Flux-branes and the Dielectric Effect in String Theory

Abstract

We consider the generalization to String and M-theory of the Melvin solution. These are flux p-branes which have (p+1)-dimensional Poincare invariance and are associated to an electric (p+1)-form field strength along their worldvolume. When a stack of Dp-branes is placed along the worldvolume of a flux (p+3)-brane it will expand to a spherical D(p+2)-brane due to the dielectric effect. This provides a new setup to consider the gauge theory/gravity duality. Compactifying M-theory on a circle we find the exact gravity solution of the type IIA theory describing the dielectric expansion of N D4-branes into a spherical bound state of D4-D6-branes, due to the presence of a flux 7-brane. In the decoupling limit, the deformation of the dual field theory associated with the presence of the flux brane is irrelevant in the UV. We calculate the gravitational radius and energy of the dielectric brane which give, respectively, a prediction for the VEV of scalars and vacuum energy of the dual field theory. Consideration of a spherical D6-brane probe with n units of D4-brane charge in the dielectric brane geometry suggests that the dual theory arises as the Scherk-Schwarz reduction of the M5-branes (2,0) conformal field theory. The probe potential has one minimum placed at the locus of the bulk dielectric brane and another associated to an inner dielectric brane shell. Comment: v2 Major Additions: dielectric radius in gravity solution matches exactly stable and unstable points of Myers probe potential, as well as the upper bound on D-brane charge. New section probing dielectric brane that clarifies dual field theory discussion and is consistent with Scherk-Schwarz reduction of (2,0) M5-brane theory. References added. 43 pages, 4 figures

Full text

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arXiv:hep-th/0105023v2 16 Aug 2001
LPTENS–01–16
DAMTP–2001–32
Flux–branes and the Dielectric Effect
in String Theory
Miguel S. Costa†1, Carlos A.R. Herdeiro‡2and Lorenzo Cornalba†3
†Laboratoire de Physique Th´ eorique de l’´Ecole Normale Sup´ erieure
24 rue Lhomond, F–75231 Paris Cedex 05, France
‡D.A.M.T.P. – University of Cambridge
Center for Mathematical Sciences
Wilberforce Road, Cambridge CB3 0WA, UK
Abstract
We consider the generalization to String and M-theory of the Melvin solution.
These are flux p–branes which have (p+1)–dimensional Poincar´ e invariance and are
associated to an electric (p + 1)–form field strength along their worldvolume. When
a stack of Dp–branes is placed along the worldvolume of a flux (p + 3)–brane it will
expand to a spherical D(p + 2)–brane due to the dielectric effect. This provides a
new setup to consider the gauge theory/gravity duality. Compactifying M–theory
on a circle we find the exact gravity solution of the type IIA theory describing the
dielectric expansion of N D4–branes into a spherical bound state of D4–D6–branes,
due to the presence of a flux 7–brane. In the decoupling limit, the deformation of the
dual field theory associated with the presence of the flux brane is irrelevant in the
UV. We calculate the gravitational radius and energy of the dielectric brane which
give, respectively, a prediction for the VEV of scalars and vacuum energy of the
dual field theory. Consideration of a spherical D6–brane probe with n units of D4–
brane charge in the dielectric brane geometry suggests that the dual theory arises as
the Scherk–Schwarz reduction of the M5–branes (2,0) conformal field theory. The
probe potential has one minimum placed at the locus of the bulk dielectric brane and
another associated to an inner dielectric brane shell.
1miguel@lpt.ens.fr
2car26@damtp.cam.ac.uk
3cornalba@lpt.ens.fr

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1Introduction
The realization that p–branes play a fundamental role in the understanding of String and
M-theory played a central role in the developments of the past years. In particular, these
p–branes admit a gravitational description in the low energy supergravity limit of String
and M-theory [1]. They can carry either the magnetic or electric charges associated with
the form gauge potentials of these theories. With the discovery of D–branes in perturbative
string theory [2] it was possible to use string theory to study black holes [3], which later
led to the famous AdS/CFT duality [4, 5, 6].
Another geometry that appears in the Einstein-Maxwell theory is the so called Melvin
Universe [7] (see also [8 − 11]). This geometry represents a magnetic flux tube in four
dimensions. In the cases that the gauge field arises from a Ka? lu˙ za–Klein compactification
the higher dimensional space for the magnetic Melvin solution is flat with some non-trivial
identifications [12, 13, 14]. This fact led to a construction of the type IIA flux–brane
from the compactification of M-theory on a circle [15, 16] (see also [17 − 21] for further
work on the Melvin solution in String theory). This geometry has 8-dimensional Poincar´ e
invariance and therefore is a flux 7–brane (the Melvin fluxtube is a flux string in 4D). It
has an additional SO(2) invariance associated to the spherical symmetry in the transverse
plane to the brane. The natural question that arises is to generalize this solution to the
case of forms with different rank in String and M-theory. This is one of the purposes of
this paper.
In analogy with the type IIA flux 7–brane, the flux p–branes in D–dimensional space–
time are associated to a flux of a (D−p−1)–form field strength along the transverse space to
the brane. If one considers electric variables then they are associated to an electric (p+1)–
form field strength along the brane worldvolume. This raises the issue of stability of the
flux p–branes. In fact, one expects to have Schwinger production of spherical (p−1)–branes
as shown in [14]. As such, to find a string theory dual of the flux p–branes in the same spirit
of the AdS/CFT duality becomes problematic. However, we can try to stabilize the flux
p–branes. Consider a Ramond–Ramond flux (p + 3)–brane and place on its worldvolume
a stack of Dp–branes. Due to the coupling of the Dp–branes to the electric (p + 4)–form
field strength the brane will expand to a dielectric 2–sphere forming a D(p + 2)–brane. In
other words, the dielectric effect is at work [22]. We shall argue that, in the decoupling
limit, the presence of the Dp–branes stabilizes the flux (p + 3)–brane.
To be more precise let us briefly describe the work of Myers on dielectric branes [22].
Following general principles such as gauge invariance and T–duality invariance, he found
new couplings to the bulk fields in the non–abelian Dp–brane action both in the Born–
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Infeld and in the Chern–Simons pieces of the action. These new couplings would not be
taken into account by just replacing abelian by non–abelian gauge fields in a Dp–brane
worldvolume action and by taking a gauge trace. In particular, this work led to a new
proposal for the Chern–Simons term of the Dp–brane action bosonic sector with the form
SCS= µp
?
Tr
?
P
?
ei2πα′iΦiΦ??Ane−B??
e2πα′F?
.(1.1)
P[...] denotes the pull–back of the background Kalb–Ramond 2–form B, and Ramond–
Ramond n–form potentials An. The novelty of this proposal is the exponential containing
the operator iΦ, denoting interior multiplication by the transverse space vector Φ associated
to displacements of the branes. Since this reduces the degree of a differential form, it allows
for Wess–Zumino couplings of a Dp–brane to Ramond–Ramond forms of degree greater
than p + 1.
A physical effect arising from this new coupling can be seen by considering a collection
of N D0–branes in a constant background electric field described by a Ramond–Ramond
4–form field strength. Then the scalar potential for this collection of branes is
V (Φ) = −(2πα′)2T0
?1
4Tr
?
[Φi,Φj]2?
+i
3Tr
?
ΦiΦjΦk?
Ftijk
?
,(1.2)
where T0is the D0–brane mass and Ftijk= Eǫijkalong three transverse directions. An
analysis of the potential extrema shows that the ground state is described by
Φi=E
4αi,(1.3)
where the αibelong to the N × N irreducible representation of the SU(2) algebra, with
the value for the potential at large N reading
VN= −π2α′3/2
96g
E4N3.(1.4)
This non–commutative configuration represents a single, somewhat granular, spherical D2–
brane with N D0–branes bound to it. For large N this sphere has a ‘radius’
rs=π
2α′|E|N .(1.5)
Of course, by T–duality a Dp–brane immersed in a (p+4) Ramond–Ramond electric field
will expand to a D(p+2)–brane with worldvolume geometry Mp+1×S2. The action of the
background electric field is to create a dipole moment (and higher multipole moments) with
respect to D(p + 2)–brane charge. In close analogy with classical electrodynamics Myers
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dubbed these branes as Dielectric Branes. Now we see that the source for the external
electric field can be taken to be a flux brane.
In a beautiful paper [23], Polchinski and Strassler realized that each of the vacua of
the N = 1∗Super Yang–Mills theory, obtained by adding finite mass terms to the N = 4
theory, is dual to the gravity background associated to a dielectric brane source (further
work can be found in [24 − 27]). In this general class of theories one can break all the
supersymmetry, for example, by adding a mass term to the gluino. However, in constructing
the gravitational dual, they considered a spherical distribution of D3–branes where they
placed a D5–brane source, and then solved the gravity equations to first order in the
mass perturbation. A natural question that arises is to fully describe the gravitational
background for the dielectric branes. We shall address this problem in this paper. By
placing a Dp–brane in a Ramond–Ramond flux (p + 3)–brane one expects to generate a
dielectric brane. The presence of the Dp–brane charge will stabilize the system. In fact,
after taking the decoupling limit this configuration is expected to be stable since the dual
field theory is on the general class of the theories analyzed by Polchinski and Strassler. In
particular, we expect tachyons to be absent.
To construct the gravitational background for a dielectric brane, the key idea is to
consider the Ramond–Ramond magnetic flux 7-brane that arises from the reduction of
M-theory flat space–time with some non–trivial identifications [15, 16]. We know that
the double dimensional reduction of the M–theory 5–brane gives the type IIA D4–brane
[28]. Hence, it is natural to suspect that implementing the Ka? lu˙ za–Klein Melvin twisted
reduction to the M5–brane will give a D4–brane in a magnetic flux 7-brane, or equivalently,
a D4–brane placed in a electric field. Then the coupling of the D4–brane to the dual RR
7–form potential will give rise to the Myers effect. Thus, after the reduction we expect to
describe within gravity a D4–brane expanded into a 2–sphere, i.e. a D6–brane.
This paper is organized as follows. In section 2 we make the ansatz for the flux p–
branes. This geometry has the desired (p + 1)–dimensional Poincar´ e invariance together
with SO(D −p −1) spherical symmetry. The associated (p + 1)–form field strength has a
electric component along the brane worldvolume. The corresponding system of differential
equations does not decouple in terms of non-interacting Liouville systems as it is usually
the case for black holes. We investigate the asymptotics of the flux branes geometry.
In section 3 we construct the gravity solution for a D4-brane expanded into a 2–sphere
due to the dielectric effect. Firstly we consider the twisted dimensional reduction of the
non–extremal M5–brane with a double analytic continuation. The ten–dimensional config-
uration is interpreted as a D4–brane immersed in a flux 7–brane with maximum magnetic
field parameter. Although this value of the magnetic field is unphysically large, we chose,
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for the sake of clarity, to consider this case first since the corresponding geometry has all
the correct features to be interpreted as a dielectric brane. Then we consider the general
case of arbitrary magnetic field which amounts to compactify a rotating M5–brane after a
similar double analytic continuation.
Section 4 is devoted to the study of the dielectric brane geometry in the decoupling
limit. The resulting geometry has the same asymptotics as the D4-brane geometry in the
decoupling limit, which shows that the deformation of the dual theory associated to the
coupling of the D4–brane worldvolume theory to the flux 7–brane is irrelevant in the UV.
Using the gravitational description we calculate the scalars VEV and vacuum energy of
the field theory where the deformation becomes relevant.
In section 5 we probe the dielectric brane geometry using a spherical D6–brane probe
with n units of D4–brane charge. First we consider the probe in the flux 7–brane back-
ground and obtain similar results to those of Myers [22], but now taking into account the
backreaction of the background electric field on the geometry. This simple case is very
useful to understand the stability of the dielectric brane geometry before taking the de-
coupling limit. We proceed with the study of the probe in the dielectric brane geometry in
the decoupling limit. It is seen that either far away or inside the dielectric brane the probe
potential has the expected form to be associated with the Scherk–Schwarz [29] reduction of
the M5–branes (2,0) low energy conformal field theory. This potential has a minimum at
the locus of the bulk dielectric brane and another minimum in its interior that is associated
to an inner dielectric brane shell.
We give our conclusions in section 6.
2 Flux–branes
We shall consider the following general action in D–dimensional space–time
S =
1
2κ2
?
dDx√−g
?
R −1
2(∂φ)2−
1
2d!eaφF2
?
,(2.1)
where κ is the gravitational coupling and F is a generic d–form field strength. The above
action can be regarded as a consistent truncation of either String or M-theory low energy
actions, where F represents any of the field strengths or electromagnetic dual in these
theories. Then the equations of motion read
2φ =
a
2d!eaφF2,d
?
eaφ⋆ F
?
= 0 ,Rab= τab,(2.2)
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where the tensor τ = τφ+ τFtakes the form
τφ
ab=1
2∂aφ∂bφ ,
τF
ab=eaφ
2
?
1
(d − 1)!Fac1···cd−1F
c1···cd−1
b
−
d − 1
(D − 2)d!gabF2
?
.
(2.3)
In order to describe a flux–brane with ISO(1,d− 1) × SO(D − d) invariance, we shall
make the following ansatz
ds2= e2A(r)ds2?
F = E ǫ
?
metric element on the unit˜d = (D − d − 1)–sphere. We have conveniently multiplied the
line element of this sphere by E−1for dimensional reasons. Let us note that the form F
is closed since E is constant and also automatically solves the corresponding equation of
motion. Notice that we refer to this electric field as constant due to the independence
on the radial coordinate (however it is not covariantly constant). Alternatively, one could
consider the electromagnetic dual of F which reads
˜F = E1−˜de−dA+C+˜dBdr ∧ ǫ
?
The Ricci tensor for the above geometry is
Rµν= −e2A−2C?
Rij= −e2B−2C?
Rrr=˜d
Md?
+ e2C(r)dr2+ e2B(r)E−2dΩ2
Md?
˜d,
?
,φ = φ(r) ,
(2.4)
where ǫ
Md?
is the volume form of d–dimensional Minkowski space Mdand dΩ˜dis the
?
S
˜d?
, (2.5)
where˜F is a (D − d)–form and ǫS˜d?
is the volume form on the unit˜d–sphere.
A′′+ d(A′)2− A′C′+˜dA′B′?
B′′+˜d(B′)2− B′C′+ dA′B′?
B′C′− B′′− (B′)2?
ηµν,
hij+ E2?˜d − 1
A′C′− A′′− (A′)2?
?
,
hij,
?
+ d
?
(2.6)
where′denotes differentiation with respect to the radial coordinate r. We are using coor-
dinates xµalong the worldvolume directions of the brane Mdand θion the˜d–dimensional
unit sphere S˜d. The only non–vanishing component of the tensor τφis
τφ
rr=1
2(φ′)2,(2.7)
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and those of τFare
τF
µν= −E2
2
eaφ
?
?d − 1
D − 2
?d − 1
D − 2
˜d
D − 2
?
?
e−2(d−1)Aηµν,
τF
ij=E2
2
eaφ
e2B−2dAhij,
τF
rr=E2
2
eaφ
?
e2C−2dA.
(2.8)
In the ansatz (2.4) there is a freedom of reparametrization of the radial coordinate r.
We conveniently choose the gauge
dA +˜dB = C , (2.9)
in which the equations of motion simplify to
φ′′= −aE2
?
?d − 1
D − 2
2
˜d
eaφ+2˜dB,
A′′=E2
2D − 2
eaφ+2˜dB+ E2?˜d − 1
?
eaφ+2˜dB,
B′′= −E2
2
?
?
e2dA+2(˜d−1)B,
(2.10)
together with a zero–energy constraint from the rr component of Einstein equations. This
constrained system of differential equations can be derived from the Lagrangian L = T −V
where
T = −1
and
V = −E2˜d(˜d − 1)e2dA+2(˜d−1)B−E2
with the zero–energy constraint T + V = 0. In the case of black holes one can usually
decouple this system in terms of non-interacting Liouville systems related through the zero
energy condition (see for example [30]). However, in this case it is not possible to decouple
the system, which makes the solution of the problem much harder. For this reason we were
not able to find a general analytic solution but will investigate the asymptotics of the flux
branes.
2(φ′)2+ d(d − 1)(A′)2+˜d(˜d − 1)(B′)2+ 2d˜dA′B′, (2.11)
2e2˜dB+aφ,(2.12)
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To analyze the differential equation (2.10) we define
f = aφ + 2˜dB ,
g = 2dA + 2
?
?˜d − 1
?
?
B ,
h =
˜d
D − 2
φ + aA .
(2.13)
These functions satisfy the following system of differential equations
h′′= 0 ,
f′′= c1ef+ c2eg,
g′′= c3ef+ c4eg,
(2.14)
where the constant coefficients cihave the form
c1= −E2
?˜d − 1
2
?
a2+ 2
˜d(d − 1)
D − 2
c3= E2,
?
≡ −λ2,
c2= 2˜d
?
E2,c4= 2
?˜d − 1
?2E2.
(2.15)
We shall see that, in the cases of interest in String and M–theory, the constant c1simplifies
to c1= −2E2.
2.1Dilatonic Melvin
For the usual dilatonic Melvin solution [31] one has˜d = 1 and the transverse sphere is
a circle which is flat. As a consequence c2 = c4 = 0 and the equations (2.14) simplify
considerably. In this case c3f′′= c1g′′, and the general solution is given by
ef=


α1
cosh
?α1
√2(λr + α2)
?


2
,g =c3
c1f + α3+ α4Er , h = α5+ α6Er , (2.16)
where the αi’s are dimensionless constants of integration. Notice that not all these con-
stants are independent because of the zero–energy constraint.
The case we consider here is that of the RR flux 7-brane of type IIA strings studied in
[16]. In this case we have a = −3/2 and d = 8. This solution corresponds to choosing the
constants of integration in (2.16) such that
ef=
1
cosh2(Er),eg= 1 + e2Er,h = 0 .(2.17)
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Then the coordinate transformation Er = ln(Eρ/2) will bring the metric, field strength
and dilaton field to the form
ds2= Λ1/8?
F = E ǫ
The space–time metric is for ρE ≪ 1 approximately flat. We can regard this as the
boundary conditions for the flux 7-brane which fixes the constants of integration in (2.16).
This fact is also true for the other flux branes and was explored in [33].
The general solution (2.16) contains singular geometries for which the Ricci scalar blows
up. These are naked singularities. Only for the flux 7–brane solution (2.18) is the geometry
non-singular. One should regard the flux–branes as non-supersymmetric vacua where the
energy density of the electromagnetic field spreads to infinity. In this sense the flux branes
do not represent localized lumps of energy and are not asymptotically Minkowskian as the
usual branes. Far away the energy associated with the constant electric field will dominate.
To analyse the geometry (2.18) we should multiply the metric by the conformal factor
eφ/2to change to the string frame. It turns out that this geometry is quite different than
the usual 4D Melvin Universe [7]. In the latter the orbits of ∂/∂φ have vanishing length at
large radial distance. In the former type IIA case the length of the ∂/∂φ orbits lφscales in
terms of proper radial distance u as lφ∼ u1/3. This means that as u → ∞ space-time does
not close. Also, this means that while in the 4D Melvin we have a quantization condition
for the flux of ⋆F through the transverse space [11], this no longer happens for the IIA
flux 7–brane (2.18).
ds2?
M8?
+ dρ2?
e4φ/3= Λ ≡ 1 + (Eρ)2/4 .
+ Λ−7/8ρ2dϕ2,
?
M8?
,
(2.18)
2.2 Flux–branes in type II Strings
Now we turn to the flux branes of type II String theory. We shall consider the case of
RR flux branes. In this case the coupling a = (5 − d)/2,˜d = 9 − d and the coefficient
c1in (2.15) simplifies considerably to c1= −2E2. The cases of NSNS flux branes can be
obtained by a S-duality transformation on the IIB flux 2– and 6–branes.
For the RR flux branes the metric functions and dilaton are related to the functions f,
g and h by the expression (2.13) that reads
?5 − d
2
f =
?
φ + 2(9 − d)B ,
g = 2dA + 2(8 − d)B ,
?9 − d
8
h =
?
φ +
?5 − d
2
?
A .
(2.19)
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The functions f and g satisfy the system of differential equations
f′′= −2E2ef+ 2(9 − d)(8 − d)E2eg,
g′′= E2ef+ 2(8 − d)2E2eg.
(2.20)
A particular solution to this system of equations can be found by setting eg= ζef[32].
Then from f′′= g′′we must have
ζ =
3
2(8 − d).(2.21)
Solving for f we find
ef=
2
(25 − 3d)
1
(Er)2,(2.22)
and we also set h = 0. A rather tedious calculation gives the functions A, B, and C that
appear in the Einstein metric (2.4)
2A =
(9 − d)2
8(25 − 3d)lnζ +
(d − 5)2
8(25 − 3d)lnζ +
2C =(9 − d)(25 − d)
8(25 − 3d)
9 − d
8(25 − 3d)f ,
25 − d
8(25 − 3d)f ,
lnζ +25(9 − d)
8(25 − 3d)f .
2B =
(2.23)
This solution satisfies the zero–energy constraint defined above. We can change to the
string frame, and write the metric in coordinates such that the radial coordinate is the
proper radial distance. The final result for the metric and dilaton field is
ds2= β1/5(Eu)2/5ds2?
Md?
+ du2+β
ζu2dΩ2
9−d,
e
10
d−5φ= ζ(Eu)2.
(2.24)
where
β =
52
23(25 − 3d).(2.25)
This metric describes a geometry with a naked singularity at the origin. However, its
asymptotics are those of the RR flux (d−1)–brane [32, 33], for which we expect a smooth
geometry. Far away the energy density associated with the electric field dominates and
determines the asymptotics4. A comparison with flat space–time is helpful; for example,
4Notice that for d < 8 the powers of u in the metric (2.24) are independent of d. On the other hand
for d = 8, the Melvin case, the powers of u are different.
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the Schwarzchild black hole with negative mass has a naked singularity and is unphysical,
however, it converges to flat space–time at infinity where the energy density vanishes.
Similarly the solution (2.24) has the correct asymptotics for the flux (d − 1)–brane. For
d < 5, i.e. for the flux p-branes with p < 4, the dilaton field converges to zero at infinity and
we are in the perturbative string theory regime. For d > 5, the string coupling diverges.
The case d = 5 gives the non–dilatonic (and self–dual) flux 4–brane of the type IIB theory.
This is analogous to what happens with the D3–brane. Finally, notice that while for the
flux 7–brane the flux of ⋆F along the transverse space is convergent, for the flux branes
analyzed here it diverges.
2.3 Flux–branes in M-theory
In M-theory there are flux 3– and 6–branes, both non–dilatonic. We have D = 11 and
a = 0. The functions f and g are related to those appearing in the metric by
f = 2(10 − d)B ,
g = 2dA + 2(9 − d)B .
(2.26)
These functions satisfy the system of differential equations
f′′= −2E2ef+ 2(10 − d)(9 − d)E2eg,
g′′= E2ef+ 2(9 − d)2E2eg.
(2.27)
To find the asymptotics of the M flux–branes we set eg= ηef, which gives
η =
3
2(9 − d).(2.28)
In the case of the flux 3–brane (d = 4) the function f reads
ef=
1
8(Er)2,(2.29)
and the metric functions A, B, and C become
2A =1
4lnη +1
24f ,2B =1
6f ,2C = lnη +7
6f .(2.30)
This gives the following metric written in terms of the proper radial distance coordinate u:
ds2=
?2
9
?1/4
(Eu)1/2ds2?
M4?
+ du2+20
27u2dΩ2
6.(2.31)
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For the flux 6–brane (d = 7) we have
ef=
2
7(Er)2, (2.32)
and
2A =1
7lnη +1
21f ,2B =1
3f ,2C = lnη +4
3f . (2.33)
The corresponding metric element can be written in the form
ds2=
?7
18
?1/7
(Eu)2/7ds2?
M7?
+ du2+14
27u2dΩ2
3.(2.34)
As for the flux p–branes of String theory with p < 7, the flux along the transverse space
diverges for the M flux–branes.
2.4Stability of the flux–branes
The flux p–branes described above are not stable. They will decay through the nucleation of
spherical (p−1)–branes as described generally in [14]. If we compactify (p−1) directions of
the flux p–branes then they will decay through the usual Schwinger production of (p − 1)
brane/anti–brane pairs. Similarly to the RR flux 7–brane case [16], one can consider
a (p − 1)–brane probe at the core of the flux p–brane and calculate the action for the
instanton associated with this decay process. This gives the well known result for the
nucleation rate Γ
Γ ∼ e−I,I = πMp−1
|E|
, (2.35)
where Mp−1is the mass of a (p − 1)–brane. It is expected that this calculation can be
reproduced using the Euclidean Quantum gravity approximation for the nucleation of
brane pairs. One would need to find the instanton for the nucleation process with the
same asymptotics of the flux–branes described above. This calculation could confirm the
expected periodicity of the electric (or magnetic) field parameter. As explained in [16], the
existence of a maximum electric field for a generic p–form is expected on the basis of String
duality and of the analysis of the RR flux 7–brane case from a M-theory perspective. An
interesting physical interpretation for this maximum electric field was given in [33]: since
the typical distance for nucleation is of order 1/E, for larger values of E the black hole
horizons will touch and the pair production will cease to exist.
A very interesting question is to consider the string theory duals of the flux brane
geometries in some decoupling limit. This can be problematic because, as explained above,
these geometries are not stable, which makes the duality difficult to establish (see [33] for
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a discussion of this point). However, one can try to stabilize the flux branes. As explained
in the Introduction this can be done by considering the dielectric effect in String theory
[22].
Consider the case of a RR flux (p+3)–brane and place a stack of N Dp-branes along its
worldvolume. Then the Dp-branes will couple to the electric RR (p+4)-form field strength
expanding into a D(p + 2)-brane with geometry Mp+1× S2. Now, the presence of the
N Dp–branes changes the asymptotics of the geometry. Far away, we have the geometry
for the flux–brane together with a charge due to the N Dp–branes. We would then need
to find an instanton with these asymptotics, representing the instability of space–time.
We know from the perturbative String theory description of the dielectric effect that this
system is locally stable, and therefore such an instanton represents a quantum tunneling
effect. Moreover, we shall argue that, the geometry in the decoupling limit has the same
asymptotics as the usual D–branes without external electric field. Therefore, in this limit
the instanton instability no longer exists and the configuration is stable. Note that the
case of p = 3 is nothing but a D3–brane expanding to a spherical D5–brane due to the
dielectric effect. These type of configurations have already made their appearance in the
gauge theory/gravity duality of the theories of the type analyzed by Polchinski and Strassler
[23], which are stable.
In the following sections we shall treat the case of p = 4, where one can find the exact
gravitational description by using the M-theory reduction of the M5–brane to the type IIA
theory and hence confirming the aforementioned expectations.
3 Dielectric branes
Because of the complexity of the gravitational background presented below it is important
to set our conventions for the bosonic sector of the eleven–dimensional supergravity action:
1
2κ
11
where κ11is the eleven–dimensional gravitational coupling and F = dA with A a 3–form
field potential. Reduction to the type IIA theory is achieved through the ansatz
10+ e4φ/3?
We shall present the construction of the dielectric branes in two steps. First we consider a
D4–brane placed in a flux 7–brane with maximum magnetic (or electric) field. This value
of the magnetic field is unphysically large but the solution is simpler and retains all the
correct features. Then we consider the case of arbitrary magnetic field.
S =
2
??
d11x√−g
?
R −
1
2 · 4!F2
?
+1
6
?
F ∧ F ∧ A
?
, (3.1)
ds
2
11= e−2φ/3ds
2
dx11+ Aadxa?2
,
A = A3+ B ∧ dx11.(3.2)
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3.1 Maximal magnetic field
Consider the non–extremal M5–brane solution obtained by a double analytic continuation
from the usual solution. The metric and 3–form potential read
ds2?
+H2/3
f
ds
2
11= H−1/3?
?dr2
M5?
+ fdτ2?
+
+ r2?
dθ2+ sin2θd˜ ϕ2+ cos2θdΩ2
2
??
,
A = −r3
Hsinhα coshα cos3θd˜ ϕ ∧ ǫ(S2) ,
(3.3)
where 0 ≤ ˜ ϕ < 2π, 0 ≤ θ ≤ π/2. The unusual parameterization of the transverse 4–sphere
is standard for rotating black holes [34]. In this coordinate system the transverse space
naturally splits into the θ = 0 three–plane orthogonal to the θ = π/2 two–plane. The
functions f and H have the form
H = 1 +
?R
r
?3
≡ 1 +
?rH
r
?3
sinh2α ,f = 1 −
?rH
r
?3
. (3.4)
The M5–brane charge quantization gives the condition
r3
Hsinhα coshα = πNl3
P, (3.5)
where lP is the eleven–dimensional Planck length and N the number of M5–branes. In
order to avoid a conical singularity, the Euclidean time direction τ has periodicity given
by
2πR11=4π
3rHcoshα ,
and it is related to the ten–dimensional type IIA string coupling and tension by R11= g√α′.
If we compactify along the killing vector ∂/∂τ there will be a set of fixed points at
r = rH, spanning a 4–sphere in the transverse space. The reduced space will be singular
on such a 4–sphere. One can instead compactify along the killing vector field
(3.6)
q =∂
∂τ+ B∂
∂ ˜ ϕ,
B =
1
R11
, (3.7)
which corresponds to the maximum value for the magnetic field. Notice that the parameter
B is related to the electric field E in the two previous sections by E = 2B. The fixed points
of this isometry are at r = rH, θ = 0 corresponding to a 2–sphere on the transverse space
(see figure 1). At each point of this 2–sphere the action of the isometry is the same as for a
Ka? lu˙ za–Klein monopole. Hence, the reduced space will be singular on a 2–sphere that we
identify with the D4–branes expanded into a D6–brane. Asymptotically space–time will
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S2
rH
θ=0
˜ r,τ
S4
τ
˜ r
S2
rHcosθ
θ
θ=π/2
˜ ϕ
•
•
•
Figure 1: Left: The (˜ r,τ) Euclidean plan with ˜ r2= r2− r2
of radius rH; Right: The parameterization of the 4–sphere; the subscript on the S2denotes the
radius.
H. At ˜ r = 0 there is a 4–sphere
look like the flux 7–brane with maximal magnetic field parameter B = 1/R11, together
with the D4–brane charge. We have chosen this particular value for B because, as will be
seen below, any other choice would lead to a conical singularity for the ten–dimensional
geometry. For this value of the magnetic field, perturbative string theory will hold only
for r ≪ R11, while the eleventh direction remains unobservable for r ≫ R11. We shall
consider this unphysical case first because it is much simpler and retains all the features
of the general case. In the next subsection we shall allow for general and physical values
of the magnetic field by considering the rotating M5–brane geometry.
To perform the reduction we change to the azimuthal angle ϕ = ˜ ϕ − Bτ, then τ has
period 2πR11and ϕ has the standard period of 2π. A straightforward calculation gives the
following type IIA background fields:
ds
2
10=
?Σ
H
?1/2
ds2?
M5?
+ (ΣH)1/2
?dr2
f
+ r2?
dθ2+ cos2θdΩ2
2
??
+
+
?H
Σ
?1/2
B = −r3
A3= −r3
f r2sin2θdϕ2,
e2φ= Σ3/2H−1/2,
HB sinhα coshα cos3θǫ
?
S2?
,
A1= BΣ−1Hr2sin2θdϕ ,
Hsinhα coshα cos3θdϕ ∧ ǫ
?
S2?
.
(3.8)
The function Σ has the form
Σ ≡ f + H(Brsinθ)2,(3.9)
and B = 1/R11= 1/(g√α′). Notice that g is the asymptotic value of the string coupling
along the θ = 0 three–plane.
15