#### On the oscillation of species

Iosif Bena
3
Simon F. Ross
1
Nicholas P. Warner
0
2
3
0
Institut des Hautes Etudes Scientifiques
, 91440 Bures-sur-Yvette,
France
1
Centre for Particle Theory, Department of Mathematical Sciences, Durham University
, South Road, Durham DH1 3LE,
U.K
2
Department of Physics and Astronomy, University of Southern California
,
Los Angeles, CA 90089, U.S.A
3
Institut de Physique Theorique
, CEA Saclay, CNRS-URA 2306, 91191 Gif sur Yvette,
France
We describe a new class of BPS objects called magnetubes: their supersymmetry is determined by their magnetic charges, while their electric charges can oscillate freely between different species. We show how to incorporate these objects into microstate geometries and create BPS solutions in which the charge densities rotate through different U(1) species as one moves around a circle within the microstate geometry. Our solutions have the same time-like supersymmetry as five-dimensional, three-charge black holes but, in various parts of the solution, the supersymmetry takes the null form that is normally associated with magnetic charges. It is this property that enables the species oscillation of magnetubes to be compatible with supersymmetry. We give an example in which the species oscillate non-trivially around a circle within a microstate geometry, and yet the energy-momentum tensor and metric are completely independent of this circle: only the amplitude of the oscillation influences the metric.
1 Introduction
Merging the two types of supersymmetry
2.1 Time-like and null supersymmetry of M2 and M5 branes
2.2 The classic solutions with time-like supersymmetry
2.3 Merging the two classes of supersymmetry
2.4 A degenerate limit and a class of null BPS solutions
3 Supergravity coupled to four vector multiplets
3.1 The supergravity action
3.2 The supersymmetry conditions
3.3 Adding species on the T 4
4 Solutions in six and ten dimensions
4.1 Lifting to six dimensions 4.2 The ten-dimensional IIB configuration 5 7
5.1 AdS3 S2
5.2 Adding the supertube/magnetube
5.3 Regularity near sources
5.4 Geometry of the regular magnetube
5.5 AdS3 S3 asymptotics at infinity
5.6 Global regularity
6 Species oscillation
6.1 Some simple oscillating solutions
6.2 Regularity
6.3 Solutions with a -independent metric The oscillating modes
7.1 The Greens function
7.2 The Fourier modes
7.2.1 The explicit modes
7.2.2 On-axis limit
7.2.3 Examples
7.3 Asymptotics of the modes at infinity 1 2 5
Conclusions
8.1 Comments on supersymmetric species oscillation
8.2 Non-BPS, asymptotically flat solutions 33 33 34
Introduction
Microstate geometries play a double role in the study of solutions in supergravity. First,
microstate geometries involve a mechanism that supports time-independent, smooth,
solitonic, horizonless solutions within massless supergravity theories and so such solutions can
potentially represent completely new classes of end-states for stars. The second role is that
there are vast classes of such solutions and these classes come with extremely large moduli
spaces. As a result, the fluctuations of microstate geometries can capture a significant
portion of the microstate structure of the black hole that possesses the same asymptotic
structure at infinity. To understand the extent to which such classical solutions can capture
microstate details it is important to understand and classify the moduli spaces of solutions
and find all the ways in which such solutions can fluctuate. This has been the focus of much
recent work and in this paper we will exhibit a new and broader class of BPS fluctuations
that we call species oscillation.
The key building block of solutions with species oscillations is a new object of string
theory, which we will call the magnetube. The magnetube has M5 and momentum charges
(which are magnetic in five dimensions), as well as several electric M2 charge densities that
can oscillate between positive and negative values along the M5-P common direction. The
original motivation for studying this object, and species oscillation in general, was the idea
that it can be used to create microstate geometries for Schwarzschild (electrically neutral)
black holes [1]. The specific magnetube considered in [1] involved a KK-monopole with
(NS5, D5) charge specified by a vector at each point along a line in the KKM worldvolume.
This charge vector rotates in the (NS5, D5) charge space as one goes around this line so
that the resulting configuration has no overall (NS5, D5) charge. At each point on the
line the configuration looks like a piece of a BPS supertube, but at different locations
the D5 charge of this object can have opposite signs, and so it does not preserve any
supersymmetry associated with these charges.
Our first result is to show that the object found in [1], as well as the class of M5-P
magnetubes that we construct in this paper, are secretly supersymmetric. Normally one
does not expect this, because the negative and positive M2 (or D5) charge densities at two
different locations will attract and will generically want to collapse into each other.
However, the story is much richer: because of the presence of extra species one can arrange the
M2-brane densities so that, at each point along the direction in which the charge densities
vary, the Killing spinors are those of the (magnetic) M5 branes and momentum, and are
not affected at all by the presence of (electric) M2 branes. Hence, the physics of the
magnetube is the mirror of the physics of normal supertubes [2]: for the magnetube the magnetic
(M5 and P) charges control the supersymmetry, and the contributions of the fluctuating
electric charges to the Killing spinor equations cancel; for normal supertubes the (electric)
M2 branes control the supersymmetry and the (magnetic) M5 and momentum charges can
fluctuate arbitrarily, as their contribution to the Killing spinor equations cancel each other.
The original spirit of [1] was to try to bend the infinite magnetube with oscillating
species (whose solution they constructed) into a round magnetube, which would be neutral
and hence give microstate geometries for the Schwarzschild black hole. It is clear that this
would break the supersymmetry of the infinite magnetube, as the resulting configuration
will have mass and no charge. However there is also an indirect and more useful way
to see how supersymmetry is broken: bending an infinite tube in R3 S1 into a round
one in R4 can be realized by adding a Taub-NUT center to the solution and bringing this
center close to the tube (in the vicinity of the Taub-NUT center the metric is R4). For
normal supertubes this procedure is supersymmetric [35], because the Killing spinors of
the Taub-NUT space are compatible with the M2-brane Killing spinors of the supertube.
However, the magnetube has M5 and P Killing spinors, and those are not compatible with
the Taub-NUT ones,1 and hence trying to create a neutral configuration results necessarily
in the breaking of the supersymmetry preserved by the infinite magnetube solution.
Our second result is to find a way to embed M5-P magnetubes into some of the known
supersymmetric solutions and to obtain new classes of BPS solutions with species
oscillation. Using magnetubes and species oscillation to create more BPS solutions may appear
contrary to the original spirit of [1], which proposed these objects as building blocks for
non-extremal microstate geometries. However, this is not so: our result strengthens the
evidence for the existence of magnetubes, and proves that there is no obstruction to bending
them, and hence to using them in principle for the purpose originally intended in [1].
Embedding a magnetube into a supersymmetric solution that has M2 Killing spinors
may, at first, seem impossible because the magnetic Killing spinors of the magnetube are not
compatible with the electric M2 Killing spinors preserved by the background. Specifically,
given a Killing spinor, , one can define the vector K = and this is always a Killing
vector that represents the time-translation generated by the anti-commutator of two
supersymmetries. In electric BPS solutions this Killing vector is time-like and in magnetic
BPS solutions this Killing vector is null: we therefore use the terminology time-like or
null supersymmetry to refer to these two distinct and apparently incompatible classes.
One can also verify that the supersymmetry projectors associated with the underlying
magnetic and electric charges are also incompatible because the projectors do not commute
with one another.
On the other hand, when lifted to six dimensions all supersymmetric solutions give rise
to null Killing isometries and so, from this perspective, there should be some possibility
of interpolation between such supersymmetries. Indeed, even in five dimensions there is a
crucial loop-hole in the structure of the electric BPS supersymmetries: the four-dimensional
spatial bases are allowed to be ambi-polar, which implies that while the Killing vector
associated with the supersymmetry is time-like almost everywhere (and particularly at
1One can see this easily by compactifying to type IIA string theory, where the Taub-NUT space becomes
a D6 brane, and the M5 and P become D4 and D0 branes respectively.
infinity), there are critical surfaces, where the signature of the base space changes from
(+, +, +, +) to (, , , ), and where the Killing vector becomes null. We will show that
the usual time-like supersymmetries of electric charges actually get infinitely boosted and
become magnetic or null supersymmetries on the critical surfaces. It is therefore possible
to introduce species oscillation on magnetubes localized right on top of a critical surface,
where the Killing spinors of the background become magnetic and are therefore compatible
to those of the magnetubes, preserving supersymmetry globally.
Besides establishing the existence of magnetubes as fundamental building blocks of
black hole microstates, the new families of supersymmetric solutions that we construct are
interesting in their own right. For example, in four dimensions the uniqueness theorems
are very stringent but in five and six dimensions there are huge families of microstate
geometries with the same asymptotic charges. It is thus interesting to explore the range of
possibilities and our new classes of solution will have electric fields that fluctuate and yet
the metric either does not fluctuate at all, or the fluctuation response is locally suppressed
due to coherent combinations of fluctuating charge densities. This class of solutions is, in
this sense, similar to Q-balls [6, 7]. The latter solutions are supported by time-dependent
matter fluctuations arranged so that the energy-momentum tensor and hence the metric
are both time-independent. The solutions we construct here are not time-dependent but
the fields fluctuate as a function of some S1-coordinate, , and yet the energy-momentum
tensor and metric can be arranged to remain invariant along . More generally, one can
arrange the fields to fluctuate in such a way that the leading-order perturbations to the
energy-momentum tensor near the source remain -independent with the fluctuations only
becoming visible at sub-leading orders.
We will formulate species oscillation in five-dimensions using a T 6 compactification
of M-theory that is essentially described in [8]. There will be one spectator M2 brane
wrapping a fixed T 2 and the species oscillation will take place between four classes of
M2branes that wrap the remaining T 4 in different ways. One can also reduce this to IIA by
compactifying on an S1 inside the T 2 of the spectator M2, converting it to an F1 string
while the oscillating M2s become oscillating D2s. If one T-dualizes on the circle wrapped
by the F1, one obtains a IIB compactification in which the F1 has become momentum
charge and the oscillating species are simply three sets of D3-branes that intersect along
a common circle while wrapping the T 4 in exactly the same manner as the original M2s.
The importance of the IIB frame is that it is only in this frame that our solutions will
be completely smooth microstate geometries. Our formulation also has the advantage of
having all the oscillating species originating from the same type of branes. It is also possible
to relate some of our magnetube solutions to the smooth IIB frame magnetubes obtained
in [1], by a duality sequence that we will discuss in section 4.2.
In section 2 we discuss the two classes, time-like and null, of supersymmetry and their
relation to electric and magnetic BPS solutions in M-theory. We also show how both types
of supersymmetry can be present in BPS solutions with ambi-polar base metrics. In
section 3 we discuss the relevant five-dimensional N = 2 supergravity theories and their origins
in M-theory. In section 4 we discuss the six-dimensional uplifts and their relation to IIB
supergravity. We also discuss the regularity conditions that make supertubes into smooth
microstate geometries in six dimensions. Section 5 contains a template microstate
geometry which is a standard, non-oscillating microstate geometry with two Gibbons-Hawking
(GH) geometric charges and a supertube on the critical surface between the GH charges.
In reality this supertube is actually a magnetube because it preserves a null
supersymmetry. Indeed, the constructions of supertubes and magnetubes are mathematically parallel
and, following the logic of section 2, one can think of magnetubes as infinitely-boosted
supertubes. In section 5 we also check the regularity and asymptotic structure of this
magnetube solution. In section 6 we introduce species oscillation into the template
magnetube and give an example in which the metric does not oscillate. We also see that the
metrics of the oscillating solutions are almost identical to those of the template solutions
and are therefore regular. In section 7 we give details of the Greens functions and explicit
mode functions that are the essential, though technical, part of our solutions with species
oscillation. While our examples involve only very simple microstate geometries, they will
nevertheless provide a good local model of what we expect from species oscillation in a
generic bubbling solution. Finally we make some concluding remarks in section 8.
Merging the two types of supersymmetry
In this section we illustrate one of the key ingredients of species oscillation: the fact that
one can mix both time-like and null supersymmetric components within a single,
fivedimensional solution. We will assume some familiarity with the five-dimensional microstate
geometries that have been constructed over the last few years. One can find discussions of
this in, for example, [1821]. In section 3 we will discuss the general form of five-dimensional
N = 2 supergravity coupled to an arbitrary number of vector multiplets and this can
also serve as something of a review of the relevant supergravity structure but in greater
generality than is needed to understand the merging of the two types of supersymmetry.
Since the latter idea is something rather new, we have chosen to show how it works here
first before diving more deeply into the technicalities of the more complicated supergravity
theories that are needed to implement species oscillation. For newcomers to the microstate
geometry program it might be useful to review section 3 first.
Time-like and null supersymmetry of M2 and M5 branes
In both eleven-dimensional and five-dimensional supergravity there are two distinct classes
of supersymmetry, which are classified by whether the associated time translation
invariance is actually time-like or null. More precisely, if is the residual supersymmetry then
the vector
is necessarily a Killing vector, and it can either be time-like or null [9, 10]. It is relatively
easy to see that the vector is dominated by its time component because, in frame indices,
one has 0 = . To investigate the other components one needs to know a little more
about the structure of the supersymmetry.
One the simplest descriptions of BPS three-charge black holes in five dimensions is
obtained by compactifying eleven-dimensional supergravity on a six-torus, T 6. The charges
are then carried by three sets of mutually BPS M2-branes and thus the supersymmetries
obey the projection conditions
= ,
where the five-dimensional space-time is coordinatized by (t = x0, x1, x2, x3, x4) and the
T 6 is coordinatized by (x5, . . . , x10). The microstate geometries corresponding to such
black holes are, by definition, required to have the same supersymmetries, in that they also
satisfy (2.2). One should also note that because 01...9 10 = , the conditions (2.2) also imply
= ,
which constrains the holonomy of the spatial base.
One can now insert the 0cd into (2.1) and commute through the a. The fact that a,
for a 6= 0, anti-commutes with at least one of the 0cd means that a = 0 for a 6= 0 and so the
Killing vector for such black holes and their microstate geometries is necessarily time-like.
The BPS three-charge black holes and their microstate geometries are dominated by their
electric charge structure but can carry non-zero, dipolar magnetic charge distributions
whose fields fall off too fast to give any net charge at infinity.
There is a simple magnetic dual of this picture in which the M2 branes are replaced
by M5 branes. We will take the M5 branes to have a common direction, = x4, which,
for the present, will be flat and either infinite or a trivially fibered S1. The M5-brane
supersymmetries obey the projection conditions
= ,
= .
Again one can insert the 0 into (2.1) and commute through the a and conclude that
a = 0 for a 6= 0, . Moreover (2.5) implies 0 = and so the Killing vector is necessarily
null. Thus these systems of M2 branes and M5 branes are exemplars of the two classes of
supersymmetric systems in five and eleven dimensions.
For our purposes, it is important to note that, just as the BPS three-charge black holes
and microstate geometries can be given non-zero, dipolar magnetic charge distributions
whose fields fall off too fast to give any net charge at infinity, the M5 brane system can be
given non-zero, dipolar electric charge distributions whose fields fall off too fast to give
any net charge at infinity. In this way one can make BPS magnetubes that have no net
electric charge and this observation will lie at the heart of supersymmetric species oscillation.
The classic solutions with time-like supersymmetry
So far we have considered rather simple classes of BPS configurations and, as we will see,
more complex BPS backgrounds can lead to mixtures of both types of supersymmetry.
Specifically we want to consider backgrounds that have become the standard fare for the
construction of microstate geometries.2 The eleven-dimensional metric has the form:
ds121 = (Z0Z1Z2) 3 (dt + k)2 + (Z0Z1Z2) 3 ds42 + (Z0Z2Z12) 13 (dx52 + dx62)
2 1
where the four-dimensional space-time metric has the standard Gibbons-Hawking (GH)
form:
ds42 = V 1 (d + A)2 + V d~y d~y ,
The eleven-dimensional Maxwell three-form potential is given by
with the five-dimensional Maxwell fields, A(I), given by:
A(I) = ZI1 (dt + k) +
LI = `I0 +
j=1 rj
j=1 rj
KI = k0I +
M = m0 +
j=1 rj
where rj |~y ~y(j)|.
The electrostatic potentials and warp factor functions, ZI , are given by
ZI =
CIJK V 1 KJ KK
2 V
(KI ~LI LI ~KI ) .
2For later convenience we are going to label the species of M2 branes and the corresponding fields by
0, 1, 2 and not use the more usual labeling, 1, 2, 3. We have therefore taken the standard description and
replaced the label 3 by 0 everywhere.
Merging the two classes of supersymmetry
One can uplift the five-dimensional supergravity solutions to six dimensions by using one of
the vector fields to make a KK fiber. We will discuss this extensively in section 5 but here
we simply want to note that in the six-dimensional uplift there is only one kind of
supersymmetry [22, 23]: the null supersymmetry with = 0. The two types of supersymmetry
in five dimensions must therefore emerge from the details of the compactification to five
dimensions. In this sense one might naturally expect to unify the two classes of
supersymmetry, and we will now elucidate how this can be seen directly from the five-dimensional
perspective.
As have been extensively noted elsewhere [1721] the metric on the four-dimensional
base is allowed to be ambi-polar, that is, it is allowed to change signature from (+, +, +, +)
to (, , , ). In spite of this, the five-dimensional and eleven-dimensional metrics are
smooth and Lorentzian. In particular, this means that V is allowed to change sign and
the surfaces where V = 0 are called critical surfaces. One can make a careful examination
of the metric (2.6) and Maxwell fields (2.10) and show that the apparently singular terms
involving negative powers of V actually cancel out on critical surfaces, leaving a smooth
background. It is also important to note that the functions ZI V must satisfy
= (Z0Z1Z2) 32 = ((Z0V )(Z1V )(Z2V )) 32 V 2 .
Since the ZI V are globally positive, this means that K is time-like except on critical
surfaces, where it becomes null. Thus the supersymmetries are time-like almost everywhere,
except on critical surfaces where they momentarily become null.
It is therefore possible, in principle, to have both types of supersymmetry within
one class of solutions. Indeed, black-hole microstate geometries have the same time-like
supersymmetries at infinity as a black hole and generically have critical surfaces in the
interior of the solution where the supersymmetries become null. It is instructive to see how
this comes about in detail.
Introduce the obvious frames for the five-dimensional metric:
e0 Z1 (dt + k) ,
e1 (ZV ) 21 V 1 (d + A) ,
ea+1 (ZV ) 21 dya ,
Note that we have chosen to write these quantities in terms of the positive functions, ZI V ,
so that the fractional powers are unambiguous and have no branch cuts. Also observe that
the frames e0 and e1 are, respectively, degenerate or singular on critical surfaces.
Define null frames:
ds52 = (e0)2 +
It is also convenient to introduce the quantity
Q Z0Z1Z2V
V CIJK LI LJ LK + CIJK CIMN LJ LK KM KN .
In particular, it follows from (2.25) that Q is smooth across the critical (V = 0) surfaces.
The whole point is that while the frames e0 and e1 are singular on the critical surfaces,
e+ and e are smooth on these surfaces. To see this, note that
e = (ZV )1(dt + ) + (ZV )1
It follows from (2.13) and (2.15) that ZV and V 2 are well behaved on critical surfaces
and thus e+ is manifestly well behaved. Now observe that
Q =
K = (K0K1K2) 13
and note that global positivity of ZI V implies that K0K1K2 > 0 on critical surfaces and
so K is real and smooth across these surfaces. As V 0, one finds
e
K2(dt + ) + 12 K5 Q (d + A) ,
which are clearly regular as V 0.
Finally, define new canonical Lorentzian frames via
= 0 = 0 = Z1 = (ZV )1 V ,
where we are using frame indices based on (2.19). This implies that in this singular frame
basis the magnitude of vanishes as O(|V |1/2) when V 0. Passing to the non-singular
frames (2.32) requires a Lorentz boost that acts on :
= exp
= 12
= ,
remain finite. Thus the Killing spinor is time-like everywhere except where V = 0, where
it becomes momentarily null.
We therefore conclude that if V 6= 0 then the supersymmetry is precisely that of the
three-charge M2-brane system, but on the V = 0 surfaces this supersymmetry becomes
compatible with the supersymmetry of M5 branes wrapped on the -circle and with
momentum charge on that circle.
A degenerate limit and a class of null BPS solutions
The fact that, despite appearances, the BPS solutions described in section 2.2 are regular
across V = 0 surfaces enables one to take this somewhat further and find classes of BPS
solutions by taking V 0 everywhere. As one might anticipate from the discussion above,
these solutions have null supersymmetry and are sourced primarily by M5 branes and
momentum.
To take this limit one can rescale V V everywhere in the solutions of section 2.2
and take to zero. The metric (2.6) simplifies to
(dx72 + dx82) +
(dx92 + dx120) ,
(dx52 + dx62)
where K is defined in (2.29) and Qb is simply Q with V 0:
Qb = 2 K3 M 14 (KI LI )2 + 14 CIJK CIMN LJ LK KM KN .
The angular momentum vector, , is now determined by the simpler equation
1
~ ~ = 2 (KI ~LI LI ~KI ) .
The electromagnetic fields (2.10) reduce to the purely magnetic forms:
A(0) =
A(1) =
A(2) =
K0 L0 K1 L1 K2 L2 d + ~(0) d~y ,
K1 L1 K2 L2 K0 L0 d + ~(1) d~y ,
K2 L2 K1 L1 K0 L0 d + ~(2) d~y .
This is the solution that corresponds to the infinite magnetube and, as expected, the Killing
vector, = t , is manifestly null everywhere in the metric (2.37).
Supergravity coupled to four vector multiplets
The supergravity action
Species oscillation requires the addition of extra vector multiplets to the N = 2 supergravity
theory employed in section 2 and we will therefore summarize the relevant aspects of these
theories. Our conventions and normalizations will be those of [24, 25].
The action of N = 2, five-dimensional supergravity coupled to N U(1) gauge fields is
given by:
S =
1 Z
CIJK XI XJ XK = 1 .
It is also convenient to define
Using the constraint (3.5), one can show that the inverse, QIJ , of QIJ is given by:
CIJK = II0 JJ0 KK0 CI0J0K0 .
QIJ = 2 XI XJ
6 CIJK XK ,
and one can show that
1 CIJK XI XJ XK = 1 . (3.8)
6 27
We are simply going to follow [8, 28] and consider eleven-dimensional supergravity
reduced on a T 6. The Maxwell fields descend from the tensor gauge field, C(3), via harmonic
2-forms on T 6. The structure constants CIJK are then simply given by the intersection
form of the dual homology cycles and the XI are moduli of the T 6.
The supersymmetry conditions
We start with the most general stationary five-dimensional metric:
ds52 = Z2 (dt + k)2 + Z ds42 ,
where Z is simply a convenient warp factor. Supersymmetry implies, via the
condition (2.3), that the metric ds24 on the spatial base manifold, B, must be hyper-Kahler.
One now defines N + 1 independent functions, ZI by
and then (3.8) implies
ZI = 3 Z XI ,
Z = 1 CIJK ZI ZJ ZK . (3.11)
6
It is more convenient to think of the solution as parametrized by the N + 1 independent
scalars, ZI , and that the warp factor is determined by (3.11).
Supersymmetry requires that the Maxwell potentials all have the form
where B(I) are purely magnetic components on the spatial base manifold, B. One defines
the magnetic field strengths accordingly:
Having made all these definitions, the BPS equations take on their canonical linear
form [29]:
where ?4 is the Hodge dual in the four-dimensional base metric ds24, and (24) is the
(fourdimensional) Laplacian of this metric.
We are, once again, going to take the metric on B to be a GH metric (2.7) and so we will
also decompose the vector k according to (2.14). The GH metric and the magnetic fluxes
will also be as before and thus we will take V and KI to have the form (2.11). However, we
will allow the rest of the solution (ZI , and ~) to depend upon all four variables, (, ~y).
The ZI s will still have the form (2.13) but now the LI are general harmonic functions on
the base B:
(24)LI = 0 . (3.17)
The BPS solutions in these circumstances have been discussed in [16, 30]. The last BPS
equation, (3.16), can be written as
3
(D~ V V D~ ) + D~ ~ + V ~ = V X ZI ~ V 1KI ,
This BPS equation, (3.16), has a gauge invariance: k k + df which translates to
The four-dimensional Laplacian can be written as
(24)F = V 1 V 2 2F + D~ D~ F .
(24)LI = (24)M = 0 .
If one takes the covariant divergence of (3.18) (using D~ ) and uses the Lorentz gauge choice,
one obtains
This equation is still solved by
D~ ~ + V ~ = V D~ M M D~ V + 2 I=1
1 N+1
KI D~ LI LI D~ KI .
One can verify that the covariant divergence (using D~ ) generates an identity that is trivially
satisfied as a consequence of (2.8), (3.21), (3.24) and
Adding species on the T 4
From the eleven-dimensional perspective, we are going to add extra Maxwell fields coming
from three-form potentials with two legs on the T 4 defined by (x5, x6, x7, x8) but leave the
fields on the other T 2, defined by (x9, x10) unchanged and supporting only one Maxwell
field, which we have labelled as A(0). Thus the only non-zero components of intersection
product, CIJK , are
for some matrix, CbJK = CbJK . One can easily see that (3.5) implies that, as matrices,
Cb3 = Cb and so, assuming that Cb is invertible, we have
C0JK = CbJK = CbKJ ,
The four-torus, T 4, has six independent harmonic forms and by generalizing the
Ansatz (2.9) these forms can give rise to six vector fields in five dimensions. However,
some of these vector fields belong to N = 2 gravitino multiplets and do not lie in an
N = 2 supergravity theory coupled to vector multiplets. Indeed, the vector fields in such
an N = 2 supergravity theory are associated with forms in H(1,1)(T 4, C) [8, 26, 27] for a
suitably chosen complex structure. We will take this complex structure to be defined by
w1 = x5 + ix6 ,
w2 = x7 + ix8 ,
w0 = x9 + ix10 .
give rise to the other two vector fields, (apart from A(0)), in (2.9). There are two more
forms in H(1,1)(T 4, C):
1 1
(dw1 dw2 + dw1 dw2) = (dx5 dx7 + dx6 dx8) ,
3 2 2 2
i 1
(dw1 dw2 dw1 dw2) = (dx5 dx8 dx6 dx7) .
4 2 2 2
This leads to an intersection matrix
0 0 0 1
which satisfies (3.28). Indeed the normalizations in (3.31) are set so as to satisfy (3.5)
and (3.28) and yield canonical and uniform normalization of the Maxwell fields.
The eleven-dimensional three-form potential has therefore the form:
C(3) = A(0) dx9 dx10 +
This is a very modest generalization of the class of theories considered in [8, 28]. Indeed, we
can easily truncate down to A(0), A(1), A(2), A(4) by imposing invariance under the discrete
inversion (t, , ~y, x5, x7, x9) (t, , ~y, x5, x7, x9).
Observe that (3.11) now implies that the space-time metric warp factor, Z, is given by
Z3 = 1
2
Z0 CbIJ ZI ZJ = Z0
It is also convenient to define the quadratic combination:
P CbIJ ZI ZJ =
The eleven-dimensional metric in this truncation is given by
ds121 = Z2(dt + k)2 + Z ds42 + Z
Z01 |dw0|2 + P 1 Z2 |dw1|2 + P 1 Z1 |dw2|2
Solutions in six and ten dimensions
We are ultimately seeking microstate geometries with species oscillation. This is most
easily achieved by using charge density modes on supertubes and for such geometries to be
smooth we must go to six-dimensional supergravity. Indeed, one must choose one of the
vector fields to become the geometric KK fiber and the remaining N vector fields become
encoded in one self-dual and N 1 anti-self-dual tensor multiplets in the six-dimensional
theory. We will not need all the details in six dimensions because the smooth Maxwell fields
in five dimensions will become smooth tensor gauge fields in six dimensions. The only detail
we have to worry about is the singular Maxwell field coming from the supertube source in
five dimensions and how this becomes smooth geometry in six dimensions.
Lifting to six dimensions
The six-dimensional metric is
1
ds62 = 2H1(dv + ) du + k 2 Z0 (dv + )
H ds42
where the four-dimensional base is exactly that of the five-dimensional theory. The
functions Z0 and K0 define the geometric KK vector field and so the vector potential, , in (4.1)
is
V (d + A) + ~ d~y ,
The six-dimensional warp factor, H, is defined by the quadratic in (3.35)
H
P =
lim r2 Q = 0 , (4.8)
r0
where Q is defined in (2.24) and expressed in simplified form in (2.25).
The two conditions, (4.7) and (4.8), guarantee that a supertube smoothly caps off the
spatial geometry, up to orbifold singularities [1113].
The ten-dimensional IIB configuration
The six-dimensional KK uplift above is most simply related to a T 4 compactification of
IIB supergravity. It is relatively easy to see what the corresponding brane configuration is
by compactifying and T-dualizing the M2 and M5 configurations described in section 3.
If one compactifies on x10 then the M2-branes of species 0 become F1-strings
wrapping x9 and all the other M2-branes become D2-branes. The M5-dipole charge of species
0 becomes an NS5-dipole charge wrapping , x5, . . . , x8, where is the GH fiber, while
all the other M5-dipole charges become D4-dipole charges. Performing the T-duality on
x9 converts the F1-strings and NS5-branes to momentum and KK monopole charges
respectively, and all the D2-charges become D3-charges intersecting on a common x9 and
wrapping the T 4 defined by (x5, . . . , x8) in exactly the same manner as the original
M2charges. Similarly, the D4-dipole charges become D3-dipole charges intersecting on the
common -circle and wrapping the T 4 defined by (x5, . . . , x8) in exactly the same manner
as the original M5 dipole charges.
Thus we obtain a IIB configuration in which all the M2 species have become either
D3branes or momentum. One can perform two more T-dualities on the T 4, for example along
(x5, x6), thereby converting two of the sets of D3-charges into D1- and D5-charges. This
T-duality changes the other D3-charge species into D3-charges with different orientation,
and introduces additional NS-NS B-fields on the T 4. Thus one can obtain the canonical
D1-D5-P system decorated with additional D3-charges. This last T-duality does not affect
the space-time metric and its regularity.
There is another way to dualize our configurations to magnetubes with D1 and D5
charges. If one starts from a magnetube whose D1 and D5 charges oscillate into F1 and
NS5 charges [1], there is a duality sequence, given by equation (3.1) in [8], that takes this
solution into a solution of the type we construct here but where the species corresponding
to A3 and Z3 is absent. Hence, if one applies the inverse of this duality sequence on a
solution with A3 = 0 = Z3 one obtains the magnetube of [1]. If one applies the inverse
duality sequence to a more general solution, where the third species is also present, the
resulting magnetube will have two extra charges, corresponding to D3 branes with two legs
along the T 4 and one along the common D1-D5 circle.
The non-oscillating template
We now describe in detail a simple example of a smooth magnetube geometry, which we
will construct by putting a single supertube (with no species oscillation) on the V = 0
surface in the simplest ambipolar Gibbons-Hawking space. The discussion here follows
closely previous work on supertubes in such backgrounds, but there is some novelty in
dealing with putting the supertube on the V = 0 surface. Our aim in this section is to
show that the conditions for smoothness of the supertube are unchanged by placing it
on the V = 0 surface, and hence translate directly in the conditions for smoothness of
a magnetube. The non-oscillating solution provides a simple context to deal with these
issues and set up the framework before we move on to introduce species oscillation. We will
see later that for the solutions with species oscillation we can choose the charge densities
such that the angular momentum density remains constant, and then the structure of the
metric in the oscillating solutio n is almost identical to that of the non-oscillating solution.
We start from the simplest solution with a non-trivial critical surface, a two centered
metric. We first review the background solution before we introduce the supertube; we
consider a particularly simple example in which the five-dimensional geometry is simply
AdS3 S2 [1416]. This solution is axisymmetric and so it is more convenient to write the
four-dimensional base in terms of cylindrical polars on the flat R3 slices:
ds42 = V 1 d + A)2 + V (d2 + dz2 + 2d2) .
For simplicity we take the two GH charges to be equal and opposite, so
V = rq+ rq ,
for some parameter a. In this simple two-centered solution, the solution will have the
classic form of section 2.2 with two vector multiplets, except we will use the labeling
{I, J, K} = {0, 1, 2} instead of the more common usage ({I, J, K} = {1, 2, 3}).
We initially take the magnetic flux functions, KI , to have the simple form
where r p2 + z2. The electric source functions, LI , are set to
1
r
The coefficients of the r1 terms in the LI have been fixed by the usual regularity condition
for bubbled geometries in five-dimensions: ZI should be finite at r = 0.
The harmonic function M in the angular momentum is
Again the coefficients of the r1 terms in M have been fixed by the usual five-dimensional
regularity conditions for bubbled geometries: must be finite at r = 0. The parameter
m is introduced to ensure smoothness and lack of Dirac strings. Finally, the 3-dimensional
angular-momentum vector, , is determined using (2.16):
To make sure that there are no CTCs near r = 0 and to remove Dirac strings, we
must impose the five-dimensional bubble equations: (r = 0) = 0. For this two-centered
solution this requires only that
In particular, one has
Rescaling and shifting the remaining variables according to
the five-dimensional metric then takes the standard global AdS3 S2 form:
ds52 R1 cosh2 d 2 + d2 + sinh2 d12
2
R1 = 2R2 = 4k .
(5.14)
One should note that an observer whose world-line has tangent, u , i.e. the observer
is fixed in the GH spatial base, follows a curve with
which means that the proper 4-velocity has norm: 4k2 cos2 . Thus this GH-stationary
observer follows a time-like curve everywhere except on the equator ( = 2 ) of the S2 where
the observers trajectory becomes null. This illustrates the general discussion in section 2.3:
in patches that are smooth across the critical surface V = 0, the GH-stationary observers
are being boosted from time-like to null trajectories as V 0.
Adding the supertube/magnetube
k2
L0 = q
1
r
1
r
The coefficients of the r1 terms in the LI are as before. The sources at r = 0 in K0,
L1 and L2 correspond to those of a supertube with dipole charge and electric charges
1 and 2. We want to demonstrate that for appropriate choices of these parameters the
addition of this supertube leads to a smooth six-dimensional geometry that asymptotically
approaches an AdS3 (S3/Z2k+) geometry.
The harmonic function M in the angular momentum is now
Again the coefficients of the r1 terms in M are as before. The parameter m0 represents
the angular momentum of the supertube while the parameter m is introduced to ensure
smoothness and lack of Dirac strings.
Finally, the 3-dimensional angular-momentum vector, , is determined using (2.16)
and we now find
2k3 2 + (z a + r+)(z + a r)
= qa r+r
+ a1 m0q + 2kq2 2 + (z arr++r+)(z r) + 2 + (z + rr)r(z + a r)
Regularity near sources
To make sure that there are no CTCs near r = 0 and to remove Dirac strings, we must
impose the five-dimensional bubble equations: (r = 0) = 0. This now imposes two
constraints:
Since K1 = K2, the absence of Dirac strings at the magnetube is already guaranteed
by (5.20). This is, of course, to be expected since a Dirac string must have two ends and
we have eliminated them everywhere else.
The other supertube regularity condition (4.8) is easily computed from (2.25)
particularly since V = 0 and L3 = 0 at r = 0. One finds that this condition is satisfied if
which is the standard relationship between the charges and angular momentum of a
supertube. The free parameters are then just the charges , , as is standard for a supertube.
Geometry of the regular magnetube
For the later discussion of the oscillating solution, it is useful to summarize here the
functions appearing in the six-dimensional metric (4.1) for the non-oscilating magnetube
solution and to note explicitly where the parameter enters into the solution. The KK electric
function Z0 is independent of the magnetube parameters,
Z0 = V + L0 = V + L,
where K, L are given by (5.4) and (5.5). The other electric functions are
so the warp factor H is
K3 K2 3k2 k2
= V 2 + r V 2 2q2 K 2q2 r + M,
where M is given by (5.18) with the parameters fixed by (5.21) and (5.23). Thus, the
electric charge parameter, , only enters the metric (4.1) directly through the warp factor,
H, and appears indirectly in the one-form k through the dependence of m0 on from (5.23).
Of course, will also appear directly in the gauge fields A(1) and A(2).
AdS3 S3 asymptotics at infinity
One can use the spherical bipolar coordinates (5.9) to study the asymptotic form of our
solution. The warp factor H will behave as
H
To obtain the leading asymptotics, we only need to consider the first term. As , the
leading-order behaviour of the metric is
2k(2k + ) d2 + sin2 d22 +
One can, in fact, allow equality here but then the metric becomes asymptotic to some form
of null wave in which only the d du term survives at infinity. If < 0 then the -circles
become closed time-like curves. We assume that (5.32) is true.
The leading-order asymptotics of the matter fields can similarly be calculated; the two
non-KK scalar fields are
X1 =
so asymptotically
X1 = X2 = 1 + + O(e4). (5.34)
2k
The background value of the scalar fields is rescaled along with the AdS3 radius of
curvature, and there is no sub-leading part at order e2, so there is no vev for the dual
operators. The two non-KK vector fields are
2 4 q k2(2k + ) + 2 k2 (a2kq + ) u
KI
A(I) = ZI1(dt + k) +
(d + A) + ~(I) d~y , I = 1, 2 ,
so we get the same component on the sphere, consistent with the fact that the spheres
radius of curvature is unchanged in the metric (5.29). The second term is locally pure
gauge, but as is a compact direction it introduces a Wilson line.
In AdS3, the asymptotic behavior of a massless gauge field is A A(0) + j ln r, where
A(0) is interpreted as the boundary gauge field and j is the vev of the boundary conserved
current. The absence of a logarithmic term in r (a linear term in ) in the expansion (5.36)
thus indicates that the introduction of the magnetube does not produce any charge density
from the point of view of the dual CFT; the only effect of is to turn on a Wilson line
along the circle.
Global regularity
As we remarked earlier, the two conditions (4.7) and (4.8) guarantee that a supertube
smoothly caps off the spatial geometry [1113] and so, in principle, (5.20), (5.21) and (5.23)
guarantee that the supertube is smooth in six dimensions. However, we actually have a
magnetube: that is, a supertube located at the critical (V = 0) surface. While this should
not affect the arguments of [12, 13], we will now examine the metric in more detail to
confirm that the magnetube limit of the supertube does not add any further subtleties.
Even though the metric appears to be singular at the critical surfaces (V = 0), it is well
known that it is actually smooth across such surfaces (see, for example, [20]). There is also
an apparent singularity at r = 0 but this is resolved by the standard change of coordinate
that shows that the supertube is smooth in six dimensions [1113]. In the solutions we are
considering, these two apparent singularities coincide and while the methods of resolution
of the two types of apparent singularity are not expected to interfere with each other, it is
important to make sure.
Expanding (4.2) in spherical polar coordinates around r = 0, one obtains the leading
order behavior:
1
(2k + ) k2 2 + (2k ) q2 2 .
Provided that Q0 > 0, the metric (5.37) is manifestly a time and S1 fibration defined by
(u, 1) over a four-dimensional spatial base defined by (r, , 2, v). The apparent singularity
at r = 0 in this spatial base is resolved in the usual manner by changing to a new coordinate,
R, defined by r = 14 R2. Thus this spatial base is an orbifold of R4, and the whole metric is
smooth up to such orbifold singularities. Indeed if one restricts to slices of constant (u, 1),
then one has d = 21q d and the four dimensional base metric becomes
ds42 2ka dR2 + 14 R2 d2 + sin2
If one remembers that has period 4 then one sees that this is precisely the metric of
flat R4 for q = = 1. Other integer values of and q thus lead to orbifold singularities.
Smoothness around r = 0 this requires Q0 > 0. If 2k > then this is manifest, but if
not one can use (5.32) to replace 2 and conclude that
4 k4 32 k6
Q0 > q2 a3 (2k + ) + (2k ) (4k + ) = q2 a3 ,
which establishes positivity.
Global regularity of the solution also requires that the functions HV , Z1Z21 and
Z0H1 are globally positive and well-behaved, except for possible singularities allowed by
supertubes. In particular, this means that all the functions ZI V must be globally positive.
This is trivial to see for Z0V since it may be explicitly written as
Z0 V =
r+ r
4k2 r + k (r+ + r) q (r+ r)
4k2 r + k (r+ + r) |q (r+ r)|
4k2 r + k (r+ + r) k (2k + ) | (r+ r)| .
where we have used the fact that (5.32) implies |q | < k(2k + ). Suppose (r+ r) > 0,
then we may write this as
2 k2 (r + r a) + 2 k2 (r + a r+) + 2 k r
0 ,
Similarly one can write
ZI V =
1
r r+ r
1
r r+ r
1
>
r r+ r
g t t = gtt = ((Z0V )(Z1V )(Z2V ))1/3(Q 2) > 0 ,
where is squared using the R3 metric. Thus we need to verify the global positivity of
(Q 2).
For large values of r, one finds
where is defined by (5.31) and hence (5.32) guarantees that (Q2) is positive as one goes
to infinity. Beyond this, we have examined this quantity in several examples of solutions
satisfying (5.32) and found them to be stably causal. A typical example is shown in figure 1.
Species oscillation
Our primary contention in this paper is that species oscillation can be done on any V = 0
surface in any microstate geometry based upon a generic GH base. However, the essential
Greens functions for such generic microstate geometries are not explicitly known. On the
other hand, the introduction of species oscillation involves a localized source and the
twocentered solution considered in section 5 provides an excellent local model of a typical
V = 0 surface in a generic microstate geometry. Moreover, the Greens functions for this
relatively simple system are known [16] and so the solutions can be constructed explicitly.
Given that species oscillation can be implemented, in a very straightforward manner,
to produce new classes of microstate geometries within such a local model, we believe
that the generalization to any microstate geometry should present no difficultly apart from
the fact that explicit analytic examples may not be available.
Some simple oscillating solutions
We will thus construct the simplest example of a non-trivial solution with species
oscillation: we add a magnetube with an oscillating charge distribution to the simple two-centre
background from the previous section. The first step will be to allow the electric charge
sources in (5.17) to be -dependent much as in [16], however here these charges will
oscillate into other species of M2-brane charge. To enable this species oscillation we add two
vector multiplets as discussed in section 3 but only give allow them to have electric charges.
We therefore take the GH base with V still given by (5.2) and keep the same magnetic
sources as in (5.16):
K3 = K4 = 0 .
The electric source functions, LI , are set to:
k2
L0 = q
1
r
I = 3, 4 ,
for some charge-density functions, I (). As before, this corresponds to a supertube that
wraps around the circle and now carries four kinds of electric charges, I (), as well
as the magnetic dipole charge . We will assume that the integral over of the
chargedensity functions I () vanishes, so that (as we will show later for the asymptotic charges)
the supertube is carrying no net electric charge; the electric charge oscillates between the
different species as we go around the circle. This is the key difference from previous
studies with varying charge density [16], and is possible because we take the supertube on
the V = 0 surface.
Since K3 = K4 0, it follows that
It was shown in [16] that for the -dependent solutions we are considering here, supertube
regularity can be analyzed locally in and that regularity is, once again, ensured by
imposing (4.7) and (4.8). As before, (4.7) guarantees the absence of Dirac strings and,
equivalently, this relevant condition can be obtained from (3.25) by requiring that ~ has
no sources of the form of a constant multiple of ~ 1r . Using the fact that K1 K2,
K3 = K4 0 and V and L0 vanish at r = 0, this Dirac string condition implies that the
singularities in L1 and L2 must be equal and opposite at r = 0 and thus:
This equation is the generalization of (5.20) to a magnetube with species oscillations. On
the other hand, one can also recover this equation by starting from a supertube that is not
Using (6.9) and K1 K2, K3 = K4 0 and the vanishing of V and L0 at r = 0, the
second regularity condition collapses in the relatively simple condition:
This implies that
2 () = 1() 2() 12 (3()2 + 42()) = 1()2 +
which is the analog of (5.23). A similar relation between () and the other charge densities
was also found in [16]. Note that while the integral of I () vanishes, the integral of ()
cannot (for non-trivial charge densities) and so the supertube will carry a net angular
momentum. A regular supertube is then parametrized by the constant dipolar magnetic
charge and three independent electric charge densities, 1(), 3() and 4().
For general charge densities, the example constructed above will have a metric that is a
nontrivial function of , although it will become -independent asymptotically. Remarkably,
however, we can choose the oscillating charge densities in such a way that the metric is
completely independent of . As described in the Introduction, this works in a manner
rather reminiscent of Q-balls: the species fluctuate but the energy-momentum tensor, and
hence the metric does not.
The basic idea is to arrange that
1(, ~y)2 + 1 3(, ~y)2 + 4(, ~y)2 = (~y)2 ,
2
for some function (~y). From the asymptotic properties of the I , it follows that we must
have
as this must be satisfied for all ~y, and we only have the freedom to choose the source
functions that are functions only of . However, the isometry of the base metric along
means that the solution can be decomposed into Fourier modes, and if we take the source
charge densities to involve a single Fourier mode, then we can take solutions I (, ~y) that
take a product form, I (, ~y) = Fn(~y)I (), and the ~y dependence factors out of (6.12)
which reduces simply to (6.13).3 Thus, we can satisfy (6.12) by taking, for example,
where is a constant and Fn is normalized so that Fn 1r as r |~y| 0.
In this way, species 1 is locked to species 2 and they oscillate into species 3 and 4.
(There is obviously a family of choices of how to distribute the oscillations amongst species
3 and 4 in such a way as to satisfy (6.13).) We will compute the details of the functions Fn
in section 7, but before going into the technicalities we can make some observations about
the solution.
First note that (6.13) and (6.11) imply that () is constant, and the identity (6.11)
collapses precisely to that of the template solution (5.23). This means that the function M
is exactly as in (5.18). Note, in particular, that M still knows about the amplitude, , of
the species oscillation. Furthermore, having K1 K2, K3 = K4 0 and 1 = 2 means
that the I cancel out in the function in (2.15) and so it is exactly the same function as
it was in the template solution of section 5. Similarly, the I cancel out in (3.25) and the
fact that M is independent of means that satisfies exactly the same equation as in the
template solution and so is identical with the solution (5.19).
The only difference between the metric of the oscillating solution and the
metric of the template arises in the warp factor, H, given by (4.5). It follows
from (2.13), (6.1), (6.2), (6.5) and (6.12) that
H2 =
The important point is that this function is independent of and thus is still a Killing
vector of the metric even though the Maxwell fields and scalars are -dependent. Because
of (6.13) the function H also has exactly the same behavior as r 0 as its
counterpart (5.26) in the template solution. Thus the local regularity of the metric is guaranteed
by that of the template solution.
The fact that the metric is unchanged apart from this change in the warp factor H
also implies that leading-order asymptotics of the metric are also the same as for the
nonoscillating solution. Indeed, the asymptotics is independent of the assumption that the
full metric is -independent. Any -dependence in H is subleading at large distances, and
thus so long as (6.13) is satisfied so that the angular momentum is -independent, the
leading-order asymptotics will be the same as in the non-oscillating solution. Thus, for
all solutions satisfying (6.13), the overall amplitude of the oscillating charge densities is
bounded by (5.32).
Furthermore, the (~y)2 in (6.15) will not contribute to the asymptotic stress tensor:
as we we will see in the next subsection, the functions Fn(~y) represent higher multipole
moments and therefore fall off faster than 1/r, so (~y)2 in (6.15) falls off faster than 1/r2.
Thus this term falls off too fast to contribute to the sub-leading part of the metric that
gives the asymptotic charges. For the gauge fields, there was no contribution to charges in
the non-oscillating case due to the absence of a logarithmic term in (5.36); the faster fall-off
in the oscillating solution does not change this. Thus, the asymptotics of solutions with
species oscillation where the angular momentum density is a constant is very similar to the
non-oscillating case. The only notable difference in the asymptotics is that the suppression
of Fn(~y) implies that the Wilson line noted in (5.36) is absent in the oscillating solution.
The oscillating modes
The Greens function
To obtain the explicit details of our oscillating solution we need to solve (3.17) on the
base defined by (2.7) with (5.2). Fortunately the relevant Greens function was computed
in [16]. The expression for this function was very complicated but here there are significant
simplifications because we have put the source point, (0, ~y), at ~y = 0.
Define the following combinations of coordinates:
(2 u2 1 + y2)
p1 y2 p(2 u2 1)2 y2 ,
Thus the end result can be expressed as a rational function of these variables. In particular,
it is useful to note that
p(2 u2 1)2 y2 =
The Greens function can then be written as [16]
Gb = a u1 y Re
w(1 y w) 1
(w w+)(w w) pu2 y w
1 w
= 2 a u y (w w+)(w w)
where we have used the fact that w = w1.
There are several important things to note: (i) r = 0 corresponds to u = 1 and y = 1
and we are going to work in the region r > 0 in which y1u2 > 1. (ii) The branch points
of pu2 y w are at infinity and at w = y1u2 > 1 and since |w| = 1, the branch cuts can
be arranged to be safely outside the unit circle. (iii) The fact that w+w = 1 means that
one of the poles at w is outside the unit circle while the other is inside. We are choosing
signs of square roots so that |w| 1 |w+|. (iv) The poles at w coincide at 1 if and
only if y = 1. This is the critical surface, and a careful examination of (7.7) shows that
Gb = 0 when y = 1 and is indeed well-behaved across this surface.
The Fourier modes
The explicit modes
One can obtain the Fourier modes from this Greens function by integrating against
ein( 21q 00) = wnein( 21q ). Dropping the phase factors of ein( 21q ), the functions of
where the contour is taken counterclockwise around the unit circle and
1 I
|w|=1 w
dw wn F (w) + F (w1) ,
Since F (w1) has a branch cut inside the unit circle, we make the inversion, w w1 in
this integral (remembering that the inversion also inverts the orientation of the contour)
and write it as
1 I
|w|=1 w
dw (wn + wn) F (w) .
Note that for n = 0 this simply gives r1 and this has been used to set the normalization
of Fn in general. For general n we can write this in terms of Chebyshev polynomials:
pu2 y w =
Fn(w) =
(w+n + wn) .
where Tn is the nth Chebyshev polynomial of the first kind and x is given by (7.3).
where Up(x) are Chebyshev polynomials of the second kind. This series converges for
|w| < |w| and hence near w = 0. The residue at w = 0 can then be written as
1 uy2 w
Thus the solutions to (6.3) can be written as linear combinations of the modes:
Fn ein( 21q ) = Fn(0) + Fn(w) ein( 21q ) .
with Fn(0) and Fn(w) given by (7.15) and (7.13). In particular, Fn(w) gives rise to the
singular source at r = 0 and the remaining parts are essential corrections as we now discuss.
On-axis limit
As we have already seen,
|w|=1 w
dw (wn + wn) = a (2u12 1) n,0 .
and this was used to set the normalization of the modes.
The first few Chebyshev polynomials are
T0 = 1 ,
T1 = x ,
T2 = 2x2 1 ;
U0 = 1 ,
U1 = 2x ,
U2 = 4x2 1 .
F0 =
F1 =
r+ + r 2r
F1
F2
where we have used (7.4) and (7.5). Similarly, one has
F2 =
4(2r a) 6
(r+ + r + 2a)2 + (r+ + r + 2a)
Note that both F1 and F2 vanish on the axis, where r+ + r = 2r. Moreover they both
have a canonically normalized source at r = 0:
Asymptotics of the modes at infinity
From the explicit examples above one sees that the modes Fn 1/rn+1 as would be
expected from charge multipoles. We can easily prove this form in general by recalling that
the Greens function (7.7) was obtained in [16] by taking the Greens function on the AdS3
S2 space (5.12) and Kaluza-Klein reducing to obtain the Greens function on the ambi-polar
base. The asymptotic behaviour of the Fourier modes of the Greens function on the AdS3
S2 space is then easily obtained by a conventional spherical harmonic analysis on the S2.
From the perspective of AdS3 S2, since 2 = 21q + 2 , our Fourier expansion
G = Pn Fn(u, y)wn corresponds to an expansion
The Greens function can be decomposed in terms of the spherical harmonics Ylm(, 2)
on the S2; in this decomposition, Fn will only involve spherical harmonics with l n.
Doing a Kaluza-Klein reduction on the S2, the massless wave equation on AdS3 S2
becomes a massive wave equation on AdS3 for each spherical harmonic mode with a mass
m2 = l(l + 1)/R22 = 4l(l + 1)/R12. Thus, Fn = Pl>n Fln where at large
sinh 21 sinh 2Fln 4l(l + 1)Fln,
so Fln e2(l+1), that is at most Fn 1/rn+1. From the perspective of the AdS3 S2
asymptotics, the species oscillation of the supertube is introducing a variation of the fields
along the 2 direction in the S2, so it corresponds to exciting higher KK harmonics, which
produce vevs for higher-dimension operators in the dual CFT rather than exciting local
charge densities of the conserved currents.
Comments on supersymmetric species oscillation
Our aim has been to show that the object underlying the species oscillation idea introduced
in [1] is a supersymmetric magnetube: a combination of (M5 and P) magnetic charges as
well as several oscillating (M2) electric charges, that carries the same supersymmetries as
M5 branes and momentum regardless of the oscillations of its electric charges. Trying to
bend this object into a ring in R4 will break these supersymmetries, and will result in
neutral configurations that have been proposed in [1] to describe microstate geometries of
neutral Schwarzschild black holes.
We have also succeeded in realizing the species oscillation idea in a supersymmetric
context, by embedding the magnetube into supersymmetric solutions. In doing this we have
uncovered a remarkable unification of the timelike and null types of supersymmetries
directly in the framework of five-dimensional supergravity; the timelike supersymmetry
on an ambi-polar background becomes locally null on the surface V = 0. Hence an object
that would be invariant only under the magnetic-type null supersymmetries, like the
magnetube, can be placed on this surface while preserving the global electric-type
timelike supersymmetry. This enables us to construct solutions with species oscillation which
preserve supersymmetry globally.
We have carefully analyzed the smoothness conditions for magnetubes on the V = 0
surface, initially for a magnetube without species oscillation and then for a magnetube
with M2 electric charge densities that vary around the tube in such a way that the net M2
charges vanish. We found that this magnetube can be obtained by moving a supertube
onto the V = 0 surface (which corresponds to giving it an infinite boost) and that, in
the V 0 limit, the smoothness conditions of the supertube are enough to ensure the
smoothness of the magnetube. We constructed a simple example of a solution with species
oscillation and gave an explicit description of its structure. In this example, the geometry
is independent of the coordinate; the gauge fields are oscillating but their combination
in the stress tensor is -independent, as in the exact near-tube solution discussed in [1].
The simple example we have considered is asymptotically AdS3 S2, and it would be
interesting to understand the interpretation of the solution with species oscillation from
the point of view of the dual CFT. A key aspect is that when we introduce the -dependent
source, the non-diagonality of the metric in the angular coordinates makes the solution a
function of 2q , as can be seen in the Greens function (7.6). This combination becomes an
angle in the S2 factor in the asymptotic solution (5.12). Thus, from the point of view of the
AdS3 S2 background, the species oscillation is not introducing dependence on the angular
coordinate in the AdS3 (shifting 1 at fixed 2 remains a symmetry of the solution) but on
one of the angular coordinates on the S2. Thus, understanding the interpretation of our
solution in the dual CFT will involve a Kaluza-Klein decomposition along the lines of [31].
It will obviously be interesting to further exploit this new freedom in constructing
supersymmetric smooth microstate geometries. It gives us a new possibility to introduce
dipole charges. Examples of solutions with varying charge densities were previously
constructed in [16], but we can now construct solutions with zero net charge. Another
interesting direction for further development is to consider species oscillation on supersymmetric
black strings and black rings.
Non-BPS, asymptotically flat solutions
Following the direction of [1], probably the most interesting question to ask is if one can
deform our supersymmetric solutions to construct non-supersymmetric asymptotically-flat
solutions, as a step towards constructing microstates for black holes like Schwarzschild.
The solution we constructed above is asymptotically AdS3 S2, but one should be able
to construct similar supersymmetric solutions that are asymptotically flat in five dimensions
by considering a supertube/magnetube at the V = 0 surface in a more general
GibbonsHawking base that gives rise to an asymptotically flat background. We have also seen
that the oscillating charge distributions do indeed make no net contribution to the charges
measured from infinity. On the other hand, the size of the oscillation is still bounded by
the background magnetic fluxes and hence by the asymptotic charges.
It is remarkable that one can have stationary asymptotically flat solutions; one might
physically expect that a varying charge density on a rotating object would give rise to
time-dependent multipole moments, which would lead to electromagnetic radiation.
However, the resolution is a familiar effect in supertubes; the supertube is rotating, but the
charge on the supertube corresponds to a charge density wave traveling around the tube
in the opposite direction, so that we get a standing wave. As a result, the multipole
moments of our supersymmetric solutions, whose oscillating parts may be viewed as
superubes/magnetubes, will all be time-independent.
On the other hand, this cancellation of the time-dependence coming from having a
standing wave is a fine-tuned phenomenon. If we excite the supertube/magnetube by
adding some energy, we will need to make it rotate faster to maintain the stabilization by
angular momentum, and it is possible that the counter-rotating charge density wave will
then give rise to time-dependent multipole moments. If true, this would a real obstacle to
finding stationary non-supersymmetric solutions. Note also that the non-stationarity is
associated with the emission of electromagnetic radiation, so the natural timescale for the
decay is likely to be much faster than the gravitational timescales associated with a black hole.
The solution discussed in [1] is vulnerable to a similar problem. In the exact near-tube
solutions the authors introduce dependence on a null coordinate t a, corresponding to a
charge density that is a function of t a. In the near-tube solution is a coordinate along
a straight black string, and this is related to a solution with charge density depending just
on the position along the string by an (infinite) boost. But in the full asymptotically-flat
solution this corresponds to a truly time-dependent charge distribution, and we expect
that this will lead to electromagnetic radiation in the sub-leading corrections to the
solution, introducing time-dependence in the metric on a timescale set by the electromagnetic
radiation reaction.
Thus we believe that species oscillation has opened up an even larger, new moduli space
of BPS microstate geometries but remain uncertain whether this idea can be adapted to
yield stationary non-BPS configurations.
Acknowledgments
We would like to thank Samir Mathur and David Turton for extensive discussions and for
providing us with an advance copy of [1]. We would also like to thank Stefano Giusto,
Rodolfo Russo and Masaki Shigemori for useful discussions. NPW is grateful to the IPhT,
CEA-Saclay and to the Institut des Hautes Etudes Scientifiques (IHES), Bures-sur-Yvette,
for hospitality while this work was done. NPW would also like to thank the Simons
Foundation for their support through a Simons Fellowship in Theoretical Physics. SFR would like
to thank the UPMC for hospitality while this work was done. NPW and SFR are also
grateful to CERN for hospitality while this work was initiated, and we are all grateful to the
Centro de Ciencias de Benasque for hospitality at the Gravity- New perspectives from strings
and higher dimensions workshop. The work of IB was supported in part by the ANR
grant 08-JCJC-0001-0, and by the ERC Starting Independent Researcher Grant
240210String-QCD-BH and by a grant from the Foundational Questions Institute (FQXi) Fund,
a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal
FQXi-RFP3-1321 to the Foundational Questions Institute. This grant was administered by
Theiss Research. The work of NPW was supported in part by the DOE grant
DE-FG0384ER-40168. SFR was supported in part by STFC and the Institut Lagrange de Paris.
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