Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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The details require some careful inspection, but

we shall stop our analysis here (Giveon an d Kutasov

1999).

More general gauge configurations can be realized

by adding more parallel NS5 branes, and thus

obtaining product groups. Adding orientifold planes,

one can change gauge groups as explained in the

previous section (Figure 8).

Finally, we can take a further step towards more

physical models, constructing N = 1 gauge theories.

For example, this can be achieved from the previous

N = 2 model, rotating the second NS5 brane from

the (x, v) position, to the (x, w) position, where

w = (x

, x

)(Figure 9). Then a new brane projection

condition appears (

=





), breaking super-

symmetry down to N = 1.

In this case, one could also obtain chiral matter,

adding, for example, orientifold planes.

Quantum Corrections from M-Theory

Up to this point we have considered classical gauge

configurations. Quant um corrections could be com-

puted switching on brane fluctuations. However, it

is an amusing fact that working with M-theory one

can obtain exact quantum results. As an example,

let us sketch how the exact Seiberg–Witten solution

can be obtained for the N = 2 model described in the

previous section, in the simplest case without

matter.

The full web of dualities suggests the existence of

a unique unifying theory called M-theory. At low

energies, M-theory appears as the strong-coupling

limit of type IIA strings. In such a limit, D0 branes

become the dominant objects and the corresponding

states can be interpreted as Kaluza–Klein modes

coming from an eleventh dimension x

compacti-

fied on a circle S

(Figure 10).

Thus, M-theory manifests itself as an 11-dimensional

supergravity. In particular, it can be shown that there

can be only a unique 11-dimensional supergravity. As

said, here the nonperturbative objects are two- or five-

dimensional membranes.

From the M-theory point of view, the D4 branes

considered in our model appear as M5 membranes

wrapped on the eleventh direction S

(Figure 11).

Because quantum corrections are no longer negligi-

ble, we can no longer think of these branes as

stretched in the x

direction, but v must also be

considered. Thus, the M5 membranes will describe,

in R

 S

, a region R

 S, where R

are the x

coordinates, and S is a Riemann surface immersed in

Q  S

, Q being spanned by the (v, x

) coordinates.

In fact, supersymmetry constrains the surface to be a

holomorphic curve, so that to describe it, it is

convenient to collect v = (x

, x

) and (x

, x

) into

complex coordinates v = x

þ ix

and s = x

þ ix

To compute quantum fluctuations, let us note that

the end of a D4 brane over an NS5 brane is free to

move along the v directions. A fully free end of a

brane would satisfy a free wave equation. However,

as x

is constrained in all directions but the v ones, it

will simply satisfy a Laplace equation in two

dimensions: 

= 0. Let us solve it, for a fixed

NS5 brane. It will be (at least for large values of v)

ðvÞ¼k

i¼1

log jv  v

ðÞ

jk

i¼1

log jv  v

ðÞ

j½21

where n

is the number of D4 branes ending on

the left-hand side of the NS5 brane, in the positions

()

, and similar for the R index, which refers to

Higgs

branch

NS5

Matter

Figure 7 Permitting Higgs phases.

NS5

D4 D4

, n

)

Figure 8 N = 2 four-dimensional super Yang–Mills theory with U(n

)  U(n

) gauge group and matter. Strings crossing the central

NS5 brane give matter in the (n

, n

) representation.

Brane Construction of Gauge Theories 365

the right-hand side. Here () refers to the th NS5

brane, and k is an integration constant.

Because x

is the real part of a holomorphic field,

whose imaginary part is compactified on a circle of

ray R

, we then find

sðvÞ¼R

i¼1

log



v  v

ðÞ



 R

i¼1

log



v  v

ðÞ



½22

This descr ibes the quantum fluctuations of the NS5

brane as seen in M-theory. In particular, because of

the imaginary part of s, the ends of the D4 branes

appear as vortices on the NS5 brane. In place of s,it

is now convenient to introduce a new field

t := exp (s=R

) so that

tðvÞ¼

i¼1



v  v

ðÞ



i¼1



v  v

ðÞ



½23

Before continuing, let us look a bit again at the

classical limit. In this case, a fixed value of v will

correspond to the position of a D4 brane, whereas a

fixed value of s will correspond to the fixed position

of an NS5 brane. The classical configuration is then



s  s

ð1Þ



s  s

ð2Þ



i¼1

ðv  v

Þ¼0 ½24

Here s

()

are the positions of the NS5 branes, and

the positions v

of the D4 branes coincide for both

the NS5 branes. Also, for large values of v, one has

(1)

 v

and t

(2)

 v

n

Quantum mechanically, the configuration is

determined in terms of v and t by the holomorphic

curve S, which can be described as an algebraic

curve F(v, t) = 0, generalizing the classical configura-

tion. As there are two NS5 branes and n D4 branes,

F must be a polynomial of degree 2 in t,

Fðv; tÞ¼A

ðvÞt

þ A

ðvÞt þ A

ðvÞ½25

where A

, a = 1, 2, 3, are all polynomials of degree n.

Note that values of v such that A

vanishes give the

solution t = 0, which corresponds to sending the right-

hand side NS5 brane to 1. Similarly, A

= 0sendsthe

other NS5 brane to 1. To avoid these undesirable

configurations, we can set A

= A

= 1. For A

,we

can take the most general choice, up to an eventual

shift in v, giving the quantum configuration

þ v

þ a

n2

þþa

v þ a



t þ 1 ¼ 0 ½26

This realizes a quantum-mechanical correspondence

between the M5 membrane configurations described

by the given polynomials, and the N = 2 super

Yang–Mills vacua. But this is also the claimed

Seiberg–Witten curve. In particular, M-theory gives

a concrete physical meaning for the support Rie-

mann surfaces of the Seiberg–Witten solutions.

To conclude, let us make some further comments.

It is clear how the construction can be extended for

involving more configurations, for example, with

more NS5 branes, or adding matter.

Also, we have seen that the geometrical picture

which branes give of gauge theories extends at the

quantum level.

A similar construction can be made for the N = 1

model, which also permits a full geometrical proof

of the Seiberg duality at both classical and quantum

levels.

Finally, we should note that there are also

other methods, which work in spacetimes where extra

dimensions are compactified. There, the branes wrap

around certain singular loci which contain information

about gauge symmetries (Lerche 1997).

See also: AdS/CFT Correspondence; Compactification of

Superstring Theory; Gauge Theories from Strings;

Noncommutative Geometry from Strings; Seiberg–Witten

Theory; Supergravity; Superstring Theories;

Supersymmetric Particle Models.

NS5

NS5′

Matter

Figure 9 Going down to N = 1 supersymmetry.

(x, x

)

(v, y)

Figure 10 In M-theory one can think as if at any ten-dimensional

spacetime point, there is attached an S

circle of ray R

(x, x

)

(v, y)

D4 brane M5 membrane

Figure 11 D4 branes become M5 membranes in M-theory.

366 Brane Construction of Gauge Theories

Further Reading

Giveon A and Kutasov D (1999) Brane dynamics and gauge

theory. Reviews of Modern Physics 71: 983.

Johnson CV (2003) D-Branes.Cambridge: Cambridge University

Press.

Lerche W (1997) Introduction to Seiberg–Witten Theory and Its

Stringy Origin. Nucl. Phys. Proc. Suppl. B 55: 83.

Polchinski J (1998) String Theory. Vol. 1: An Introduction to the

Bosonic String. Cambridge: Cambridge University Press.

Polchinski J (2004) String Theory. Vol. 2: Superstring Theory and

Beyond. Cambridge: Cambridge University Press.

Witten E (1997) Solutions of four-dimentional field theories via

M-theory. Nuclear Physics B 500: 3.

Zwiebach B (2004) A First Course in String Theory. Cambridge:

Cambridge University Press.

Brane Worlds

R Maartens, Portsmouth University, Portsmouth, UK

Introduction

At high enough energies, Einstein’s classical theory

of general relativity breaks down, and will be

superseded by a quantum gravity theory. The

singularities predicted by general relativity in grav-

itational collapse and in the hot big bang origin of

the universe are thought to be artifacts of the

classical nature of Einstein’s theory, which will be

removed by a quantum theory of gravity. Develop-

ing a quantum theory of gravity and a unified theory

of all the forces and particles of nature are the two

main goals of current work in fundamental physics.

The problem is that general relativity and quantum

field theory cannot simply be molded together.

There is as yet no generally accepted (pre-)quantum

gravity theory.

The quest for a quantum gravity theory has a long

and thus far not very successful history. Many

different lines of attack have been developed, each

having a different way of dealing with the classical

singularities that arise from point particles and

smooth spacetime geometry. String theory does

away with zero-dimensional point particles, and

particles are modeled as different states of new

fundamental objects, the one-dimensional strings. It

turns out, however, that there is a price to pay – the

number of sp acetime dimensions must be greater

than four for a consistent theory . When fermions are

included, which leads to superstring theory, the

required number of dimensi ons is ten – one time and

nine space dimensions.

There are in fact five distinct ð1þ9Þ-dimensional

superstring theories. In the mid-1990s, duality

transformations were discovered that relate these

superstring theories to each other and to the ð1þ10Þ-

dimensional supergravity theory. This led to the

conjecture that all of these theories arise as different

limits of a single theory, which has come to be

known as M theory. It was also discovered that

extended objects of higher dimension than strings

play a fundamental role in the theory. These objects

are known as ‘‘branes’’ (from membranes), and the

relation between them and strings leads to a new

picture of how gravity and matter may be connected

in the universe. Roughly speaking, open strings

describe the particles of the nongravitational sector,

and their ends are attached to branes, while closed

strings, which describe the graviton and associated

particles of the gravitational sector, can move freely

in all dimensions.

Thus, the obs ervable universe could b e a

ð1 þ3Þ-surface – a ‘‘brane,’’ embedded i n a

ð1 þ3 þdÞ-dimensional spacetime – the ‘‘bulk,’’

with standard-model particles and fields trapped on

the brane, while gravity is free to access the bulk.

Brane-world models offer a phenomenological way to

test some of the novel predictions and corrections to

general relativity that are implied by M theory.

Higher-Dimensional Gravity

Brane worlds can be seen as reviving the original

higher-dimensional ideas of Kaluza and Klein in the

1920s, but in a new context of quantum gravity. An

important consequence of extra dimensions is that

the four-dimensional Planck scale M

 M

(4)

1.2  10

GeV is no longer the fundam ental energy

scale of gravity. The fundamental scale is instead

(4þd)

. This can be seen from the modification of

the gravitational potential. For an Einstein–Hilbert

gravitational action,

gravity

2

ð4þdÞ

x d

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



ð4þdÞ



ð4þdÞ

R  2

ð4þdÞ

½1

we have the higher-dimensional Einstein field

equations,

ð4þdÞ



ð4þdÞ



ð4þdÞ

¼

ð4þdÞ

þ 

ð4þdÞ

½2

Brane Worlds 367

where x

= (x

, y

, ..., y

) and 

(4þd)

is the gravita-

tional coupling constant given by



ð4þdÞ

¼ 8G

ð4þdÞ

8

2þd

ð4þdÞ

½3

The static weak field limit of the field equations

leads to the ð4þdÞ-dimensional Poisson equation,

whose solution is the gravitational potential

VðrÞ/



ð4þdÞ

1þd

½4

In the simplest scenario, we can assume a

toroidal configuration for the d extra dimensions,

with each compactified on the same length scale L.

Then on scales r . L, the potential is ð4þdÞ -

dimensional, V  r

(1þd)

. By contrast, on scales

large relative to L, where the extra dimensions do

not contribute to variations in the potential, V behaves

like a four-dimensional potential, V  L

d

1

.This

means that the usual Planck scale becomes an effective

coupling constant, describing gravity on scales much

larger than the extra dimensions, and related to the

fundamental scale via the volume of the extra

dimensions:

 M

2þd

ð4þdÞ

½5

Large Extra Dimensions

If the extra-dimensional volume is significantly

above the Planck scale, then the true fundamental

scale M

(4þd)

can be much less than the effective scale

 M

d

) M

ð4þdÞ

 M

½6

In this case, we understand the weakness of gravity

as due to the fact that it ‘‘spreads’’ into extra

dimensions, and only a part of it is felt in four

dimensions.

A lower limit on M

(4þd)

is given by null results in

table-top experiments to test for deviations from

Newton’s law in four dimensions, V / r

1

. These

experiments currently probe submillimeter scales,

and find no detectable deviation, so that

L . 10

1

mm ð10

15

TeVÞ

1

) M

ð4þdÞ

& 10

ð3215dÞ= ðdþ2Þ

TeV ½7

Stronger bounds can be derived from null results in

particle accelerators in some brane-world mod els, or

from constraints imposed by observations of super-

novae or of light-element abundance.

Brane worlds, arising in the framework of string

theory, thus incorporate the possibility that the

fundamental scale is much less than the Planck

scale felt in four dimensions. This emerges by virtue

of the larg e size of the extra dimensions. It is not

necessary for all extra dimensions to be of equal size

for this mechanism to operate. There are string

theory solutions (Horava–Witten solutions) with

two ð1þ9Þ -branes located at the boundaries of the

bulk, at the endpoints of an S

orbifold, that is,

a circle folded on itself across a diameter. The

orbifold extra dimension is the large one, whereas

the other six extra dim ensions on the branes are

compactified on a very small scale, close to the

fundamental scale, and their effect on the

dynamics is felt through ‘‘moduli’’ fields, that is,

five-dimensional scalar fields.

These solutions can be thought of as effectively

five dimensional, with an extra dimension that can

be large relative to the fundamental scale. They

provide the basis for the Randall–Sundrum 1 (RS1)

phenomenological models of five-dimensional grav-

ity. The single-brane Randall–Sundrum 2 (RS2)

models with infinite extra dimension arise when

the orbifold radius tends to infinity. The RS models

are not the only phenomenological realizations of M

theory ideas. They were preceded by the brane-

world models of Arkani-Hamed, Dimopoulos, and

Dvali (ADD), which put forward the idea that a

large volume for the compact extra dimensions

would lower the effective Planck scale M

(4þd)

.If

(4þd)

is close to the electroweak scale, M

, then

this would address the long-standing ‘‘hierarchy’’

problem, that is, why there is such a large gap

between M

 1 TeV and M

 10

TeV.

In the ADD models, more than one extra

dimension is required for agreement with experi-

ments, and there is ‘‘democracy’’ among the equiva-

lent extra dimensions, which, in addition, are flat.

By contrast, the RS models have a ‘‘preferred’’ extra

dimension, with other extra dimensions treated as

ignorable (i.e., stabilized except at energies near the

fundamental scale). Furthermore, this extra dimen-

sion is curved or ‘‘warped’’ rather than flat: the bulk

is a portion of anti-de Sitter (AdS

) spacetime. The

RS branes are Z

-symmetric (mirror symmetry), and

have a tension, which serves to counter the influence

on the brane of the negative bulk cosmological

constant. This also means that the self-gravity of the

branes is incorporated in the RS models. The novel

feature of the RS models compared to previous

higher-dimensional models is that the obs ervable

three dimensions are protected from the large extra

dimension (at low energies) by curvature (warping),

rather than straightforward co mpactification.

The RS brane worlds provide phenomenological

models that reflect at least some of the features of

368 Brane Worlds

M theory, and that bring exciting new geometric

and particle physics ideas into play. The RS2

models also provide a framewor k for e xpl oring

holographic ideas that have emerged in M theory.

Roughly speaking, holography suggests that

higher-dimensional dynamics may be determined

from a knowledge of the fields on a lower-

dimensional boundary. The AdS/CFT c orrespon-

dence is an example in which the classical

dynamics of the higher-dimensional A dS gravita-

tional field are equivalent to the quantum

dynamics of a conformal field theory (CFT) on

the boundary.

Kaluza–Klein Modes

The dilution o f gravity via extra dimensions not

only weakens gravity, it also broadens the range of

graviton modes felt on the brane. The graviton is

more than just the four-dimensional massless mode

of four-dimensional gravity – other modes, with an

effective mass on the brane, arise from the fact

that the graviton is a (4þd)-dimensional massless

particle. These extra modes on the brane are

known as Kaluza–Klein ( KK) modes of the

graviton.

For simplicity, consider a flat brane with one flat

extra dimension, compactified through the identi-

fication y $ y þ 2nL,wheren = 0, 1, 2, ... .The

perturbative five-dimensional graviton is defined

via

ð5Þ



ð5Þ



þ h

½8

where

(5)



is the five-dimensional Minkowski metric

and h

is a small transverse traceless perturbation. Its

amplitude can be Fourier expanded as

hðx

; yÞ¼

iny=L

ðx

Þ½9

where h

are the amplitudes of the KK modes, that

is, the effective four-dimensional modes of the five-

dimensional graviton. To see that these KK modes

are massive from the brane viewpoint, we start from

the five-dimensional wave equation that the massless

five-dimensional field h satisfies (in a suitable

gauge):

ð5Þ

h ¼ 0 )

h þ @

h ¼ 0 ½10

It follows that the KK modes satisfy a four-

dimensional Klein–Gordon equation with an effec-

tive four-dimensional mass, m

¼ m

; m

½11

The massless mode, h

, is the usual four-

dimensional graviton mode. But there is a tower

of massive modes, L

1

,2L

1

, ...,which

imprint the effect of the five-dimensional gravita-

tional field on the four-dimensional brane. Com-

pactness of the extra dimension leads to

discreteness of the spectrum. For an infinite

extra dimension, L !1, the separation between

the modes disappears and the tower forms a

continuous spectrum.

Randall–Sundrum Brane Worlds

RS brane worlds do not rely on compactification to

localize gravity at the brane, but on the curvature of

the bulk. What prevents gravity from ‘‘leaking’’ into

the extra dimension at low energies is a negative

bulk cosmological constant,



ð5Þ

¼

‘

¼6

½12

where ‘ is the curvature radius of AdS

and  is the

corresponding energy scale. The bulk cosmologi cal

constant with its repulsive gravity effect acts to

‘‘squeeze’’ the gravitational potential closer to the

brane. We can see this clearly in Gaussian normal

coordinates x

= (x



, y) based on the brane at y = 0,

for which the metric takes the form

ð5Þ

¼ dy

þ e

2jyj=‘









½13

with 



the Minkowski metric. The ex ponential

warp factor reflects the confining role of the bulk

cosmological constant. The Z

-symmetry about the

brane at y = 0 is incorporated via the jyj term. In the

bulk, this metric is a solution of the five-dimensional

Einstein equations,

ð5Þ

¼

ð5Þ

½14

that is,

(5)

= 0 in eqn [2]. The brane is a flat

Minkowski spacetime, g



,0)= 











, with

self-gravity in the form of brane tension.

The two RS models are distinguished as follows:

RS1 There are two branes in RS1, at y = 0 and

y = L, with Z

-symmetry identifications

y $y; y þ L $ L  y ½15

The branes have equal and opposite tensions, ,

where

 ¼

4

‘

½16

The positive-tension ‘‘TeV’’ brane has fundamental

scale M

(5)

 1 TeV. Because of the exponential

Brane Worlds 369

warping factor, the effective scale on the negative

tension ‘‘Planck’’ brane at y = L is M

. On the

positive tension brane,

¼ M

ð5Þ

‘ 1  e

2L=‘

½17

So RS1 gives a new approach to the hierarchy

problem. Because of the finite separation between

the branes, the KK spectrum is discrete.

RS2 In RS2, there is only one, positive-

tension, brane. This may be thought of as arising

from sendin g the ne gative ten sion brane off to

infinity, L !1. Then the energy scales are

related via

ð5Þ

‘

½18

On the RS2 brane, the negative 

(5)

is offset by

the positive brane tension . The fine-tuning in eqn

[16] ensures that there is zero effective cosmological

constant on the brane, so that the brane has the

induced geometry of Minkowski spacetime. To see

how gravity is localized at low energies, we consider

the five-dimensional graviton perturbations of the

metric:

ð5Þ

þh

¼ 0 ¼ h



¼ @





½19

We split the amplitude h into three-di mensional

Fourier modes, and the linearized five-dimensional

Einstein equations lead to the wave equatio n (y > 0)

2y=‘

€

h þ k

¼ h



‘

½20

Separability means we can write

hðt; yÞ¼

’

ðtÞh

ðyÞ½21

and the wave equation reduces to

€

’

þðm

þ k

Þ’

¼ 0 ½22



‘

þ e

2y=‘

¼ 0 ½23

The zero-mode solution is

’

ðtÞ¼A

0þ

þikt

þ A

0

ikt

½24

ðyÞ¼B

þ C

4y=‘

½25

and the massive KK mode (m > 0) solutions are

’

ðtÞ¼A

mþ

exp þi

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ k



þ A

m

exp i

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ k



½26

ðyÞ¼e

2y=‘

mle

y=‘

h

þC

mle

y=‘

i

½27

where J

, Y

are Bessel functions.

The boundary condition for the perturbations is

(t,0)= 0, which implies

¼ 0; C

¼

ðm‘Þ

½28

In the RS1 model, we have a further boundary

condition, h

(t, L) = 0, which leads to a discrete

eigenspectrum, namely the masses m that satisfy

m‘e

L=‘



ðm‘ÞY

m‘e

L=‘



ðm‘Þ¼0 ½29

The zero mode is normalizable, since

2y=‘



< 1½30

Its contribution to the gravitational potential

V = ð1/2Þh

gives the four-dimensional result, V /

1

. The contribution of the massive KK modes sums

to a correction of the four-dimensional potential.

For r  ‘, one obtains

VðrÞ

1 þ

‘





GM‘

½31

which simply reflects the fact that the potential

becomes truly five dimensional on small scales. For

r  ‘,

VðrÞ

1 þ

2‘



½32

which gives the small correction to four-dimensional

gravity at low energies from extra-dimensional effects.

Cosmological Brane Worlds

The RS models contain vacuum (Minkowski)

branes. In order to pursue brane-world ideas in

cosmology, we need to generalize the RS models to

incorporate cosmological branes with matter and

radiation on them. The effective field equations on

the brane are the vehicle for brane-bound observers

to interpret cosmological dynamics. They arise from

projecting the five-dimensional field equations onto

the brane, via the Gauss–Codazzi equations. These

equations involve also the extrinsic curvature K



the brane, which determines how the brane is

imbedded in the bulk.

The stress-energy on the brane (tension, matter,

radiation) means that there is a jump in K



across

370 Brane Worlds

the brane. More precisely, the junction conditions

across the brane are



 g





¼ 0 ½33



 K





¼

ð5Þ

brane





brane



½34

where

brane



¼ T



 g



½35

is the total energy–momentum tensor on the brane

and T

brane

= g



brane



. The Z

-symmetry means that

when approaching the brane from one side and

going through it, one emerges into a bulk that looks

the same, but with the normal reversed. This implies

that





¼K



½36

so that we can use the junction condition (eqn [34])

to determine the extrinsic curvature:



¼



ð5Þ



ð  TÞg





½37

where T = T



, we have dropped the (þ) and we

evaluate quantities on the brane by taking the limit

y !þ0.

Together with the Gauss–Codazzi equations, eqn [37]

leads to the induced field equations on the brane:



¼g



þ 



þ 6



z



E



½38

where



 

ð4Þ



ð5Þ

½39

  

ð4Þ



ð5Þ

þ 





½40















 T



½41

and



ð5Þ

ACBD





½42

where n

is the unit normal to the brane and

(5)

ACBD

is the Weyl tensor in the bulk.

The induced field equations [38] show two key

modifications to the standard four-dimensional Einstein

field equations arising from extra-dimensional effects.





 (T



)

is the high- energy correction term,

which is negligible for   , but dominant for

   (where  is the energy density):

j



=j

j















½43





, the projection of the bulk Weyl tensor on the

brane, encodes corrections from KK or five-

dimensional graviton effects. From the brane-

observer viewpoint, the energy–momentum

corrections in S



are local, whereas the KK

corrections in E



are nonlocal, since they

incorporate five-dimensional gravity wave

modes. These nonlocal corrections cannot be

determined purely from data on the brane. In

the perturbative analysis of RS2 which leads to

the corrections in the gravitational potential, eqn

[32], the KK modes that generate this correction

are responsible for a nonzero E



; this term is

what carries the modification to the weak-fiel d

field equations.

The effective field equations are not a closed system.

One needs to supplement them by five-dimensional

equations governing E



,whichareobtainedfromthe

five-dimensional Einstein equations.

Cosmological Dynamics

A(1þ4)-dimensional spacetime with spatial

4-isotropy (four-dimensional spherical/ plane/

hyperbolic symmetry) has a natural splitting into

hypersurfaces of symmetry, which are (1 þ3)-

dimensional surfaces with 3-isotropy and

3-homogeneity, that is, Friedmann–Robertson–

Walker (FRW) surfaces. In particular, the AdS

bulk of the RS2 brane world, which admits a

foliation into Minkowski s urfaces, also admits a n

FRW foliation since it is 4-isotropic. T he general-

ization of AdS

that preserves 4-isotropy and

solves the five-dimensional Einstein equation is

Schwarzschild AdS

, and this bulk therefore

admits an FRW foliation. It follows that an

FRW cosmological brane world can be embedded

in Schwarzschi ld A dS

spacetime.

The black hole in the bulk is felt on the brane

via the E



term. The bulk black hole gives rise to

‘‘dark radiation’’ on the brane via its Coulomb

effect. The FRW brane can be thought of as

moving radially along the fifth dimension, with the

junction conditions determining the velocity via

the Friedmann equation. Thus, one can interpret

the expansion of the universe as motion of the

brane through the static bulk. In the special case

of no black hole and no brane motion, the brane is

empty and has Minkowski geometry, that is, the

original RS2 brane world is recovered, in different

coordinates.

An intriguing aspect of the cosmological metric is

that five-dimensional gravitational wave signals can

take ‘‘shortcuts’’ through the bulk in traveling

Brane Worlds 371

between points A and B on the brane. The travel

time for such a graviton signal is less than the time

taken for a photon signal (which is stuck to the

brane) from A to B.

Cosmological dynamics on the brane are governed

by the modified Friedmann equation:



 1 þ



2



 

½44

where H = ˙a=a is the Hubble expansion rate, a(t)is

the scale factor, K is the curvature index, and m is

the mass of the bulk black hole.

The 

= term is the high-energy term. When  

, in the early universe, then H

/ 

. This means

that a given energy density produces a greater rate of

expansion that it would in standard four-dimen-

sional gravity. As a consequence, inflation in the

early universe is modified in interesting ways, some

of which may leave a signature in cosmological

observations.

The m=a

term in eqn [44] is the ‘‘dark

radiation,’’ so called because it redshifts with

expansion like ordinary radiation. But, unlike

ordinary radiation, it is not a form of detec table

matter, but the imprint on the brane of the

gravitational field in the bulk (the Coulomb effect

of the bulk black hole). This additional effective

relativistic degree of freedom is constrained by

nucleosynthesis in the early universe. Any extra

radiative energy not thermally coupled to radiation

affects the rate of production of light elements, and

observed abundances place tight constraints on

such extra energy. The dark radiation can be no

more than 3% of the radiation energy density at

nucleosynthesis:





nuc

. 0:03 ½45

The other modification to the Hubble rate is via

the high-energy correction =. In order to recover

the observational successes of general relativity, the

high-energy regime where significant deviations

occur must take place before nucleosynthesis, that

is, cosmological observations impose the lower

limit

>ð1MeVÞ

) M

ð5Þ

> 10

GeV ½46

This is much weaker than the limit imposed by

table-top experiments, which limit the curvature

radius to ‘ . 0.2 mm, leading to

>ð100 GeVÞ

) M

ð5Þ

> 10

GeV ½47

The high-energy regime during radiation domina-

tion is short-lived. Since 

= decays as a

8

during the

radiation era, it will rapidly drop below one, and the

universe will enter the low-energy four-dimensional

regime. However, traces of the high-energy era may be

left in the perturbation spectra that leave an imprint in

the cosmic microwave background radiation.

In conclusion, simple brane-world models of RS2

type provide a rich phenomenology for exploring

some of the ideas that are emerging from M theory.

The higher-dimensional degrees of freedom for the

gravitational field, and the confinement of standard

model fields to the visible brane, lead to a complex

but fascinating interplay between gravity, particle

physics, and geometry, which enlarges and enriches

general relativity in the direction of a quantum

gravity theory. High-pre cision astronomical data

mean that cosmology is a potential laboratory for

testing and constraining these brane worlds. The

models predict extra-dimensional signatures in the

cosmic microwave background and other observa-

tions, and these predictions can in principle be tested

against data.

See also: String Theory: Phenomenology; Supergravity;

Superstring Theories.

Further Reading

Brax P and van de Bruck C (2003) Cosmology and brane worlds:

a review. Classical and Quantum Gravity 20: R201 (arXiv:

hep-th/0303095) (arXiv: hep-th/0303095).

Cavaglia M (2003) Black hole and brane production in TeV

gravity: a review. International Journal of Modern Physics

A18: 1843 (arXiv:hep-ph/0210296).

Langlois D (2003) Cosmology in a brane-universe. Astrophysics

and Space Science 283: 469 (arXiv:astro-ph/0301022).

Maartens R (2004) Brane-world gravity. Living Reviews in

Relativity 7: 7 (arXiv:gr-qc/0312059).

Quevedo F (2002) Lectures on string/brane cosmology. Classical

and Quantum Gravity 19: 5721 (arXiv:hep-th/0210292).

Rubakov V (2001) Large and infinite extra dimensions. Physics-

Uspekhi 44: 871 (arXiv:hep-ph/0104152).

Wands D (2002) String-inspired cosmology. Classical and

Quantum Gravity 19: 3403 (arXiv:hep-th/0203107).

372 Brane Worlds

Branes and Black Hole Statistical Mechanics

S R Das, University of Kentucky, Lexington, KY, USA

Introduction

In classical general relativity, a black hole is a

solution of Einstein’s equations with a region of

spacetime which is causally disconnected from the

asymptotic region at infinity. The boundary of such

a region is called the ‘‘event horizon.’’ The spacetime

around the simplest black hole in three space

dimensions is described by the Schwarzschild metric

¼ 1 

2GM



þ 1 

2GM



1

þ r

d

½1

where G is Newton’s gravitational constant, c is the

velocity of light, and we have used spherical

coordinates with d the line element on an S

nonrotating, uncharged star which is too massive to

form a neutron star will eventually collapse, and at

late times the metri c will be given by [1]. The

horizon is a null surface S

 t and the radius of the

is r

horizon

= 2GM= c

. The Schwarzschild solution

has generalizations to black holes with charge and

angular mom entum and no-hair theorems guarantee

that a black hole has no other characteristic property.

All these solutions can be generalized to other

theories like supergravity in various dimensions.

In 1974, Hawking showed that due to pair

production of particles near the horizon, black

holes radiate thermally. Hawking’s calculation is

valid for black holes whose masses are much larger

than the Planck mass: for such black holes, the

curvature at the horizon is weak and normal

semiclassical quantization is valid. Remarkably, the

properties of Hawking radiation are quite universal.

A black hole can be characterized by an entropy

called the Bekenstein–Hawking entropy. The leading

result for the entropy S

for all black holes in any

theory with the standard Einstein–Hilbert action is

given by

½2

where A

denotes the area of the horizon. The

temperature T

is given by



2

½3

where  is the surface gravity at the horizon. The

principle of detailed balance further ensures that the

radiation rate of some species of particle i, 

(k),

in some given momentum range ( k, k þ dk) is related

to the corresponding absorption cross section 

(k)by

ðkÞ¼



ðkÞ

!=T

 1

ð2Þ

½4

where ! is the energy and d denotes the number of

spatial dimensions. The  sign refers to fermions

(bosons), respectively. A nontrivial k dependence of



signifies a departure from black-body behavio r.

Consequently, 

(k) is often called a grey-body

factor. Equations [2] and [3] may be derived by

combining Hawking’s calculation of the radiation

with standard thermodynamic relations. Alterna-

tively, they follow from the leading semiclassical

approximations of path-integral formulations of

Euclidean gravity based on the standard Einstein–

Hilbert action. For an account of black-hole

thermodynamics, see Wald (1994).

Unlike usual thermodynamic systems, black holes

appear to pose a deep puzzle. In usual systems,

thermodynamics is a coarse-grained description of a

system which is in a highly degenerate state.

Typically, such systems are described in terms of a

few macroscopic parameters such as the total

energy, the total volume, the total charge. For each

set of values of these macroscopic parameters, there

are a large number of microscopic states which can

be described in terms of the constituents such as

atoms or molecules. This degener acy manifests itself

as an entropy S which is related to the number of

microscopic states for a given set of value s of the

macroscopic parameters,  by Boltzmann’s relation

S ¼ logðÞ½5

where units have been chosen such that the

Boltzmann constant is unity. For a black hole, the

macrostates are specified by its mass, charge, and

angular momentum. No-hair theorems, however,

seem to suggest that there are no other properties

and hence no obvious candidate for microstates. In

the absence of such a statistical basis, one would be

inevitably led to the conclusion that there is loss of

information in processes involving black holes.

In a consistent quantum theory of gravity, there

would be such a statistical basis since quantum

mechanics is unitary. String theory is a strong

candidate for a unified theory which contains

gravity. Indeed, string theory provides a microscopic

description for a class of black holes.

Branes and Black Hole Statistical Mechanics 373

Black Hole Solutions in String Theory

Perturbatively, the basic excitations of string theory

are f undamental closed and open strings character-

izedbyastringtensionT

and hence a length scale,

the string length l

= 1=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2T

. Consistency requires

that the string should be able to propagate in ten

spacetime dimensions and should be supersym-

metric at the fundamental level. Formulated in

this fashion, there are several consistent string

theories: type IIA, type IIB, and heterotic string

theory (which contain only closed strings perturba-

tively) and type I theory (which contains both open

and closed strings).

At energies much smaller than 1=l

,onlythe

massless modes of the string can be excited. For all

these string theories, the massless spectrum of closed

strings contains the graviton and the low-energy

dynamics is given by the appropriate supersymmetric

generalization of general relativity, supergravity. In

addition, the closed-string spectrum contains a

neutral scalar field, the dilaton , whose expectation

value gives rise to a dimensionless parameter govern-

ing interactions, called the string coupling g

¼ e

<>

½6

The ten-dimensional gravitational constant is given

¼ 8

½7

Ten-dimensional supergravity has a wide va riety of

black hole solutions, the simplest of which is the

straightforward generalization of the Schwarzschild

solution.

Black p-Brane Solutions

More significant ly, there are solutions which are

charged with respect to the various gauge fields that

appear in the supergravity spectrum. Generically,

these charged solutions represent extended objects.

For accounts of such solutions, see Maldacena

(1996).

Consider, for example, the supergravity which

follows from type IIB string theory. This theory has

a pair of 2-form gauge fields B

and B

and a

4-form gauge field A

MNPQ

with a self-dual field

strength. Just as an or dinary point electric charge

produces a 1-form gauge field, a (p þ 1)-form gauge

field may be sourced by an electrically charged

p-dimensional extended object. The corresponding

field strength is a (p þ 2)-form, whose Hodge dual in

d spacetime dimensions is a (d  p  2) form. This

shows that there should be magnetically charged

(d  p  4)-dimensional extended objects as well.

These extended objects are called ‘‘branes.’’

In the type IIB example, there should be two

kinds of one-dimensional extended objects

which carry electric charge under B

, B

called the F-string and the D-string, respectively.

There are also two k inds of five-dimensional

branes which carry magnetic charges under

, B

, called the NS 5- brane and D5 br ane,

respectively. Finally, there should be a 3-brane,

since the corresponding 5-form field strength is

self-dual as well as a D7 brane. A similar catalog

can be prepared for other string theories, as well

as for 11-dimensional supergravity, which is the

low-energy limit of M-theory.

The classical solutions for a set of p-branes of the

same kind generally have inner and outer horizons

which have the topology t  S

8p

 R

. The outer

horizon is then associated with a Hawking tempera-

ture and a Bekenstein–Hawking entropy. Of parti-

cular interest are extremal limits. In this limit, the

inner and outer horizons coincide and the mass

density is simply pro portional to the charge. Given

some charge, the extremal solution has the lowest

energy. Extremal limits are interesting because in

supergravity these correspond to solutions in which

some of the supersymmetries (in this case, half of the

supersymmetries) are retained – such solutions are

called Bogomolny–Prasad–Sommerfeld (BPS) satu-

rated solutions. The charge in question appears as a

central charge in the extended supersymmetry

algebra. This fact may be used to show that such

BPS solutions are absolutely stable. Indeed, for the

particular solution considered here, the Hawking

temperature T

! 0, so that there is no Hawking

radiation, as required by stability. Furthermore, the

entropy S

! 0. The horizon shrinks to a point

which appears as a naked null singularity.

All the ten dimensions of string theory need not be

noncompact. In fact, to describe the real world, one

must have a solution of string theory in which six of

the dimensions are wrapped up and form a compact

space. In principle, however, one can compactify

any number of dimensions. In the above example

of a p-brane, it is trivial to compactify the

directions along which the brane is extended to a

p-dimensional torus, T

, which can be chosen to be

a product of p circles each of radius R. At length

scales much smaller than R, the theory then becomes

a (10  p)-dimensional theory. The p-brane appears

as a black hole with a spherical horizon and,

since the original p-form gauge field now behaves

as an ordinary 1-form gauge field with a nonzero

time component, this is an electrically charged

black hole.

374 Branes and Black Hole Statistical Mechanics