UG-FT-216/07
CAFPE-86/07
FFUOV-07/03
PUPT-2231
April 2007
A KK-monopole giant graviton in AdS5 × Y5
Bert Janssena,1, Yolanda Lozanob,2 and Diego Rodŕıguez-Gómezc,3
a Departamento de F́ısica Teórica y del Cosmos and
Centro Andaluz de F́ısica de Part́ıculas Elementales
Universidad de Granada, 18071 Granada, Spain
b Departamento de F́ısica, Universidad de Oviedo,
Avda. Calvo Sotelo 18, 33007 Oviedo, Spain
c Department of Physics, Princeton University,
Princeton, NJ 08540, U.S.A.
ABSTRACT
We construct a new giant graviton solution in AdS5 × Y5, with Y5 a quasi-regular Sasaki-Einstein
manifold, consisting on a Kaluza-Klein monopole wrapped around the Y5 and with its Taub-NUT
direction in AdS5. We find that this configuration has minimal energy when put in the centre
of AdS5, where it behaves as a massless particle. When we take Y5 to be S
5, we provide a
microscopical description in terms of multiple gravitational waves expanding into the fuzzy S5
defined as an S1 bundle over the fuzzy CP 2. Finally we provide a possible field theory dual
interpretation of the construction.
1E-mail address: bjanssen@ugr.es
2E-mail address: yolanda@string1.ciencias.uniovi.es
3E-mail address: drodrigu@princeton.edu
http://arxiv.org/abs/0704.1438v3
1 Introduction
As it is well-known, giant gravitons are stable brane configurations with non-zero angular momen-
tum, that are wrapped on (n− 2)- or (m− 2)-spheres in AdSm×Sn spacetimes and carry a dipole
moment with respect to the background gauge potential [1, 2, 3, 4]. They are not topologically
stable, but are at dynamical equilibrium because the contraction due to the tension of the brane
is precisely cancelled by the expansion due to the coupling of the angular momentum to the back-
ground flux field. These spherical brane configurations turn out to be massless, conserve the same
number of supersymmetries and carry the same quantum numbers of a graviton.
Giant graviton configurations were first proposed as a way to satisfy the stringy exclusion
principle implied by the AdS/CFT correspondence [1]. The spherical (n− 2)-brane expands into
the Sn part of the geometry with a radius proportional to its angular momentum. Since this
radius is bounded by the radius of the Sn, the configuration has associated a maximum angular
momentum. The (m − 2)-brane giant graviton configurations [2, 3], on the other hand, expand
into the AdSm part of the geometry, and they do not satisfy the stringy exclusion principle. For
a discussion on the degeneracy of these two types of giant gravitons and the point-like graviton,
we refer for instance to [5] and references therein.
The construction of giant gravitons has also been generalised to AdS5 × Y5 spacetimes, where
Y5 is a Sasaki-Einstein manifold. In [6] and [7] a D3-brane wrapped around the angular S
3 of the
AdS5 and moving along the Reeb vector of the Sasaki-Einstein space was considered, yielding a
dual giant graviton. A generalisation of giant graviton configurations preserving 1/4 or 1/8 of the
supersymmetries has also been considered by Mikhailov [8] in the AdS5 × T 1,1 spacetime.
In this paper we find a new giant graviton configuration in AdS5 × Y5, which consists of
a Kaluza-Klein (KK) monopole with internal angular momentum, wrapping the Y5 part of the
geometry and with Taub-NUT direction in the AdS5 part. This solution has distinguishing features
with respect to the previous giant graviton solutions constructed in the literature. First, the
monopole does not couple to the 4-form potential of the background and the configuration is
therefore not at a dynamical equilibrium position. Still it is stable because by construction it is
wrapped around the entire Y5. Secondly, it has a fixed size L, the “radius” of the Y5, independent
of the momentum of the configuration. In fact the energy of the monopole depends only on its
position in the AdS5 part of the spacetime, and it is minimised when the monopole sits at the
centre of AdS5, where it behaves as a massless particle. In this sense this new giant graviton
configuration does not provide a realisation of the stringy exclusion principle. However, its mere
existence is sufficiently surprising to motivate a closer look at the configuration. Furthermore, the
fact that the giant graviton is built up from a Kaluza-Klein monopole, could lead to interesting
view points in the context of the AdS/CFT correspondence.
The organisation of this paper is as follows. In section 2 we present the Kaluza-Klein monopole
giant graviton solution. We start by introducing our probe monopole and then construct an action
suitable to describe it. We then calculate the energy of the configuration and show that when
the monopole sits at the centre of AdS5 it behaves as a massless particle. In section 3 we move
to consider the microscopical description of this configuration in terms of expanding gravitational
waves. Given that the fuzzy version of an arbitrary Sasaki-Einstein manifold is not known, we
particularise to the case in which Y5 = S
5. The fuzzy 5-sphere that we consider is defined as an S1
bundle over the fuzzy CP 2. This fuzzy manifold has been successfully used in the microscopical
description of 5-sphere giant gravitons [9, 10], and of the baryon vertex with magnetic flux [11].
In these examples the fibre structure of the S5 plays a crucial role in the construction. Finally in
section 4 we present a candidate description of our configuration in the field theory side. We end
with some conclusions in section 5.
2 A new giant graviton solution
2.1 The Kaluza-Klein monopole probe
Consider the AdS5 × Y5 spacetime, with Y5 a quasi-regular five-dimensional Sasaki-Einstein man-
ifold. All these Sasaki-Einstein manifolds have a constant norm Killing vector, called the Reeb
vector. For the cases we are interested in, the U(1) action of the Reeb vector is free and the
quotient space is (at least locally) a four-dimensional regular Kähler-Einstein manifold M4 with
positive curvature. In that case the metric on Y5 can (at least locally) be written as a U(1) fibre
bundle over the M4,
ds2Y = ds
M + (dψ +B)
2, (2.1)
where ds2M is the metric on the M4 and the Killing vector k
µ = δ
is the Reeb vector. The Kähler
form on M4 is related to the fibre connection B via ωM =
dB.4 The AdS5 × Y5 background
contains as well a non-vanishing 4-form RR-potential.
Using the U(1) decomposition above, the metric of AdS5 × Y5 can be written as
ds2 = −(1 + r
)dt2 +
(1 + r
dΩ22 + (dχ+A)
ds2M + (dψ +B)
, (2.2)
where we have used global coordinates in the AdS part and written the angular S3 contained in
AdS5 as a U(1) fibre over S
2. A and B stand for the connections of the S3 and Y5 fibre bundles
respectively. In these coordinates, the fibre directions χ and ψ are clearly globally defined isometry
directions.
Consider now a KK-monopole wrapped on the Y5, with Taub-NUT direction χ and propagating
along ψ. This will be our KK-monopole probe. In order to study the dynamics of this monopole
we start by constructing an action suitable to describe it.
The effective action describing the dynamics of the Type IIB Kaluza-Klein monopole was
constructed in [13]. Like the Type IIA NS5-brane, to which it is related by T-duality along the
Taub-NUT direction, the action for the monopole is described by a six-dimensional (2, 0) tensor
supermultiplet, which contains a self-dual 2-form Ŵ+
and 5 scalars {X i, ω, ω̃}. The self-dual
2-form is associated to the (S-duality invariant) configuration of the monopole intersecting a D3-
brane, wrapped on the Taub-NUT direction. The worldvolume scalars ω and ω̃ are associated
with the intersections of D5- and NS5-branes respectively, and form a doublet under S-duality.
Finally the scalars X i (with i = 1, 2, 3) are the embedding scalars, that describe the position of
the monopole in the transverse space. Note that although the worldvolume of the monopole is
six-dimensional, its position is specified by only three embedding scalars. This is because the
Taub-NUT direction is considered to be transverse, but being an isometry direction it does not
yield a dynamical degree of freedom. The KK-monopole action takes in fact the form of a gauged
sigma model, where the degree of freedom corresponding to the Taub-NUT direction is gauged
away [14]. Due to the presence of the self-dual two-form Ŵ+
, there is no straightforward covariant
formulation of the action (see for example [15]). However, like in the case of the five-brane [16, 17],
it is possible to give an approximation, expanding the action to quadratic order in the self-dual
two-form.
In our case, the situation is actually simpler. As our KK-monopole probe is wrapped around
Y5, the U(1) fibre direction ψ is contained in its worldvolume. Therefore it is possible to effectively
compactify the monopole over the fibre direction and to consider the (much simpler) action for a
wrapped KK-monopole. Moreover, momentum charge along this U(1) fibre direction can easily
be induced by switching on an appropriate magnetic flux in the worldvolume.
The field content of the wrapped monopole is given by the five-dimensional (1, 1) vector
multiplet, which contains 5 scalars and one vector, and is the dimensional reduction of the six-
dimensional (2, 0) tensor multiplet. The self-duality condition becomes a Hodge-duality condition
4For an extensive summary on the properties of Sasaki-Einstein manifolds we refer to [12].
between the vector and a two-form, which does not appear explicitly in the action.5 In this way
an action can be constructed to all orders in the field strength. In practise, the action of the
wrapped monopole is most easily constructed from the action of the Type IIA KK-monopole, as
the latter has the six-dimensional (1, 1) vector multiplet as its worldvolume field content [18]. Af-
ter T-dualising along a worldvolume direction, the resulting action describes a Type IIB monopole
wrapped along the T-duality direction, with an effectively five-dimensional worldvolume. As our
KK-monopole probe is wrapped on the S1 fibre direction of the Y5, its spatial worldvolume becomes
effectively R×M4.
2.2 The action for the wrapped monopole
The starting point is the action for the Type IIA Kaluza-Klein monopole constructed in [18], which
we compactify along a worldvolume direction and T-dualize. The resulting action describes a Type
IIB Kaluza-Klein monopole which is wrapped on the T-duality direction and has, effectively, a
five-dimensional worldvolume. The T-duality direction appears in the action as a new isometric
direction, whose Killing vector we denote by kµ. On the other hand the Killing vector associated
with the Taub-NUT direction is denoted by ℓµ. The explicit action is given by
S = −T4
d5σ e−2φkℓ2
| det(DaXµDbXνgµν + eφk−1ℓ−1Fab)|
P [ikiℓN
(7)]− P
∧ F − 1
[k(1)
∧ F ∧ F + . . .
, (2.3)
where the scalars k and ℓ are the norm of kµ and ℓµ respectively, k(1) denotes the 1-form with com-
ponents kµ and (iℓikΩ)µ1...µn = ℓ
ρkνΩνρµ1...µn . In this action the pull-backs into the worldvolume
are taken with gauge covariant derivatives
µ = ∂aX
µ − k−2kν∂aXνkµ − ℓ−2ℓν∂aXνℓµ, (2.4)
which ensure local invariance under the isometric transformations generated by the two Killing
vectors
δXµ = Λ(1)(σ)kµ + Λ(2)(σ)ℓµ . (2.5)
In this way the embedding scalars corresponding to the isometry directions are eliminated as
dynamical degrees of freedom and the action is given by a gauged sigma model of the type first
considered in [14].
The two-form field strength F is defined as
F = 2∂V (1) + P [ikiℓC(4)] , (2.6)
where the worldvolume vector field V (1) is the T-dual of the vector field of the Type IIA monopole
(or, alternatively, the dimensional reduction of the self-dual two-form Ŵ+). While in the Type
IIA monopole the vector field is associated to D2-branes wrapped on the Taub-NUT direction, in
the IIB case it is associated to D3-branes, wrapped on both Killing directions.
The action of the Type IIA monopole contains as well a worldvolume scalar associated to
strings wrapped on the Taub-NUT direction. This field gives, upon T-duality, a worldvolume
scalar ω which forms a doublet under S-duality with the T-dual ω̃ of the component of the IIA
vector field along the T-duality direction. These two scalars are necessary in order to compensate
for the two degrees of freedom associated to the two transverse scalars that have been eliminated
from the action through the gauging procedure. This scalar doublet does however not play a role
in our construction and has therefore been set to zero in our action above. The action (2.3) should
then be regarded as a truncated action suitable for the study of the wrapped monopole in the
AdS5 × Y5 background.
5This is very similar to the M5-brane case. The unwrapped M5-brane contains a self-dual 2-form in its world-
volume, whereas the M5-brane wrapped on the eleventh direction (the D4) depends on an unconstrained five-
dimensional vector field.
In the Chern-Simons part of the action we find a coupling to N (7), the tensor field dual to
the Taub-NUT Killing vector ℓµ, considered as a 1-form. The contraction iℓN
(7) is the field to
which a KK-monopole with Taub-NUT direction ℓµ couples minimally (see [18]). In (2.3) this
field is further contracted with the second isometric direction kµ, indicating that the monopole is
wrapped along this direction. More importantly for our construction below, the second coupling
in the CS action involves the momentum operator P [k(1)/k2], associated to the isometric direction
with Killing vector kµ. Therefore, it is possible to induce momentum charge in this isometric
direction, with an appropriate choice of F . As we will show below, we will make use of this
coupling to let the monopole propagate along the isometric direction ψ. Finally, the dots indicate
couplings to other Type IIB background fields which do not play a role in our construction.
2.3 The giant graviton solution
Let us now particularise the action (2.3) to our probe KK-monopole. We take our monopole
wrapped on the transverse Y5. Therefore the fibre direction of the decomposition of Y5 as a U(1)
fibre bundle over M4 is identified as the isometric worldvolume direction in (2.3), and M4 as the
effective four-dimensional spatial worldvolume. The Taub-NUT direction is taken along the S1
fibre direction of the S3 contained in AdS5. Therefore, we have explicitly
kµ = δ
, ℓµ = δµχ. (2.7)
With this choice of Killing directions the contribution of the 4-form RR-potential of the AdS5×Y5
background to the action vanishes. This is so because both couplings to C(4), in (2.6) and in (2.3),
involve directions along Y5, plus the Taub-NUT direction, χ, which lives in the AdS5 part of the
spacetime.
Furthermore, in order to induce momentum charge in the ψ direction we choose the world-
volume vector field V proportional to the curvature tensor of the Y5 fibre connection B, such
F = ∗F ,
F ∧ F = 2n2ΩM , (2.8)
where ΩM is the volume of M4 and the Hodge star is taken with respect to the metric on this
manifold6. With this Ansatz F satisfies trivially the Bianchi identities. Then, through the second
coupling in the Chern-Simons part of the action (2.3), we have that
k−2k(1)
∧ F ∧ F = n2TW
k−2k(1)
, (2.9)
where we have used the fact that the tension of the wrapped monopole is related to the tension of
the point-like object carrying momentum charge (the gravitational wave) through ΩMT4 = TW .
Therefore, with this Ansatz for F , we are dissolving in the worldvolume n2 momentum charges
in the ψ direction. Notice that the instantonic nature of (2.8) guarantees that the equations of
motion for F are satisfied.
A second remarkable property of the Ansatz (2.8) is that the determinant of (P [g]ab +Fab) is
a perfect square [9, 11], such that the Born-Infeld part of the action (2.3) gives rise to
S = −T4
dt dΩM
|gM |, (2.10)
which after integration over M4 gives rise to the following Hamiltonian
. (2.11)
6The integral above is non-zero because it is the product of two integrals,
F , over non-trivial two-cycles in M4
(see for example [19]). Since
F = 2πn due to Dirac quantisation condition, n represents the winding number of
D3-branes wrapped around each of the two cycles. For our construction we have chosen the same winding number
in both cycles in order to preserve the self-duality condition (2.8).
The energy of the configuration is therefore a function of the radial coordinate r of AdS5 and
is clearly minimised when r = 0, that is, when the monopole is sitting at the centre of AdS5.
Moreover, for this value of r the energy is given by
. (2.12)
Therefore, the configuration that we have proposed behaves as a giant graviton: it has the
energy of a massless particle with momentum Pψ but clearly has some finite radius L, as it is
wrapped around the entire Y5. Since it saturates a BPS bound it is a solution of the equations of
motion. Finally, we should note that there is no dynamical equilibrium between the brane tension
and the angular momentum, as in the traditional giant graviton configurations of [1, 2, 3, 4].
However the stability of the configuration is still guaranteed due to the fact that it wraps the
entire transverse space.
3 A microscopical description in terms of dielectric gravi-
tational waves
It is by now well-known that the traditional giant graviton configurations of [1, 2, 3, 4] can be
described microscopically in terms of multiple gravitational waves expanding into (a fuzzy version
of) the corresponding spherical brane by Myers’ dielectric effect [20]. In particular, the M5-brane
giant graviton configurations of the AdS4 × S7 and AdS7 × S4 spacetimes have been described in
terms of multiple M-waves expanding into a fuzzy 5-sphere that is defined as an S1 bundle over a
fuzzy CP 2 [9]. A non-trivial check of the validity of this description is that it agrees exactly with
the spherical brane description in [1, 2] when the number of gravitons is very large.
In this spirit one would expect that the KK-monopole giant graviton configuration constructed
in the previous section would be described microscopically in terms of dielectric Type IIB gravita-
tional waves expanding into a fuzzy Y5. The fuzzy version of general quasi-regular Sasaki-Einstein
manifolds is however not known. Therefore we will restrict to the case in which Y5 coincides with
the 5-sphere. In this case we will see that the waves expand into a fuzzy 5-sphere, defined as an
S1 bundle over a fuzzy CP 2, i.e. the same type of fuzzy manifolds involved in the description used
in [9] (see also [10, 11]).
Consider now a number N of coinciding gravitational waves in the AdS5 × S5 background.
The action describing multiple gravitational waves in Type IIB was constructed in [21, 22] and it
contains the following couplings:
SW = − TW
dτ STr
Eµν − Eµi(Q−1 − δ)ijEjkEkν
(3.1)
dτ STr
−P [k−2k(1)] − iP [(iX iX)iℓC(4)]− 12P [(iX iX)
2iℓikN
(7)] + . . .
where
Eµν = Gµν − k−1ℓ−1(ikiℓC(4))µν , Gµν = gµν − k−2kµkν − ℓ−2ℓµℓν ,
Qµν = δ
ν + ikℓ [X
µ, Xρ]Eρν , ((iX iX)iℓC4)λ = [X
ρ, Xν]ℓµCµνρλ. (3.2)
(iℓikΩ)µ1...µn = ℓ
ρkνΩνρµ1...µn ,
This action is valid to describe waves propagating in backgrounds which contain two isometric
directions, parametrised in the action by the Killing vectors kµ and ℓµ. The wave action is
actually a gauged sigma model in which the embedding scalars associated to the Killing directions
are projected out. The physical meaning of kµ is that it corresponds to the propagation direction
of the gravitational waves, while ℓµ is an isometry direction inherited from the T-duality operation
involved in the construction of the action (see [21] and [22] for more details).
Although the only non-zero term in the Chern-Simons action in the AdS5 × S5 background is
the dipole coupling to C(4), it is worth calling the attention to the quadrupole coupling to N (7).
Indeed, this coupling shows that the waves can expand via a quadrupole effect into a monopole
with Taub-NUT direction parametrised by lµ and further wrapped around the kµ direction. This
monopole will then act as the source of iℓikN
Let us now use the action (3.1) to describe microscopically the KK-monopole of the previous
section in the AdS5×S5 background. In this caseM4 = CP 2 and ds2M stands for the Fubini-Study
metric on the CP 2 (see for instance [23]). The identification of the isometry directions of (3.1)
is then obvious. In order to account for the momentum in the worldvolume of the monopole,
we identify the propagation direction of the waves with the S5 fibre direction ψ, while the extra
isometry will be identified with the Taub-NUT direction χ of the monopole:
kµ = δ
, ℓµ = δµχ. (3.3)
With this choice of Killing vectors it is clear that the contribution of C(4) to the action vanishes,
both in the BI and in the CS parts. Therefore, any dielectric effect will be purely gravitational
[24, 25, 26].
Furthermore, we will have the gravitational waves expand into the entire five-sphere, whose
fuzzy version we choose to describe as an S1 bundle over the fuzzy CP 2. Therefore, we take the
non-commutative scalars in (3.1) to parametrise the fuzzy CP 2 base of the S5.
The fuzzy CP 2 has been extensively studied in the literature. For its use in the giant graviton
context we refer to [9, 10], where more details on the construction that we sketch below can
be found. CP 2 is the coset manifold SU(3)/U(2) and can be defined as the submanifold of R8
determined by the constraints
xixi = 1 ,
j,k=1
dijkxjxk =
xi , (3.4)
where dijk are the components of the totally symmetric SU(3)-invariant tensor. A fuzzy version
of CP 2 can then be obtained by imposing the conditions (3.4) at the level of matrices (see for
example [27]). We define a set of coordinates X i (i = 1, . . . , 8) as
X i =
(2N − 2)/3
, (3.5)
where T i are the SU(3) generators in the N -dimensional irreducible representations (k, 0) or (0, k),
with N = (k + 1)(k + 2)/2. The first constraint in (3.4) is then trivially satisfied through the
quadratic Casimir (2N − 2)/3 of the group, whereas the rest of the constraints are satisfied for
any N (see [27, 9] for the details). The commutation relations between the X i are given by
[X i, Xj] =
i f ijk
(2N − 2)/3
Xk, (3.6)
with f ijk the structure constants of SU(3) in the algebra of the Gell-Mann matrices [λi, λj ] =
2if ijkλk.
Substituting the Ansätze (3.5) and (3.3) in the action (3.1), we find
S = −TW
dτ STr
3L6r2
32(N − 1)X
, (3.7)
up to order N−2. Here we have dropped those contributions to det Q that vanish when taking
the symmetrised trace, and ignored higher powers of N which will vanish in the large N limit.7
7These terms cannot be nicely arranged into higher powers of the quadratic Casimir without explicit use of the
constraints.
Taking the symmetrised trace we arrive at the following Hamiltonian
32(N − 1)
, (3.8)
which, in the large N limit, is in perfect agreement with the Hamiltonian for the spherical KK-
monopole, given by (2.11). To see this we should recall that in the macroscopical description the
momentum charge of the configuration is the number of waves dissolved in the worldvolume, and
is therefore given by n2. In the microscopical description the momentum charge is given directly
by the number of microscopic waves, N . Therefore in the large N limit N and n2 must coincide.
The Hamiltonian in (3.8) is also a function of the radial coordinate r of AdS5 and it is minimised
at r = 0 where it takes the value E = Pψ/L, thus corresponding to a giant graviton configuration.
4 A possible interpretation in the dual field theory
In this section we try to give a possible field theoretic interpretation of the giant graviton con-
figuration that we have studied, along the lines of [3] (see also [28, 29, 30, 31]). We will discuss
the giant graviton configuration in the AdS5 × S5 background, but we will later speculate on a
possible generalisation to other Sasaki-Einstein spaces.
We have learnt in the previous sections that the fibre direction in the S3 contained in AdS5
plays a crucial role in the construction of the giant graviton configuration, as it is identified with
the Taub-NUT direction of the monopole. Therefore, it is useful to work in the global patch for
It is then natural to consider the dual field theory as living in R×S3, where there is a conformal
coupling to the curvature. The bosonic piece of the action reads
dt dΩ3 Tr
µΦa +
Φ∗aΦa +
[Φa,Φ
, (4.1)
where L is the radius of the S3 and the Φa (with a = 1, 2, 3) are the complexification of the 6
adjoint real scalars X i of N = 4 SYM. After defining Φa = Xa+ iXa+3, only an SU(3) subgroup
of the original SO(6) R-symmetry group remains explicit.
Regarding S3 as an S1 bundle over S2 it seems a consistent truncation to assume that the Φa
do not depend on the fibre coordinate. Actually, this will be the field theory analogue of the fact
that this direction corresponds to the Taub-NUT direction of the monopole on the gravity side.
Taking adapted coordinates to the U(1) fibration we have
S = 2π
dΩ2 Tr
− ∂tΦ∗a∂tΦa − 4Φ∗a∆S2Φa +
Φ∗aΦa +
[Φa,Φ
, (4.2)
where ∆S2 is the Laplacian in the 2-sphere.
We can then expand the scalars in spherical harmonics Φ
a on the two-sphere. Given that
we will be interested in the lowest energy modes, we will truncate all of them except the massless
mode Φ
a , which corresponds to the constant mode on the S
2. Furthermore, we consider the
following Ansatz for the gauge and SU(3) dependence of our fields
Φ(0)a = e
if(t)Ma ⊗ Ja, (4.3)
where f(t) is an arbitrary function of time, the traceless matrix Ma is defined as
Ma = diag
M − 1 , · · · ,−
M − 1
, (4.4)
and the Ja’s are SU(2)-generators in a k-dimensional representation. Since the total rank of the
gauge group is N we should have that kM = N . Indeed, in this branch the gauge group breaks
into SU(M). The gauge transformations which are left are those of the form
Φa → gΦag†, g = diag(g̃k, · · · , g̃k) , (4.5)
where the g̃ki are SU(2) gauge transformations of dimension k.
The action then reduces to
S = 8π2
C2(k)M
M − 1
− (f ′)2v2 + v
, (4.6)
where v2 = δabvavb is to be interpreted as a non-dynamical parameter whose value will determine
the minima of the potential. In addition, C2(k) is the Casimir of the SU(2) k-dimensional repre-
sentation. In this action f(t) is a cyclic variable and therefore its conjugate momentum, p, will be
conserved. The Hamiltonian is given by
H = p2
(M − 1)
32π2v2C2(k)M
8π2MC2(k)v
(M − 1)L2 , (4.7)
which has a minimum for v2 =
(M−1)
32π2C2(k)M
pL. Remarkably the on-shell energy is precisely the
dispersion relation
. (4.8)
Therefore, the configuration (4.3) can be seen as a massless particle. Furthermore, out of the
original SU(3) rotating our Φa just an SU(2) survives, given that with our Ansatz the Φa become
a vector of SU(2). Thus, the moduli space reduces to SU(3)/U(2), which is precisely the symmetry
of CP 2 as a coset space, which is in turn the manifold wrapped by our KK-monopole.
Given that our construction of the wrapped KK-monopole works not just in the S5 case, but
also in more generic spaces, we expect a similar field theory description for the dual of a generic
Sasaki-Einstein space. In supporting this claim, let us notice that the potential term did not play
any role in the S5 case, because with the Ansatz we assumed, it vanishes. In the generic case, we
will assume a similar Ansatz for our fields, namely
Xα = e
if(t)M⊗Gα , (4.9)
where now for simplicity we take the same M matrix as before but with all the v’s identical. The
Gα are the generators of the global symmetry group G. Given this Ansatz, we expect that the
superpotential does not play any role, not even in the most generic Y p,q case. In addition, since
we take AdS in the global patch, the field theory will be defined in R × S3, so we will always
have the conformal coupling to the curvature. Just this term, together with the kinetic energy, is
enough to reproduce a dispersion relation of the form E ∼ p.
In the general case we can also regard the S3 as an S1 bundle over S2, and take our fields
independent of the U(1). This is the field theory counterpart of the presence of the Taub-NUT
direction in the gravity side. In addition, out of the full global symmetry group G we will just keep
the subgroup g compatible with our Ansatz,8 so we would expect a moduli space of the form G/g.
Let us consider for example the conifold. In this case the global symmetry group is SU(2)×SU(2).
Therefore we have to first reduce it to the diagonal SU(2) and then take the conifold scalars A
and B to be eif(t)M. This leaves an [SU(2)×SU(2)]D/U(1) moduli space, which is the symmetry
of 2 2-spheres, and this is in turn what one would get from the gravity side.
Finally, we would like to note that our construction is quite generic. From the gravity point
of view we just require that the momentum of the KK-monopole wrapping the five-dimensional
space is taken along a U(1) fibre direction, and that its Taub-NUT direction is along the S1
in the decomposition of the S3 ⊂ AdS5 as an S1 bundle over S2. In the field theory side our
requirements are also quite generic. The existence of the Taub-NUT direction is reflected on the
fact that the SCFT is defined in R×S3 and S3 is taken as S1 over S2. In addition, our description
is not sensitive to the superpotential, which we believe is the counterpart to the fact that the
KK-monopole wraps the whole 5-dimensional manifold. Then, we are left with the kinetic term
and the conformal coupling to curvature, which is enough to ensure the right dispersion relation.
Since in general our Ansatz will reduce the original global symmetry, the moduli space will be
G/g, which we believe will correspond in general to the symmetry of the 4-dimensional base which
the KK-monopole wraps.
8Note that g may involve discrete subgroups such as Zk
5 Conclusions
In this letter we have constructed a new type of giant graviton solution in AdS5 × Y5, with Y5 a
quasi-regular Sasaki-Einstein manifold. This solution consists on a Kaluza-Klein monopole with
internal momentum, wrapped around the entire Y5 and with Taub-NUT direction along the AdS5
part.
Although the dynamics of this monopole can be described using the effective action for the
Type IIB Kaluza-Klein monopole constructed in [13], this action is only known to quadratic order
in the self-dual 2-form of its six-dimensional (2, 0) tensor multiplet field content. However, given
that Y5 can be decomposed as a U(1) bundle over a four-dimensional Kähler-Einstein manifold
M4, it is possible to use the action for a monopole wrapped on a U(1) direction to describe it.
This action, having the field content of the five-dimensional (1, 1) vector multiplet, is known to all
orders. Moreover, it is possible to induce momentum charge along the U(1) direction through a
suitably chosen worldvolume vector field with non-zero instanton number. Using the action for a
U(1) wrapped monopole we have shown that the energy of the configuration depends on its radial
position in the AdS space and behaves as a massless particle when put in the origin, while having
the size of the Y5.
Given that the spherical monopole carries a non-vanishing momentum charge there should be
a microscopical description of the same configuration in terms of expanding gravitational waves.
This description would involve however the fuzzy version of Y5, which is not known in general.
Therefore, we have restricted to the case in which Y5 = S
5, and let the multiple dielectric gravita-
tional waves expand into a fuzzy 5-sphere. The fuzzy 5-sphere built up by the gravitational waves
is constructed as an Abelian fibre bundle over a fuzzy CP 2, a construction that has been used
before in the study of the traditional giant gravitons and the baryon vertex with magnetic flux.
The configuration thus obtained turns out to exactly agree in the limit of large number of waves
with the effective KK-monopole description.
We believe there are several reasons why this new giant graviton solution has not been found
earlier in the literature. First of all, since it has no relation with the stringy exclusion principle
it is not straightforward to find the corresponding state in the CFT side. Moreover, as we have
shown in section 4, our scalar field configuration breaks the R-symmetry group in a rather peculiar
way, making explicit the U(1) fibre structure of the S3. Secondly, the fact that it is built up from
a Kaluza-Klein monopole and not from a more ordinary type of brane, makes our construction
more involved.
An interesting question to answer would be whether the KK-monopole giant graviton solution
is supersymmetric or not. This is however difficult to answer, on the one hand due to the form of
the Killing spinors in the particular coordinate system that we are using and, on the other hand,
due to the fact that the kappa-symmetry for the Kaluza-Klein monopole is not known. Yet, the
fact that the configuration is massless implies that it saturates a BPS bound, which hints to the
fact that it probably preserves some fraction of the supersymmetry. We would like to leave this
problem for future investigations.
Acknowledgements
We wish to thank S.Benvenutti, B. Craps, D. van den Bleeken, T. van Proeyen and A.V. Ramallo
for the useful discussions. The work of B.J. is done as part of the program Ramón y Cajal of
the Ministerio de Educación y Ciencia (M.E.C.) of Spain. He is also partially supported by the
M.E.C. under contract FIS 2004-06823 and by the Junta de Andalućıa group FQM 101. The work
of Y.L. has been partially supported by the CICYT grant MEC-06/FPA2006-09199 (Spain) and
by the European Commission FP6 program MRTN-CT-2004-005104, in which she is associated
to Universidad Autónoma de Madrid. D.R.G. is supported by the Fullbright-M.E.C. fellowship
FU-2006-0740.
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	Introduction
	A new giant graviton solution
	The Kaluza-Klein monopole probe
	The action for the wrapped monopole
	The giant graviton solution
	A microscopical description in terms of dielectric gravitational waves
	A possible interpretation in the dual field theory
	Conclusions

  We construct a new giant graviton solution in AdS_5 x Y_5, with Y_5 a
quasi-regular Sasaki-Einstein manifold, consisting on a Kaluza-Klein monopole
wrapped around the Y_5 and with its Taub-NUT direction in AdS_5. We find that
this configuration has minimal energy when put in the centre of AdS_5, where it
behaves as a massless particle. When we take Y_5 to be S^5, we provide a
microscopical description in terms of multiple gravitational waves expanding
into the fuzzy S^5 defined as an S^1 bundle over the fuzzy CP^2. Finally we
provide a possible field theory dual interpretation of the construction.

Introduction
As it is well-known, giant gravitons are stable brane configurations with non-zero angular momen-
tum, that are wrapped on (n− 2)- or (m− 2)-spheres in AdSm×Sn spacetimes and carry a dipole
moment with respect to the background gauge potential [1, 2, 3, 4]. They are not topologically
stable, but are at dynamical equilibrium because the contraction due to the tension of the brane
is precisely cancelled by the expansion due to the coupling of the angular momentum to the back-
ground flux field. These spherical brane configurations turn out to be massless, conserve the same
number of supersymmetries and carry the same quantum numbers of a graviton.
Giant graviton configurations were first proposed as a way to satisfy the stringy exclusion
principle implied by the AdS/CFT correspondence [1]. The spherical (n− 2)-brane expands into
the Sn part of the geometry with a radius proportional to its angular momentum. Since this
radius is bounded by the radius of the Sn, the configuration has associated a maximum angular
momentum. The (m − 2)-brane giant graviton configurations [2, 3], on the other hand, expand
into the AdSm part of the geometry, and they do not satisfy the stringy exclusion principle. For
a discussion on the degeneracy of these two types of giant gravitons and the point-like graviton,
we refer for instance to [5] and references therein.
The construction of giant gravitons has also been generalised to AdS5 × Y5 spacetimes, where
Y5 is a Sasaki-Einstein manifold. In [6] and [7] a D3-brane wrapped around the angular S
3 of the
AdS5 and moving along the Reeb vector of the Sasaki-Einstein space was considered, yielding a
dual giant graviton. A generalisation of giant graviton configurations preserving 1/4 or 1/8 of the
supersymmetries has also been considered by Mikhailov [8] in the AdS5 × T 1,1 spacetime.
In this paper we find a new giant graviton configuration in AdS5 × Y5, which consists of
a Kaluza-Klein (KK) monopole with internal angular momentum, wrapping the Y5 part of the
geometry and with Taub-NUT direction in the AdS5 part. This solution has distinguishing features
with respect to the previous giant graviton solutions constructed in the literature. First, the
monopole does not couple to the 4-form potential of the background and the configuration is
therefore not at a dynamical equilibrium position. Still it is stable because by construction it is
wrapped around the entire Y5. Secondly, it has a fixed size L, the “radius” of the Y5, independent
of the momentum of the configuration. In fact the energy of the monopole depends only on its
position in the AdS5 part of the spacetime, and it is minimised when the monopole sits at the
centre of AdS5, where it behaves as a massless particle. In this sense this new giant graviton
configuration does not provide a realisation of the stringy exclusion principle. However, its mere
existence is sufficiently surprising to motivate a closer look at the configuration. Furthermore, the
fact that the giant graviton is built up from a Kaluza-Klein monopole, could lead to interesting
view points in the context of the AdS/CFT correspondence.
The organisation of this paper is as follows. In section 2 we present the Kaluza-Klein monopole
giant graviton solution. We start by introducing our probe monopole and then construct an action
suitable to describe it. We then calculate the energy of the configuration and show that when
the monopole sits at the centre of AdS5 it behaves as a massless particle. In section 3 we move
to consider the microscopical description of this configuration in terms of expanding gravitational
waves. Given that the fuzzy version of an arbitrary Sasaki-Einstein manifold is not known, we
particularise to the case in which Y5 = S
5. The fuzzy 5-sphere that we consider is defined as an S1
bundle over the fuzzy CP 2. This fuzzy manifold has been successfully used in the microscopical
description of 5-sphere giant gravitons [9, 10], and of the baryon vertex with magnetic flux [11].
In these examples the fibre structure of the S5 plays a crucial role in the construction. Finally in
section 4 we present a candidate description of our configuration in the field theory side. We end
with some conclusions in section 5.
2 A new giant graviton solution
2.1 The Kaluza-Klein monopole probe
Consider the AdS5 × Y5 spacetime, with Y5 a quasi-regular five-dimensional Sasaki-Einstein man-
ifold. All these Sasaki-Einstein manifolds have a constant norm Killing vector, called the Reeb
vector. For the cases we are interested in, the U(1) action of the Reeb vector is free and the
quotient space is (at least locally) a four-dimensional regular Kähler-Einstein manifold M4 with
positive curvature. In that case the metric on Y5 can (at least locally) be written as a U(1) fibre
bundle over the M4,
ds2Y = ds
M + (dψ +B)
2, (2.1)
where ds2M is the metric on the M4 and the Killing vector k
µ = δ
is the Reeb vector. The Kähler
form on M4 is related to the fibre connection B via ωM =
dB.4 The AdS5 × Y5 background
contains as well a non-vanishing 4-form RR-potential.
Using the U(1) decomposition above, the metric of AdS5 × Y5 can be written as
ds2 = −(1 + r
)dt2 +
(1 + r
dΩ22 + (dχ+A)
ds2M + (dψ +B)
, (2.2)
where we have used global coordinates in the AdS part and written the angular S3 contained in
AdS5 as a U(1) fibre over S
2. A and B stand for the connections of the S3 and Y5 fibre bundles
respectively. In these coordinates, the fibre directions χ and ψ are clearly globally defined isometry
directions.
Consider now a KK-monopole wrapped on the Y5, with Taub-NUT direction χ and propagating
along ψ. This will be our KK-monopole probe. In order to study the dynamics of this monopole
we start by constructing an action suitable to describe it.
The effective action describing the dynamics of the Type IIB Kaluza-Klein monopole was
constructed in [13]. Like the Type IIA NS5-brane, to which it is related by T-duality along the
Taub-NUT direction, the action for the monopole is described by a six-dimensional (2, 0) tensor
supermultiplet, which contains a self-dual 2-form Ŵ+
and 5 scalars {X i, ω, ω̃}. The self-dual
2-form is associated to the (S-duality invariant) configuration of the monopole intersecting a D3-
brane, wrapped on the Taub-NUT direction. The worldvolume scalars ω and ω̃ are associated
with the intersections of D5- and NS5-branes respectively, and form a doublet under S-duality.
Finally the scalars X i (with i = 1, 2, 3) are the embedding scalars, that describe the position of
the monopole in the transverse space. Note that although the worldvolume of the monopole is
six-dimensional, its position is specified by only three embedding scalars. This is because the
Taub-NUT direction is considered to be transverse, but being an isometry direction it does not
yield a dynamical degree of freedom. The KK-monopole action takes in fact the form of a gauged
sigma model, where the degree of freedom corresponding to the Taub-NUT direction is gauged
away [14]. Due to the presence of the self-dual two-form Ŵ+
, there is no straightforward covariant
formulation of the action (see for example [15]). However, like in the case of the five-brane [16, 17],
it is possible to give an approximation, expanding the action to quadratic order in the self-dual
two-form.
In our case, the situation is actually simpler. As our KK-monopole probe is wrapped around
Y5, the U(1) fibre direction ψ is contained in its worldvolume. Therefore it is possible to effectively
compactify the monopole over the fibre direction and to consider the (much simpler) action for a
wrapped KK-monopole. Moreover, momentum charge along this U(1) fibre direction can easily
be induced by switching on an appropriate magnetic flux in the worldvolume.
The field content of the wrapped monopole is given by the five-dimensional (1, 1) vector
multiplet, which contains 5 scalars and one vector, and is the dimensional reduction of the six-
dimensional (2, 0) tensor multiplet. The self-duality condition becomes a Hodge-duality condition
4For an extensive summary on the properties of Sasaki-Einstein manifolds we refer to [12].
between the vector and a two-form, which does not appear explicitly in the action.5 In this way
an action can be constructed to all orders in the field strength. In practise, the action of the
wrapped monopole is most easily constructed from the action of the Type IIA KK-monopole, as
the latter has the six-dimensional (1, 1) vector multiplet as its worldvolume field content [18]. Af-
ter T-dualising along a worldvolume direction, the resulting action describes a Type IIB monopole
wrapped along the T-duality direction, with an effectively five-dimensional worldvolume. As our
KK-monopole probe is wrapped on the S1 fibre direction of the Y5, its spatial worldvolume becomes
effectively R×M4.
2.2 The action for the wrapped monopole
The starting point is the action for the Type IIA Kaluza-Klein monopole constructed in [18], which
we compactify along a worldvolume direction and T-dualize. The resulting action describes a Type
IIB Kaluza-Klein monopole which is wrapped on the T-duality direction and has, effectively, a
five-dimensional worldvolume. The T-duality direction appears in the action as a new isometric
direction, whose Killing vector we denote by kµ. On the other hand the Killing vector associated
with the Taub-NUT direction is denoted by ℓµ. The explicit action is given by
S = −T4
d5σ e−2φkℓ2
| det(DaXµDbXνgµν + eφk−1ℓ−1Fab)|
P [ikiℓN
(7)]− P
∧ F − 1
[k(1)
∧ F ∧ F + . . .
, (2.3)
where the scalars k and ℓ are the norm of kµ and ℓµ respectively, k(1) denotes the 1-form with com-
ponents kµ and (iℓikΩ)µ1...µn = ℓ
ρkνΩνρµ1...µn . In this action the pull-backs into the worldvolume
are taken with gauge covariant derivatives
µ = ∂aX
µ − k−2kν∂aXνkµ − ℓ−2ℓν∂aXνℓµ, (2.4)
which ensure local invariance under the isometric transformations generated by the two Killing
vectors
δXµ = Λ(1)(σ)kµ + Λ(2)(σ)ℓµ . (2.5)
In this way the embedding scalars corresponding to the isometry directions are eliminated as
dynamical degrees of freedom and the action is given by a gauged sigma model of the type first
considered in [14].
The two-form field strength F is defined as
F = 2∂V (1) + P [ikiℓC(4)] , (2.6)
where the worldvolume vector field V (1) is the T-dual of the vector field of the Type IIA monopole
(or, alternatively, the dimensional reduction of the self-dual two-form Ŵ+). While in the Type
IIA monopole the vector field is associated to D2-branes wrapped on the Taub-NUT direction, in
the IIB case it is associated to D3-branes, wrapped on both Killing directions.
The action of the Type IIA monopole contains as well a worldvolume scalar associated to
strings wrapped on the Taub-NUT direction. This field gives, upon T-duality, a worldvolume
scalar ω which forms a doublet under S-duality with the T-dual ω̃ of the component of the IIA
vector field along the T-duality direction. These two scalars are necessary in order to compensate
for the two degrees of freedom associated to the two transverse scalars that have been eliminated
from the action through the gauging procedure. This scalar doublet does however not play a role
in our construction and has therefore been set to zero in our action above. The action (2.3) should
then be regarded as a truncated action suitable for the study of the wrapped monopole in the
AdS5 × Y5 background.
5This is very similar to the M5-brane case. The unwrapped M5-brane contains a self-dual 2-form in its world-
volume, whereas the M5-brane wrapped on the eleventh direction (the D4) depends on an unconstrained five-
dimensional vector field.
In the Chern-Simons part of the action we find a coupling to N (7), the tensor field dual to
the Taub-NUT Killing vector ℓµ, considered as a 1-form. The contraction iℓN
(7) is the field to
which a KK-monopole with Taub-NUT direction ℓµ couples minimally (see [18]). In (2.3) this
field is further contracted with the second isometric direction kµ, indicating that the monopole is
wrapped along this direction. More importantly for our construction below, the second coupling
in the CS action involves the momentum operator P [k(1)/k2], associated to the isometric direction
with Killing vector kµ. Therefore, it is possible to induce momentum charge in this isometric
direction, with an appropriate choice of F . As we will show below, we will make use of this
coupling to let the monopole propagate along the isometric direction ψ. Finally, the dots indicate
couplings to other Type IIB background fields which do not play a role in our construction.
2.3 The giant graviton solution
Let us now particularise the action (2.3) to our probe KK-monopole. We take our monopole
wrapped on the transverse Y5. Therefore the fibre direction of the decomposition of Y5 as a U(1)
fibre bundle over M4 is identified as the isometric worldvolume direction in (2.3), and M4 as the
effective four-dimensional spatial worldvolume. The Taub-NUT direction is taken along the S1
fibre direction of the S3 contained in AdS5. Therefore, we have explicitly
kµ = δ
, ℓµ = δµχ. (2.7)
With this choice of Killing directions the contribution of the 4-form RR-potential of the AdS5×Y5
background to the action vanishes. This is so because both couplings to C(4), in (2.6) and in (2.3),
involve directions along Y5, plus the Taub-NUT direction, χ, which lives in the AdS5 part of the
spacetime.
Furthermore, in order to induce momentum charge in the ψ direction we choose the world-
volume vector field V proportional to the curvature tensor of the Y5 fibre connection B, such
F = ∗F ,
F ∧ F = 2n2ΩM , (2.8)
where ΩM is the volume of M4 and the Hodge star is taken with respect to the metric on this
manifold6. With this Ansatz F satisfies trivially the Bianchi identities. Then, through the second
coupling in the Chern-Simons part of the action (2.3), we have that
k−2k(1)
∧ F ∧ F = n2TW
k−2k(1)
, (2.9)
where we have used the fact that the tension of the wrapped monopole is related to the tension of
the point-like object carrying momentum charge (the gravitational wave) through ΩMT4 = TW .
Therefore, with this Ansatz for F , we are dissolving in the worldvolume n2 momentum charges
in the ψ direction. Notice that the instantonic nature of (2.8) guarantees that the equations of
motion for F are satisfied.
A second remarkable property of the Ansatz (2.8) is that the determinant of (P [g]ab +Fab) is
a perfect square [9, 11], such that the Born-Infeld part of the action (2.3) gives rise to
S = −T4
dt dΩM
|gM |, (2.10)
which after integration over M4 gives rise to the following Hamiltonian
. (2.11)
6The integral above is non-zero because it is the product of two integrals,
F , over non-trivial two-cycles in M4
(see for example [19]). Since
F = 2πn due to Dirac quantisation condition, n represents the winding number of
D3-branes wrapped around each of the two cycles. For our construction we have chosen the same winding number
in both cycles in order to preserve the self-duality condition (2.8).
The energy of the configuration is therefore a function of the radial coordinate r of AdS5 and
is clearly minimised when r = 0, that is, when the monopole is sitting at the centre of AdS5.
Moreover, for this value of r the energy is given by
. (2.12)
Therefore, the configuration that we have proposed behaves as a giant graviton: it has the
energy of a massless particle with momentum Pψ but clearly has some finite radius L, as it is
wrapped around the entire Y5. Since it saturates a BPS bound it is a solution of the equations of
motion. Finally, we should note that there is no dynamical equilibrium between the brane tension
and the angular momentum, as in the traditional giant graviton configurations of [1, 2, 3, 4].
However the stability of the configuration is still guaranteed due to the fact that it wraps the
entire transverse space.
3 A microscopical description in terms of dielectric gravi-
tational waves
It is by now well-known that the traditional giant graviton configurations of [1, 2, 3, 4] can be
described microscopically in terms of multiple gravitational waves expanding into (a fuzzy version
of) the corresponding spherical brane by Myers’ dielectric effect [20]. In particular, the M5-brane
giant graviton configurations of the AdS4 × S7 and AdS7 × S4 spacetimes have been described in
terms of multiple M-waves expanding into a fuzzy 5-sphere that is defined as an S1 bundle over a
fuzzy CP 2 [9]. A non-trivial check of the validity of this description is that it agrees exactly with
the spherical brane description in [1, 2] when the number of gravitons is very large.
In this spirit one would expect that the KK-monopole giant graviton configuration constructed
in the previous section would be described microscopically in terms of dielectric Type IIB gravita-
tional waves expanding into a fuzzy Y5. The fuzzy version of general quasi-regular Sasaki-Einstein
manifolds is however not known. Therefore we will restrict to the case in which Y5 coincides with
the 5-sphere. In this case we will see that the waves expand into a fuzzy 5-sphere, defined as an
S1 bundle over a fuzzy CP 2, i.e. the same type of fuzzy manifolds involved in the description used
in [9] (see also [10, 11]).
Consider now a number N of coinciding gravitational waves in the AdS5 × S5 background.
The action describing multiple gravitational waves in Type IIB was constructed in [21, 22] and it
contains the following couplings:
SW = − TW
dτ STr
Eµν − Eµi(Q−1 − δ)ijEjkEkν
(3.1)
dτ STr
−P [k−2k(1)] − iP [(iX iX)iℓC(4)]− 12P [(iX iX)
2iℓikN
(7)] + . . .
where
Eµν = Gµν − k−1ℓ−1(ikiℓC(4))µν , Gµν = gµν − k−2kµkν − ℓ−2ℓµℓν ,
Qµν = δ
ν + ikℓ [X
µ, Xρ]Eρν , ((iX iX)iℓC4)λ = [X
ρ, Xν]ℓµCµνρλ. (3.2)
(iℓikΩ)µ1...µn = ℓ
ρkνΩνρµ1...µn ,
This action is valid to describe waves propagating in backgrounds which contain two isometric
directions, parametrised in the action by the Killing vectors kµ and ℓµ. The wave action is
actually a gauged sigma model in which the embedding scalars associated to the Killing directions
are projected out. The physical meaning of kµ is that it corresponds to the propagation direction
of the gravitational waves, while ℓµ is an isometry direction inherited from the T-duality operation
involved in the construction of the action (see [21] and [22] for more details).
Although the only non-zero term in the Chern-Simons action in the AdS5 × S5 background is
the dipole coupling to C(4), it is worth calling the attention to the quadrupole coupling to N (7).
Indeed, this coupling shows that the waves can expand via a quadrupole effect into a monopole
with Taub-NUT direction parametrised by lµ and further wrapped around the kµ direction. This
monopole will then act as the source of iℓikN
Let us now use the action (3.1) to describe microscopically the KK-monopole of the previous
section in the AdS5×S5 background. In this caseM4 = CP 2 and ds2M stands for the Fubini-Study
metric on the CP 2 (see for instance [23]). The identification of the isometry directions of (3.1)
is then obvious. In order to account for the momentum in the worldvolume of the monopole,
we identify the propagation direction of the waves with the S5 fibre direction ψ, while the extra
isometry will be identified with the Taub-NUT direction χ of the monopole:
kµ = δ
, ℓµ = δµχ. (3.3)
With this choice of Killing vectors it is clear that the contribution of C(4) to the action vanishes,
both in the BI and in the CS parts. Therefore, any dielectric effect will be purely gravitational
[24, 25, 26].
Furthermore, we will have the gravitational waves expand into the entire five-sphere, whose
fuzzy version we choose to describe as an S1 bundle over the fuzzy CP 2. Therefore, we take the
non-commutative scalars in (3.1) to parametrise the fuzzy CP 2 base of the S5.
The fuzzy CP 2 has been extensively studied in the literature. For its use in the giant graviton
context we refer to [9, 10], where more details on the construction that we sketch below can
be found. CP 2 is the coset manifold SU(3)/U(2) and can be defined as the submanifold of R8
determined by the constraints
xixi = 1 ,
j,k=1
dijkxjxk =
xi , (3.4)
where dijk are the components of the totally symmetric SU(3)-invariant tensor. A fuzzy version
of CP 2 can then be obtained by imposing the conditions (3.4) at the level of matrices (see for
example [27]). We define a set of coordinates X i (i = 1, . . . , 8) as
X i =
(2N − 2)/3
, (3.5)
where T i are the SU(3) generators in the N -dimensional irreducible representations (k, 0) or (0, k),
with N = (k + 1)(k + 2)/2. The first constraint in (3.4) is then trivially satisfied through the
quadratic Casimir (2N − 2)/3 of the group, whereas the rest of the constraints are satisfied for
any N (see [27, 9] for the details). The commutation relations between the X i are given by
[X i, Xj] =
i f ijk
(2N − 2)/3
Xk, (3.6)
with f ijk the structure constants of SU(3) in the algebra of the Gell-Mann matrices [λi, λj ] =
2if ijkλk.
Substituting the Ansätze (3.5) and (3.3) in the action (3.1), we find
S = −TW
dτ STr
3L6r2
32(N − 1)X
, (3.7)
up to order N−2. Here we have dropped those contributions to det Q that vanish when taking
the symmetrised trace, and ignored higher powers of N which will vanish in the large N limit.7
7These terms cannot be nicely arranged into higher powers of the quadratic Casimir without explicit use of the
constraints.
Taking the symmetrised trace we arrive at the following Hamiltonian
32(N − 1)
, (3.8)
which, in the large N limit, is in perfect agreement with the Hamiltonian for the spherical KK-
monopole, given by (2.11). To see this we should recall that in the macroscopical description the
momentum charge of the configuration is the number of waves dissolved in the worldvolume, and
is therefore given by n2. In the microscopical description the momentum charge is given directly
by the number of microscopic waves, N . Therefore in the large N limit N and n2 must coincide.
The Hamiltonian in (3.8) is also a function of the radial coordinate r of AdS5 and it is minimised
at r = 0 where it takes the value E = Pψ/L, thus corresponding to a giant graviton configuration.
4 A possible interpretation in the dual field theory
In this section we try to give a possible field theoretic interpretation of the giant graviton con-
figuration that we have studied, along the lines of [3] (see also [28, 29, 30, 31]). We will discuss
the giant graviton configuration in the AdS5 × S5 background, but we will later speculate on a
possible generalisation to other Sasaki-Einstein spaces.
We have learnt in the previous sections that the fibre direction in the S3 contained in AdS5
plays a crucial role in the construction of the giant graviton configuration, as it is identified with
the Taub-NUT direction of the monopole. Therefore, it is useful to work in the global patch for
It is then natural to consider the dual field theory as living in R×S3, where there is a conformal
coupling to the curvature. The bosonic piece of the action reads
dt dΩ3 Tr
µΦa +
Φ∗aΦa +
[Φa,Φ
, (4.1)
where L is the radius of the S3 and the Φa (with a = 1, 2, 3) are the complexification of the 6
adjoint real scalars X i of N = 4 SYM. After defining Φa = Xa+ iXa+3, only an SU(3) subgroup
of the original SO(6) R-symmetry group remains explicit.
Regarding S3 as an S1 bundle over S2 it seems a consistent truncation to assume that the Φa
do not depend on the fibre coordinate. Actually, this will be the field theory analogue of the fact
that this direction corresponds to the Taub-NUT direction of the monopole on the gravity side.
Taking adapted coordinates to the U(1) fibration we have
S = 2π
dΩ2 Tr
− ∂tΦ∗a∂tΦa − 4Φ∗a∆S2Φa +
Φ∗aΦa +
[Φa,Φ
, (4.2)
where ∆S2 is the Laplacian in the 2-sphere.
We can then expand the scalars in spherical harmonics Φ
a on the two-sphere. Given that
we will be interested in the lowest energy modes, we will truncate all of them except the massless
mode Φ
a , which corresponds to the constant mode on the S
2. Furthermore, we consider the
following Ansatz for the gauge and SU(3) dependence of our fields
Φ(0)a = e
if(t)Ma ⊗ Ja, (4.3)
where f(t) is an arbitrary function of time, the traceless matrix Ma is defined as
Ma = diag
M − 1 , · · · ,−
M − 1
, (4.4)
and the Ja’s are SU(2)-generators in a k-dimensional representation. Since the total rank of the
gauge group is N we should have that kM = N . Indeed, in this branch the gauge group breaks
into SU(M). The gauge transformations which are left are those of the form
Φa → gΦag†, g = diag(g̃k, · · · , g̃k) , (4.5)
where the g̃ki are SU(2) gauge transformations of dimension k.
The action then reduces to
S = 8π2
C2(k)M
M − 1
− (f ′)2v2 + v
, (4.6)
where v2 = δabvavb is to be interpreted as a non-dynamical parameter whose value will determine
the minima of the potential. In addition, C2(k) is the Casimir of the SU(2) k-dimensional repre-
sentation. In this action f(t) is a cyclic variable and therefore its conjugate momentum, p, will be
conserved. The Hamiltonian is given by
H = p2
(M − 1)
32π2v2C2(k)M
8π2MC2(k)v
(M − 1)L2 , (4.7)
which has a minimum for v2 =
(M−1)
32π2C2(k)M
pL. Remarkably the on-shell energy is precisely the
dispersion relation
. (4.8)
Therefore, the configuration (4.3) can be seen as a massless particle. Furthermore, out of the
original SU(3) rotating our Φa just an SU(2) survives, given that with our Ansatz the Φa become
a vector of SU(2). Thus, the moduli space reduces to SU(3)/U(2), which is precisely the symmetry
of CP 2 as a coset space, which is in turn the manifold wrapped by our KK-monopole.
Given that our construction of the wrapped KK-monopole works not just in the S5 case, but
also in more generic spaces, we expect a similar field theory description for the dual of a generic
Sasaki-Einstein space. In supporting this claim, let us notice that the potential term did not play
any role in the S5 case, because with the Ansatz we assumed, it vanishes. In the generic case, we
will assume a similar Ansatz for our fields, namely
Xα = e
if(t)M⊗Gα , (4.9)
where now for simplicity we take the same M matrix as before but with all the v’s identical. The
Gα are the generators of the global symmetry group G. Given this Ansatz, we expect that the
superpotential does not play any role, not even in the most generic Y p,q case. In addition, since
we take AdS in the global patch, the field theory will be defined in R × S3, so we will always
have the conformal coupling to the curvature. Just this term, together with the kinetic energy, is
enough to reproduce a dispersion relation of the form E ∼ p.
In the general case we can also regard the S3 as an S1 bundle over S2, and take our fields
independent of the U(1). This is the field theory counterpart of the presence of the Taub-NUT
direction in the gravity side. In addition, out of the full global symmetry group G we will just keep
the subgroup g compatible with our Ansatz,8 so we would expect a moduli space of the form G/g.
Let us consider for example the conifold. In this case the global symmetry group is SU(2)×SU(2).
Therefore we have to first reduce it to the diagonal SU(2) and then take the conifold scalars A
and B to be eif(t)M. This leaves an [SU(2)×SU(2)]D/U(1) moduli space, which is the symmetry
of 2 2-spheres, and this is in turn what one would get from the gravity side.
Finally, we would like to note that our construction is quite generic. From the gravity point
of view we just require that the momentum of the KK-monopole wrapping the five-dimensional
space is taken along a U(1) fibre direction, and that its Taub-NUT direction is along the S1
in the decomposition of the S3 ⊂ AdS5 as an S1 bundle over S2. In the field theory side our
requirements are also quite generic. The existence of the Taub-NUT direction is reflected on the
fact that the SCFT is defined in R×S3 and S3 is taken as S1 over S2. In addition, our description
is not sensitive to the superpotential, which we believe is the counterpart to the fact that the
KK-monopole wraps the whole 5-dimensional manifold. Then, we are left with the kinetic term
and the conformal coupling to curvature, which is enough to ensure the right dispersion relation.
Since in general our Ansatz will reduce the original global symmetry, the moduli space will be
G/g, which we believe will correspond in general to the symmetry of the 4-dimensional base which
the KK-monopole wraps.
8Note that g may involve discrete subgroups such as Zk
5 Conclusions
In this letter we have constructed a new type of giant graviton solution in AdS5 × Y5, with Y5 a
quasi-regular Sasaki-Einstein manifold. This solution consists on a Kaluza-Klein monopole with
internal momentum, wrapped around the entire Y5 and with Taub-NUT direction along the AdS5
part.
Although the dynamics of this monopole can be described using the effective action for the
Type IIB Kaluza-Klein monopole constructed in [13], this action is only known to quadratic order
in the self-dual 2-form of its six-dimensional (2, 0) tensor multiplet field content. However, given
that Y5 can be decomposed as a U(1) bundle over a four-dimensional Kähler-Einstein manifold
M4, it is possible to use the action for a monopole wrapped on a U(1) direction to describe it.
This action, having the field content of the five-dimensional (1, 1) vector multiplet, is known to all
orders. Moreover, it is possible to induce momentum charge along the U(1) direction through a
suitably chosen worldvolume vector field with non-zero instanton number. Using the action for a
U(1) wrapped monopole we have shown that the energy of the configuration depends on its radial
position in the AdS space and behaves as a massless particle when put in the origin, while having
the size of the Y5.
Given that the spherical monopole carries a non-vanishing momentum charge there should be
a microscopical description of the same configuration in terms of expanding gravitational waves.
This description would involve however the fuzzy version of Y5, which is not known in general.
Therefore, we have restricted to the case in which Y5 = S
5, and let the multiple dielectric gravita-
tional waves expand into a fuzzy 5-sphere. The fuzzy 5-sphere built up by the gravitational waves
is constructed as an Abelian fibre bundle over a fuzzy CP 2, a construction that has been used
before in the study of the traditional giant gravitons and the baryon vertex with magnetic flux.
The configuration thus obtained turns out to exactly agree in the limit of large number of waves
with the effective KK-monopole description.
We believe there are several reasons why this new giant graviton solution has not been found
earlier in the literature. First of all, since it has no relation with the stringy exclusion principle
it is not straightforward to find the corresponding state in the CFT side. Moreover, as we have
shown in section 4, our scalar field configuration breaks the R-symmetry group in a rather peculiar
way, making explicit the U(1) fibre structure of the S3. Secondly, the fact that it is built up from
a Kaluza-Klein monopole and not from a more ordinary type of brane, makes our construction
more involved.
An interesting question to answer would be whether the KK-monopole giant graviton solution
is supersymmetric or not. This is however difficult to answer, on the one hand due to the form of
the Killing spinors in the particular coordinate system that we are using and, on the other hand,
due to the fact that the kappa-symmetry for the Kaluza-Klein monopole is not known. Yet, the
fact that the configuration is massless implies that it saturates a BPS bound, which hints to the
fact that it probably preserves some fraction of the supersymmetry. We would like to leave this
problem for future investigations.
Acknowledgements
We wish to thank S.Benvenutti, B. Craps, D. van den Bleeken, T. van Proeyen and A.V. Ramallo
for the useful discussions. The work of B.J. is done as part of the program Ramón y Cajal of
the Ministerio de Educación y Ciencia (M.E.C.) of Spain. He is also partially supported by the
M.E.C. under contract FIS 2004-06823 and by the Junta de Andalućıa group FQM 101. The work
of Y.L. has been partially supported by the CICYT grant MEC-06/FPA2006-09199 (Spain) and
by the European Commission FP6 program MRTN-CT-2004-005104, in which she is associated
to Universidad Autónoma de Madrid. D.R.G. is supported by the Fullbright-M.E.C. fellowship
FU-2006-0740.
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	Introduction
	A new giant graviton solution
	The Kaluza-Klein monopole probe
	The action for the wrapped monopole
	The giant graviton solution
	A microscopical description in terms of dielectric gravitational waves
	A possible interpretation in the dual field theory
	Conclusions
