We can first of all try to predict what sort of bulk field corresponds to the monopole
operators discussed in section 4.2. Let us recall how the duality works in the ABJM case,
27
Page 29
when F0= 0. Consider first a monopole operator that creates one unit of field strength
for both gauge groups at a particular point. This operator has k indices under both gauge
groups, and we can make it gauge-invariant by contracting it with k bifundamentals. The
resulting bound state corresponds to a D0 brane in the gravity dual; notice that such
a brane has no tadpole on its worldsheet for the worldsheet vector potential A, as we
already saw in section 3.4. Another monopole operator that can be considered is the one
that creates one unit of flux for, say, the second gauge group. In this case, we cannot
make this operator gauge–invariant: it will have k “dangling” indices. This corresponds
to a D2 brane wrapping an internal two–cycle. As we also already saw in section 3.4, such
a brane has a tadpole on its worldsheet, coming from the term?A1F =?F2A; so one
needs to have k strings ending on it, and these k strings correspond to the k indices of
the monopole operator.
When we switch on F0, even a D0 brane will have a tadpole on its worldsheet, coming
from the coupling?F0A. On the field theory side, this corresponds to the fact that the
monopole operator that creates one unit of field strength for both gauge groups has now
k1 fundamental indices for the first gauge group and k2 antifundamental ones for the
second. This cannot be made gauge-invariant; we are always left with at least |k1− k2|
“dangling” indices. This fact was used in [12] to establish that the Romans mass integer
is the sum of the Chern–Simons couplings, so that, in the present language, n0= k1−k2;
see also [30,31]. In [18] it was similarly shown that n4is the difference between the two
gauge group ranks N2− N1.5Putting this together, we obtain a dictionary between the
flux integers and the ranks and levels of the field theory:
n0= k1− k2,n2= k2,n4= N2− N1,n6= N2. (5.35)
In section 4.2, we considered monopole operators which create k2units of field strength
for the first gauge group, and k1units of field strength for the second. We noticed that
these have k1k2bifundamental indices, and thus can be made gauge–invariant. Following
the identifications of D2 branes and D0 branes above, if we assume for example that
k1> k2, we can say that these new gauge–invariant monopoles correspond to a bound
state k2D0 branes and k1−k2D2 branes. We have already noticed in section 3.4 that such
a bound state can cancel the tadpole on the worldsheet, because it makes the prefactor
in (3.24) vanish.
Let us make this expectation more precise. Consider a D2 brane wrapped on a two–
cycle B2in the N = 2 solution. As we will see in appendix B.2, supersymmetry requires
5The relative sign between the expressions for n4and n0had not been determined so far. We made
here a choice consistent with our final result in formula (5.43).
28
Page 30
that the D2 brane lives at the North pole t = 0 or at the South pole t = π/2, and that
it wraps the S2that does not shrink there. We also need to cancel the tadpole for the
world-volume field A which arises from the Wess-Zumino coupling,
A ∧ (F2+ F0(F − B)) .
We can split B into a fiducial choice plus a zero mode, as in (5.21). The tadpole cancel-
lation requires
F − B = F − β − B0= −F2/F0.
Since B0was chosen to satisfy (5.20), we need to turn on a world–volume flux
(5.36)
(5.37)
F = β . (5.38)
There is a possible obstruction to doing this, coming from the quantization of the world–
volume flux, that says that
(2πls)2
is given by b = −n2/n0; hence in general it is rational and not an integer. So we see
that a single D2 brane is generally not consistent. We can get around this, however, by
considering n0D2–branes. In that case, the equation we want to satisfy actually reads
1
?
S2F ∈ Z. The value of b =
1
(2πls)2
?
S2β from (5.22)
F = β 1n0. (5.39)
The integral of the trace of the left hand side is the first Chern class on the world–volume,
which is the induced D0-brane charge n0. The integral of the trace of the right hand side
now gives bn0= −n2. We conclude that we can cancel the tadpole by considering a bound
state of n0D2 branes and n2D0 branes, just as in section 3.4.
Naively, one might think that the mass of a D2–D0 bound state should be at least as
heavy as a D0-brane, which in units of AdS mass is mD0L ∼ L/gs. Since this is heavy in
the limit (5.34), one might think that such a bound state can never reproduce the light
mass predicted in section 4.2.
Fortunately, such pessimism proves to be unfounded. The mass of the state is given
by
mD2L = n0L
1
(2π)2gsl3
s
?
B2
?
det(g + F − B) = n0L
1
(2π)2gsl3
s
?
B2
?
det
?
g −F2
F0
?
(5.40)
.
where we used the tadpole cancellation condition. We will take the cycle B2 to be a
representative of the non–trivial cycle, which is the diagonal of the two S2’s. Using the
explicit form for the metric in (5.4), as well as (5.12), (5.18) and (A.12), we get:
??e2B1
mD2L = 4π
n0L
(2π)2gsl3
s
4
?2
+
?k2e2B1
F0
?2
= 2π2n0l3
gs
w0(ψ1)
?
1 + ψ2
ψ2
1
1
. (5.41)
29
Page 31
The fact that the two expressions under the square root are proportional is related to the
BPS condition, as discussed in appendix B.2.
Inserting the values of l and gsfrom (5.27),(5.28) for generic fluxes we obtain
mD2L =
?n2
2
2− n0n4
?2π2w0(ψ1)
f4(ψ1)
?
1 + ψ2
ψ2
1
1
. (5.42)
Quite remarkably, the function of ψ1 in the previous formula, which can be computed
numerically, turns out to be constant with value 1. The final result for the mass formula
is then
?n2
Upon using the dictionary (5.35), this formula is identical to the field theory prediction
(4.13) in the limit where n0 ? n2. This is exactly the limit where we can trust the
supergravity solution, since, as shown in (5.27), for a generic value of ψ1, L is large only
if n2? n0. In this limit, it is also true that the dimension of the corresponding operator
is given by ∆ ∼ mD2L.
In contrast with the N = 1 results in section 3.4, and with the naive expectation
expressed earlier, we see from (5.43) that the mass of the bound state remains finite also
in the limit N ? k3/n2
of the constituent D0-brane, leaving a smaller piece that is proportional to the volume of
the shrinking S2. These are precisely the new light states that we had predicted to exist
from the field theory analysis in section 4.2.
mD2L =
2
2− n0n4
?
. (5.43)
0. A contribution from the B field cancels the large mass ∼ L/gs
Acknowledgements
O. A. is supported in part by the Israel–U.S. Binational Science Foundation, by a research
center supported by the Israel Science Foundation (grant number 1468/06), by a grant
(DIP H52) of the German Israel Project Cooperation, and by the Minerva foundation with
funding from the Federal German Ministry for Education and Research. D. J. wishes
to acknowledge funding provided by the Association of Members of the Institute for
Advanced Study. A. T. and A. Z. are supported in part by INFN and MIUR under
contract 2007–5ATT78–002.
30
Page 32
ASupersymmetry equations and pure spinors for
the N = 2 solution
We will give in this section more details about the N = 2 solution we found in section 5.
The supersymmetry parameters for compactifications of the form AdS4× M6 (or
Minkowski4× M6) decompose as
?
N
?
Here, N is the number of supersymmetries. The subscripts ± denote positive and negative
chirality spinors, in four and six dimensions; the negative chirality spinors are conjugate
to the positive chirality ones,
?1=
N
a=1
ζa
+⊗ η1a
++ ζa
−⊗ η1a
−,
?2=
a=1
ζa
−⊗ η2a
++ ζa
+⊗ η2a
−.
(A.1)
ζa
−= (ζa
+)∗,ηia
−= (ηia
+)∗. (A.2)
For each a, ζa
elements of this basis to be “Killing spinors”, which means that Dµζ+=µ
with i = 1,2, are a priori independent six–dimensional Weyl spinors. In this section, we
will consider N = 2.
A priori, one could have taken the ζain ?1and ?2to be different. This can indeed
be done for compactifications with vanishing RR flux; for example, for the usual N = 2
Calabi–Yau compactifications. To recover that case in (A.1), one can take for example
η21= η12= 0, and keep a non–vanishing η11and η22. However, in compactifications where
RR fluxes are present, the ζain ?1and ?2are required to be equal, up to a constant that
can be reabsorbed in the ηia. Hence (A.1) describes all possible N = 2 compactifications,
and is particularly appropriate for vacua with RR fluxes.
+can vary among a basis of four–dimensional Weyl spinors; we will take the
2γµζ−. The ηia
+,
Using (A.1) in the supersymmetry equations yields equations for the internal spinors
ηia. In fact, these equations do not mix the ηi1with the ηi2. In what follows, we will first
analyze the equations of the ηi1≡ ηi; we will come back to the second pair later.
We will construct a pair of pure spinors as tensor products of the supersymmetry
parameters6η1and η2
Φ±= η1
+⊗ η2†
±. (A.3)
6As usual, we left implicit a Clifford map on the left hand side, that sends dxm→ γm.
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Page 33
The type IIA supersymmetry conditions can be expressed as [32]:
(d − H∧)(eA−ϕRe (Φ−)) = 0 ,
(d − H∧)(e3A−ϕIm (Φ−)) = −3e2A−ϕµIm (Φ+) +e4A
(d − H∧)(e2A−ϕΦ+) = −2µeA−ϕRe (Φ−) ;
||Φ+|| = ||Φ−|| = eA.
(A.4a)
8
∗ λ(F) , (A.4b)
(A.4c)
(A.4d)
Here, F are the internal fluxes (which determine also the external fluxes, by self–duality).
A is the warping function, defined as ds2
in ds2
has no independent meaning, since one can reabsorb it in A. We have normalized µ = 2
in this paper. The symbol λ acts on a k–form by multiplying it by the sign (−)Int(k/2).
Finally, the norm in (A.4d) is defined as ||A||2= i(A ∧ λ(¯A))6.
The metric (5.4) can be written in terms of the vielbein
10= e2Ads2
AdS4+ ds2
6. The cosmological constant
AdS4is given by Λ = −3|µ|2. Since A is non–constant in the solution, however, this Λ
E1=
1
2√2?dt +i
?eB1sin(t)e−ia/2e1− eB2cos(t)eia/2e2
E3=1
2
8Γ(da + A2− A1),
E2=1
2
?,
?,
?eB1cos(t)e−ia/2e1+ eB2sin(t)eia/2e2
(A.5)
where ei= dθi+isinθidφiare the natural one–forms on the spheres S2
and eB2= sin(t) we recover the Fubini–Study metric of CP3with natural K¨ ahler form
J =i
2
(5.31)]). It is also convenient to use the forms
i. For eB1= cos(t)
?3
i=1Ei∧¯Eiand natural three form section Ω = E1∧E2∧E3(see for example [28,
Ji= dAi=i
2ei∧ ¯ ei (not summed) ,o ≡i
2eiae2∧ ¯ e1; (A.6)
the Jiwere already defined in (5.3). These forms satisfy
dJi= 0 ,do = i(da + A2− A1) ∧ o ,o ∧ ¯ o = −J1∧ J2. (A.7)
The generic pure spinors corresponding to an SU(3)×SU(3) structure can be written
in terms of the “dielectric Ansatz”
?
Φ−= −i
Φ+=i
8cos(2ψ)eA+iθexp
−i
?
?
j
cos(2ψ)+i
2z ∧ ¯ z
?
+ tan(2ψ)Re(ω)
?
,
,
8sin(2ψ)eA+iθz ∧ exp
−cot(2ψ)Re(ω) −
i
sin(2ψ)Im(ω)
?
(A.8)
32
Page 34
where θ and ψ are two new angular variables; one can see easily that the supersymmetry
equations (A.4) relate them by
tan(θ) = −cot(2t) sin(2ψ) .(A.9)
The one–form z and the two–forms j and ω can also be used to describe an SU(2) structure
on M6. For our solution, these forms are given by
z = −ie−iθE1,
j =i
2
ω = iE2∧¯E3.
?E2∧¯E2+¯E3∧ E3
?
,
(A.10)
We can also characterize j and ω in terms of the forms in (A.6):
−Re(ω)
j
=1
4
cos(2t) −sin(2t)
sin(2t)cos(2t)
−e2B1J1+ e2B2J2
2eB1+B2Re(o)
, Im(ω) =1
2eB1+B2Im(o) .
(A.11)
The RR fluxes are determined to be as in equation (5.12) with
k2=ce−4A
2w1
sec(2ψ)
cos(2t)(2Ct,ψ+ w1),
sec(2ψ)
cos(2t)(2Ct,ψ+ w2),g2= −ce−4A
˜k2= 2ce−4A
2w2
w1w2
(2Ct,ψ(w1+ w2) + 3w1w2),
k4= −ce−4A
˜k4=ce−4A
2w2
˜ g4= −ce−4A
k6= 6ce−4A,
4w1w2
sin(2ψ)
sin(4t)cos2(2ψ)(2Ct,ψ(w1+ w2) + w1w2),
tan(2ψ)
sin(2t)(2Ct,ψ+ 3w2),
tan(2ψ)
sin(2t)(2Ct,ψ+ 3w1),
2w1
(A.12)
where Ct,ψwas defined in (5.7). Recall also that one possible choice of NS–NS field that
satisfies the equations of motion is B0= F2/F0, as in (5.20).
So far we have described the solution as if it were an N = 1 solution: we have
only paid attention to the a = 1 part of (A.1). To show that the solution actually has
N = 2 supersymmetry, we have to provide a second pair of spinors, ηi2, that satisfies
33
Page 35
the equations of motion for supersymmetry with the same expectation values for all the
fields. In terms of pure spinors, we can now form the bilinears
˜Φ±= η12
+⊗ η22†
±
(A.13)
and require that they solve again the equations (A.4), with the same values of the fluxes
and the same metric.
In fact, one expects the two solutions Φ and˜Φ to be rotated by R–symmetry, so
that there is actually a U(1)’s worth of solutions to (A.4). To see this U(1), rotate the
two–form o in (A.6) by a phase:7
o → e−iαo ≡ oα. (A.14)
We can correspondingly define a pair of pure spinors Φα
appears. The crucial fact about the rotation of o in (A.14) is that it keeps its differential
properties (A.7) unchanged: namely, doα= i(da + A2− A1) ∧ oα. Because of this fact,
the computations to check (A.4) do not depend on α; and, since we checked already that
α = 0 gives a solution, it follows that any Φα
solution with different fluxes; but we can see from (5.12) that o never appears in Fk. We
conclude, then, that the solution we have found is an N = 2 solution.
±, by changing o → oαwherever it
±is a solution. A priori, this could be a
B BPS particles
In this section, we will give a general analysis of BPS particles in flux compactifications
(subsection B.1), and we will then apply those general results to the N = 2 background
described in section 5 and appendix A.
B.1 General considerations
We will start with some general considerations about BPS states in N = 2 backgrounds
with fluxes. These will in general be states that are left invariant by a certain subalgebra
of the supersymmetry algebra. This subalgebra is in general defined by the fact that the
two supersymmetry parameters ?iare related:
Γ??2= ?1. (B.1)
7Alternatively, one can change the vielbeine (A.5) by translating a → a + α.
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Page 36
In first approximation, Γ?is the product of the gamma matrices parallel to the brane.
When B fields or worldsheet fluxes F are present, Γ?receives additional contributions
of eF−B. We will give a definition later on, in the context needed for this paper; for
the general and explicit expression, see for example [33, Eq. (3.3)]. For an AdS4× M6
compactification, we would like to use the decomposition (A.1). For particles, this will
lead to an equation involving the four–dimensional spinors ζi
follows, the index0is meant to be a frame index. To have a chance to solve the resulting
equations, we need to postulate a relation between these spinors. One can write for
example
γ0ζa
±and γ0ζa
±; here and in what
+= Aabζb
−, (B.2)
for some matrix A. (Recall that in general the index a runs from 1 to N; for us, N = 2,
and so a = 1,2.) In fact, (B.2) is almost the most general choice one can make, compatibly
with the symmetries of the problem. The only generalization one could make would be
to multiply the left-hand side by another matrix Bab. Whenever this matrix is invertible,
one can reabsorb it by a redefinition of Aab. In this sense, we can say that (B.2) is the
“generic” Ansatz for a BPS particle.
The matrix Aabin (B.2) needs to satisfy certain conditions. Let us work for simplicity
in a basis where all the space–time gamma matrices γµ, µ = 0,...,3 are real, and the
internal γm, m = 1,...,6, are purely imaginary; the ten–dimensional gamma matrices
are then given as usual by
Γµ= eAγµ⊗ 1 ,Γm+3= γ5⊗ γm. (B.3)
It follows from these definitions that Γµare real. Let us now conjugate (B.2); using (A.2),
the fact that γ0is real, and that γ2
0= −1, we get
AabAbc= −δac.(B.4)
If we were considering an N = 1 background, Aabwould be a one–by–one matrix, and
(B.4) would have no solution. This is just what one would expect: there are no BPS
particles in a N = 1 background. For N = 2, one choice that satisfies (B.4) is
A = e−iλ
01
−10
. (B.5)
We can now use (B.3) to write
Γ?= γ0⊗ γ?, (B.6)
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Page 37
where γ?is now an element of the internal Clifford algebra; it contains the product of
all the internal gamma matrices parallel to the brane, plus additional contributions from
the worldsheet flux and B-field. Let Bp⊂ M6be the p–cycle wrapped by the brane, of
dimension p and with coordinates σα, α = 1,...,p. Then we define the natural volume
form on B to be
volB≡
One can also define similarly an “inverse volume form” as the multivector
?
det(g + F − B)dσ1∧ ... ∧ dσp. (B.7)
vol−1
B=
∂1∧ ... ∧ ∂p
?det(g + F − B)
, (B.8)
which is a section of Λp(TB). This multivector can be used to give an intrinsic definition
of γ?: here is how. We can define eFvol−1to be the multivector of mixed degree that one
obtains by contracting the indices of the form eFwith the multi–vector vol−1. Recall now
that multivectors can be “pushed forward”: if we call x : B ?→ M6the embedding map,
with components xm(σ), then x∗(eFvol−1) is a multivector in M6, obtained by contracting
all indices α on B with the tensor ∂αxm. In fact:
γ?= x∗(eF−Bvol−1
B) . (B.9)
Here, we left implicit on the right hand side a Clifford map that sends a vector ∂minto a
gamma matrix γm. We already used this map on forms (see footnote 6). One can show
that γ?is unitary:
γ†
?γ?= 1 .(B.10)
For a more explicit expression of γ?, see [33, Eq. (3.5)].
If we now use (B.2), (B.6) and (A.1) in (B.1), we get
γ?η2a
+= (A−1)baη1b
+. (B.11)
For our choice (B.5), this reads
γ?η22
γ?η21
+= −eiλη11
+= eiλη12
+, (B.12a)
+. (B.12b)
We are now left with solving (B.12), which are two purely internal equations. Each of
the two equations is formally identical to others that have already appeared [33] in the
context of BPS objects which do exist in N = 1 flux compactifications: branes which
extend along the time direction, plus one, two or three space directions. Hence we can
36
Page 38
simply follow the same steps; we will now summarize that procedure for (B.12a), and
then apply the result to (B.12b).
Let us first define the new pure spinors
Ψ+≡ η11
+⊗ η22†
±
; (B.13)
notice that these are different from the pure spinors Φ±, defined in (A.3), which entered
the supersymmetry equations (A.4). In (A.3), η1and η2were to be understood as η1a
and η2a, for a either 1 or 2. In (B.13), we are mixing a = 1 with a = 2.
A possible basis for the space of spinors of positive chirality is given by η11
Three linear combinations of the γmmake γmη11
γi, where i is a holomorphic index with respect to an almost complex structure I. Ex-
plicitly we have η11†
expand
γ?η22
The coefficients a and bm have a geometrical interpretation.
multiply (B.14) from the left by η11†
From the formula
Tr(? A? B†) =8
k!Am1...mk¯Bm1...mk,
we see that Tr(γ?Ψ†
(B.9), we see that γ?contains factors of ∂αxm; when contracting with¯Ψ+, these factors
reconstruct a pull–back of that form. In conclusion we get8
+and γmη11
−.
−vanish: they are its three “annihilators”
+ γmγnη11
+= (1+iI)mn≡ 2¯Πmn. In terms of this basis a priori one can
+= aη11
++ bmγmη11
−.(B.14)
To compute a, we can
+η11†
+ ; we get aeA= η11†
+ γ?η22
+= Tr(γ?η22
+ ) = Tr(γ?Ψ†
+).
(B.15)
+) consists of contracting the free indices of γ?with those of¯Ψ+. From
(eF−B¯Ψ+)|B=a
8eAvolB, (B.16)
where |Bdenotes the top–form part on B of the pull–back. By similarly multiplying (B.14)
from the left by η11†
− γn, we get
(dxm· eF−BΨ−)|B= −1
4bnΠnmeAvolB, (B.17)
where · denotes the Clifford product: v· = v ∧ +v?. Here, ∂m?(dxm1∧ ... ∧ dxmp) ≡
pδm[m1dxm2∧ ...dxmp].
We can now go back to (B.12a). Comparing to the expansion (B.14), we get
a = −eiλ,bm= 0 . (B.18)
8The factor of eAcomes from the fact that ∀a,i, ||ηia|| = eA/2, which follows from (A.4).
37
Page 39
Using the geometrical interpretations (B.16) and (B.17), we get
Re(−e−iλeF−BΨ+)|B=1
8eAvolB, (B.19)
and
Im(e−iλeF−BΨ+)|B= 0,
(v · eF−BΨ−)|B= 0 .
(B.20a)
(B.20b)
Actually, one can show that (B.19) is equivalent to the system (B.20). To see this,
observe that γ?is unitary, as we saw in (B.10). This implies that γ?η22
same norm as η22
+should have the
+. Since all the spinors have norm eA(see footnote 8), it follows that
|a|2+ 2bm¯bm= 1 .(B.21)
This means that imposing Re(a) = 1 is equivalent to imposing Im(a) = 0 and bm= 0.
Recalling (B.16) and (B.17), we get our claim that (B.19) is equivalent to (B.20).
This completes our analysis of (B.12a) (along the lines of [33]). For (B.12b), similar
considerations apply; we obtain
Re(e−iλeF−B˜Ψ+)|B=1
8eAvolB, (B.22)
and
Im(e−iλeF−B˜Ψ+)|B= 0,
(v · eF−B˜Ψ−)|B= 0 ,
(B.23a)
(B.23b)
for the pure spinors
˜Ψ±≡ η12
+⊗ η21†
±
. (B.24)
Let us now summarize this section: we have shown that a brane wrapping an internal
cycle B, and extended along the time direction, is BPS if and only if (B.19) (or equivalently
(B.20)) is satisfied by Ψ and, analogously, (B.22) (or equivalently (B.23)) is satisfied by
˜Ψ, where Ψ and˜Ψ are defined respectively in (B.13) and (B.24). We will now compute
these pure spinors for the solution described in section 5 and in appendix A.
B.2 D2/D0 bound states in the N = 2 solution
As discussed in the previous section, in order to study the supersymmetry of BPS particles
obtained from wrapped branes, we need to form bilinears in the supersymmetry spinors
38
Page 40
ηia
given by the pair of spinors defining the SU(3) × SU(3) structure in (A.10). Recall that
a SU(3) structure is specified by two invariant tensors (J,Ω) or, equivalently, by a spinor
η+(of norm 1) such that
+. We first need to write them explicitly. A convenient basis to expand our spinors is
η+⊗ η†
η+⊗ η†
+=1
8e−iJ,
−= −i
8Ω .
(B.25)
The SU(3) × SU(3) structure in (A.10) can be seen as the intersection of two SU(3)
structures given by (J1,Ω1) = (j +i
call the corresponding spinors η+and χ+. They are related by χ+=
denotes the Clifford multiplication by the one-form zmγm. We will need in the following
an expression for the tensor products of a generic linear combination
2z∧ ¯ z,ω∧z) and (J2,Ω2) = (−j +i
2z∧ ¯ z,−¯ ω∧z). We
1
√2z · η−, where z·
µ+= aη++ bχ+,
ν+= xη++ y χ+.
(B.26)
This is given by [34]
µ+⊗ ν†
µ+⊗ ν†
+=1
8
?
?
a¯ xe−ij+ b¯ yeij− i(a¯ yω + ¯ xb¯ ω)
i(by¯ ω − axω) + (bxeij− aye−ij)
?
e1/2z¯ z,
?
−=1
8
z .
(B.27)
We can choose the spinors for the first supersymmetry as follows
η11
+= ieA/2+iθ(eiπ
η21
4cos(ψ)η+− ie−iπ
4cos(ψ)η++ ie−iπ
4sin(ψ)χ+) ,
+= ieA/2(eiπ
4sin(ψ)χ+) .
(B.28)
It is easy to reproduce, using formula (B.27), the dielectric ansatz (A.8) for the pure
spinors .
As discussed in appendix A, there is a U(1) family of supersymmetries obtained by
rotating o → oα= e−iαo. We can conveniently choose as a second independent supersym-
metry the one with oπ= −o. This is defined by
η12
+= ieA/2+iθ(eiπ
η22
4cos(ψ) ˜ η+− ie−iπ
4cos(ψ) ˜ η++ ie−iπ
4sin(ψ) ˜ χ+) ,
+= ieA/2(eiπ
4sin(ψ) ˜ χ+) ,
(B.29)
where
˜ η+= −icos(2t)η++ isin(2t)χ+,
˜ χ+= isin(2t)η++ icos(2t)χ+.
(B.30)
39
Page 41
This reproduces the rotated pure spinors Φπ
With these ingredients, we can compute the spinors Ψ±and˜Ψ±defined in (B.13) and
(B.24) and check the BPS conditions for a D2-brane. It is easy to see that the D2-brane
considered in section 5, which wraps the diagonal S2and sits at the North or South pole,
is indeed supersymmetric. Let us consider, for definiteness, the North pole. At t = 0,
ψ = 0 and we see that ηi2
±.
+= −iηi1
+. As a consequence, at t = 0,
Ψ±= ±iΦ±,
˜Ψ±= −iΦ±, (B.31)
and we are reduced to check expressions for the pure spinors Φ±at the North pole. Taking
into account that ψ = 0 there, we have
Φ+|t=0=i
Φ−|t=0=i
8eA+iθe−iJ,
8eA+iθz ∧ ω .
(B.32)
The condition (B.20b) for Ψ−(and the analogous (B.23b) for˜Ψ−) gets contributions only
from the contraction with the vector z and it is automatically satisfied because ω vanishes
at the North pole, t = 0. It is easily seen that the conditions for Ψ+and˜Ψ+are equivalent
and it is enough to analyze those for Ψ+. Equation (B.20a) reads
Im?ei(θ−λ)e−ij?∧ eF−B|B2= 0,
and determines the world-volume field
(B.33)
F = (B + cot(θ − λ)j)|B2. (B.34)
We see that a wrapped D2 brane can be made supersymmetric by choosing an appropriate
world-volume field. However, as discussed in section 5.4, to have a consistent BPS state we
need to impose the quantization of the world-volume field and the cancellation of tadpoles.
As discussed there, the quantization condition requires to take n0D2-branes. On the other
hand, the tadpole condition requires F = β or, equivalently, F −B = −F2/F0. At t = 0,
using the explicit form for the metric in (5.4), as well as (5.12), (5.18), (A.10), (A.9) and
(A.12), we evaluate tan(θ) = −ψ1and j = −1
that J1is the volume form of one of the two S2’s, as defined in (5.3). We thus see that
the tadpole condition is satisfied by λ = 0. The mass of n0D2 branes is then obtained
by integrating the volume form in (B.19)
4e2B1J1and F2/F0= −1
4e2B1J1/ψ1. Recall
n0
?
B2
?
det(g + F − B) = n0
1
sin(θ)
?
B2
J = n01
4e2B1
?
1 + ψ2
ψ1
1
?
B2
J1.(B.35)
40
Page 42
Using this, one exactly reproduces the result (5.41) of section 5.4.
A more detailed analysis of equations (B.20) (and the analogous ones for˜Ψ±) shows
that a D2-brane sitting at t ?= 0,π/2 cannot be supersymmetric and simultaneously satisfy
the tadpole condition.
References
[1] L. J. Romans, “Massive N=2a Supergravity in Ten Dimensions,” Phys. Lett. B169
(1986) 374.
[2] A. Sagnotti and T. N. Tomaras, “Properties of Eleven–Dimensional Supergravity,”.
CALT-68-885.
[3] K. Bautier, S. Deser, M. Henneaux, and D. Seminara, “No cosmological D = 11
supergravity,” Phys. Lett. B406 (1997) 49–53, hep-th/9704131.
[4] S. Deser, “Uniqueness of D = 11 supergravity,” hep-th/9712064.
[5] J. Polchinski and E. Witten, “Evidence for Heterotic – Type I String Duality,” Nucl.
Phys. B460 (1996) 525–540, hep-th/9510169.
[6] B. E. W. Nilsson and C. N. Pope, “Hopf fibration of eleven-dimensional supergravity,”
Class. Quant. Grav. 1 (1984) 499.
[7] S. Watamura, “Spontaneous compactification and CPN: SU(3) × SU(2) × U(1),
sin2(θW), g3/g2 and SU(3) triplet chiral fermions in four dimensions,” Phys. Lett.
B136 (1984) 245.
[8] D. P. Sorokin, V. I. Tkach, and D. V. Volkov, “On the relationship between compact-
ified vacua of d = 11 and d = 10 supergravities,” Phys. Lett. B161 (1985) 301–306.
[9] O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “N=6 superconformal
Chern-Simons-matter theories, M2-branes and their gravity duals,” JHEP 10 (2008)
091, 0806.1218.
[10] A. Tomasiello, “New string vacua from twistor spaces,” Phys. Rev. D78 (2008)
046007, 0712.1396.
[11] P. Koerber, D. L¨ ust, and D. Tsimpis, “Type IIA AdS4compactifications on cosets,
interpolations and domain walls,” JHEP 07 (2008) 017, 0804.0614.
41
Page 43
[12] D. Gaiotto and A. Tomasiello, “The gauge dual of Romans mass,” JHEP 01 (2010)
015, 0901.0969.
[13] M. Petrini and A. Zaffaroni, “N = 2 solutions of massive type IIA and their Chern–
Simons duals,” JHEP 09 (2009) 107, 0904.4915.
[14] C. M. Hull, “Massive string theories from M–theory and F–theory,” JHEP 11 (1998)
027, hep-th/9811021.
[15] T. Sakai and S. Sugimoto, “Low energy hadron physics in holographic QCD,” Prog.
Theor. Phys. 113 (2005) 843–882, hep-th/0412141.
[16] B. A. Burrington, V. S. Kaplunovsky, and J. Sonnenschein, “Localized Backreacted
Flavor Branes in Holographic QCD,” JHEP 02 (2008) 001, 0708.1234.
[17] K. Behrndt and M. Cvetic, “General N = 1 supersymmetric fluxes in massive type
IIA string theory,” Nucl. Phys. B708 (2005) 45–71, hep-th/0407263.
[18] O. Aharony, O. Bergman, and D. L. Jafferis, “Fractional M2-branes,” JHEP 11
(2008) 043, 0807.4924.
[19] D. Gaiotto and D. L. Jafferis, “Notes on adding D6 branes wrapping RP3in AdS4×
CP3,” 0903.2175.
[20] M. K. Benna, I. R. Klebanov, and T. Klose, “Charges of Monopole Operators in
Chern–Simons Yang–Mills Theory,” JHEP 01 (2010) 110, 0906.3008.
[21] D. L. Jafferis, “Quantum corrections to N = 2 Chern–Simons theories with flavor
and their AdS4duals,” 0911.4324.
[22] F. Benini, C. Closset, and S. Cremonesi, “Chiral flavors and M2-branes at toric CY4
singularities,” JHEP 02 (2010) 036, 0911.4127.
[23] S. Kim and K. Madhu, “Aspects of monopole operators in N = 6 Chern–Simons
theory,” JHEP 12 (2009) 018, 0906.4751.
[24] D. L. Jafferis and A. Tomasiello, “A simple class of N = 3 gauge/gravity duals,”
JHEP 10 (2008) 101, 0808.0864.
[25] D. Martelli and J. Sparks, “Moduli spaces of Chern–Simons quiver gauge theories
and AdS4/CFT3,” Phys. Rev. D78 (2008) 126005, 0808.0912.
42
Page 44
[26] A. Hanany and A. Zaffaroni, “Tilings, Chern–Simons Theories and M2 Branes,”
JHEP 10 (2008) 111, 0808.1244.
[27] A. Kapustin, “Wilson–’t Hooft operators in four-dimensional gauge theories and S–
duality,” Phys. Rev. D74 (2006) 025005, hep-th/0501015.
[28] D. Gaiotto and A. Tomasiello, “Perturbing gauge/gravity duals by a Romans mass,”
J. Phys. A42 (2009) 465205, 0904.3959.
[29] M. Cvetic, H. Lu, and C. N. Pope, “Consistent warped–space Kaluza–Klein re-
ductions, half–maximal gauged supergravities and CPnconstructions,” Nucl. Phys.
B597 (2001) 172–196, hep-th/0007109.
[30] M. Fujita, W. Li, S. Ryu, and T. Takayanagi, “Fractional Quantum Hall Effect via
Holography: Chern–Simons, Edge States, and Hierarchy,” JHEP 06 (2009) 066,
0901.0924.
[31] O. Bergman and G. Lifschytz, “Branes and massive IIA duals of 3d CFT’s,” JHEP
04 (2010) 114, 1001.0394.
[32] M. Gra˜ na, R. Minasian, M. Petrini, and A. Tomasiello, “A scan for new N = 1 vacua
on twisted tori,” JHEP 05 (2007) 031, hep-th/0609124.
[33] L. Martucci and P. Smyth, “Supersymmetric D–branes and calibrations on general
N = 1 backgrounds,” JHEP 11 (2005) 048, hep-th/0507099.
[34] R. Minasian, M. Petrini, and A. Zaffaroni, “Gravity duals to deformed SYM theories
and generalized complex geometry,” JHEP 12 (2006) 055, hep-th/0606257.
43
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