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

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D1–D5–N System and Five-Dimensional Black

Holes

For reasons which will become clear in the next

section, it is useful to get extremal black holes with

large horizon areas, so that Hawking’s semiclassical

formulas are valid. It turns out that such solutions

involve branes of various types which intersect each

other and are suitably wrapped on compact internal

spaces. Such black holes then have necessarily

different kinds of charges. It turns out that the

simplest case is a five-dimensional black hole with

three kinds of charges, which is obtained by brane

systems wrapped on a compact five-dimensional

space. An example is a type IIB solution which has

D5 branes which are wrapped on eith er T

 S

K3  S

, together with D1 branes wrapped on the S

as well as some momentum along the S

. From the

noncompact five-dimensional point of view, this is a

black hole with three kinds of gauge charges: the D5

charge Q

, the D1 charge Q

, and a Kaluza–Klein

charge N coming from the momentum P = N=R

along the circle of radius R.

When the internal space is T

 S

the five-

dimensional Einstein frame metric is given by

¼½f ðrÞ

2=3

1 



þ½f ðrÞ

1=3

ð1  r

þ r

d



½8

where

f ðrÞ¼ 1 þ

sinh



1 þ

sinh



 1 þ

sinh



½9

and the three charges are

sinh 2

32

; Q

sinh 2 

N ¼

32

sinh 2

½10

where V is the volume of the T

and R is the radius

of the circle S

The ADM mass of the black hole is

ADM

RVr

32

½cosh 2

þ cosh 2

þ cosh 2½11

The Bekenstein–Hawking entropy is given by

RVr

8

cosh 

cosh  ½12

while the Hawking temperature is

2r

cosh 

cosh 

½13

The extremal limit of this solution is given by

! 0;

;

;!1

; Q

; N ¼ fixed

½14

The extremal solution is a BPS saturated state and

retains four of the original supersymmetries. In this

limit, the inner and outer horizons coincide. How-

ever, the horizon is now a smooth S

with a finite

area in the Einstein frame metric. Consequently, the

extremal Bekenstein–Hawking entropy is also finite

and may be seen to be

charge extremal

¼ 2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

½15

The temperature, however, is zero in this limit,

which is consistent with the stability of a BPS

saturated state.

The above five-dimensional black hole is in fact a

generalization of the Reissner–Nordtsrom black

hole. Similar solutions with large horizon areas in

the ex tremal limit can be constructed in four

dimensions. One such construction is in the IIB

theory wrapped on T

in which there are four sets of

D3 branes which wrap four different T

’s contained

in the T

. Black holes with lower supersymmetry

may be obtai ned by replacing the T

by a Calabi–

Yau space.

Duality and Branes

String theory has a rich set of symmetries called

duality symmetries which relate different kinds of

string theories that are suitably compactified.

These symmetries relate different classical solutions.

For example, application of these symmetries relate

the five-dimensional black holes above with other

five-dimensional black holes with different kinds of

charges. Furthermore, at the level of supergravity,

these various theories may be derived from

a yet unknow n 11-dimensional theory called the

M-theory whose low-energy limit is 11-dimensional

supergravity.

Branes in String Theory

For a given string theory, the perturbative spectrum

consists of strings. How ever, at the nonperturbative

Branes and Black Hole Statistical Mechanics 375

level, there are, in addition, extended objects of

other dimensionalities. Duality symmetries imply

that these extended objects are as ‘‘fundamental’’

as the strings themselves. Such extended objects are

also called branes. For an exhaustive account of

branes in string theory, see Johnson (2003).

Like their counterparts in supergravity, branes in

string theory are typically charged with respect to

some gauge fields. While supergravity solutions are

possible with any value of the charge, in string

theory the brane charges have to be quantized.

Multiple units of the minimum quantum of charge

can appear as collections of branes each with unit

charge or, alternatively, branes which wrap arou nd

compact cycles in space a multiple number of times.

D-Branes

The extended objects in string theory are described

in terms of their collective excitations. These

are best understood for the class of branes called

D-branes in the type II theory, discovered by

Polchinski. These are D1, D3, D5, and D7 branes

in type IIB and D0, D2, D4, and D6 branes in

type IIA theory. Dp branes are characterized by the

fact that they couple to, and act as sources for,

(p þ 1)-form gauge fields which belong to the

Ramond–Ramond sector of the theory. Collective

excitations of a p-dimensional extended object in

field theory are expected to be described by waves

on its (p þ 1)-dimensional world volume. The

collective coordinate action would be a quantum

field theory which has vectors, corresponding

to longitudinal oscillations of the brane, and

scalars which correspond to transverse oscillations.

For D-branes in string theory, the theory of

collective excitations is a string field theory of open

strings whose endpoints lie on the brane. (This is the

origin of the nomenclature D-brane: an open string

whose ends are constrained to lie on the brane has a

world-sheet description in which the bosonic

fields corresponding to trans verse target space

coordinates have Dirichlet boundary conditions.)

The lowest-energy states of open superstrings are

ordinary massless gauge fields and their supersym-

metric partners so that the low-energy limit of

the string field theory is a supersymmetric gauge

theory.

The fact that the underlying theory is a string

theory has an important consequence. For a system

of N parallel D-branes of the same type, one

would have open strings which join different branes

as well as the same brane. The low-energy

theory then becomes a supersymmetric nonabelian

gauge theory with gauge group U(N). In a suitable

gauge, the off-diagonal gauge fields and their super-

symmetric partners (which include scalar fields in

the adjoint representation) are the low-energy

degrees of freedom of open strings which connect

different branes.

The mass density or tension T

of a single D p

brane is given by

ð2Þ

pþ1

½16

This couples to the (p þ 1)-form gauge field with a

charge



¼ g

½17

and the Yang–Mills coupling constant for the collec-

tive theory on the brane world volume is given by

YMDp

¼ð2Þ

p2

p3

½18

The ground state of a single Dp brane is a BPS state

which preserves 16 of the 32 supersymmetries of the

original theory. One consequence of this is that two or

more parallel Dp branes of the same type form a

threshold bound state preserving the same supersym-

metries, with no net force between them. As a result, the

tension of N parallel Dp branes is simply NT

Branes of different dimensionalities can also form

bound states. Of particular interest are configura-

tions which can form threshold bound states which

preserve some supersymmetries. For example, a set

of N

parallel Dp branes can form a thres hold

bound state with a set of N

parallel D(4 þ p)

branes with all the p branes lying entirely along the

(4 þ p)-branes. This configuration is also a BPS

saturated state preserving eight of the original

supersymmetries and would have charges under

both (p þ 1)-form and (p þ 5)-form gauge poten-

tials. The BPS nature ensures that the total mass

density is the sum of the individual mass densities.

NS Branes

The other extended objects in string theory are

called NS branes since they couple to p-form

gauge fields which arise from the Neveu–Schw arz/

Neveu–Schwarz sector of the world-sheet theory.

These are present in all the five string theories and

appear in two types. The first is a macroscopic

fundamental string which may be wound around a

compact direction. The second is called a solitonic

5-brane. While the collective dynamics of a funda-

mental string is the standard world-sheet description

of string theory, the description for the NS 5-brane

is rather complicated and not known in full

detail. The rest of this article deals exclusively with

D-branes.

376 Branes and Black Hole Statistical Mechanics

D-Branes and Black Branes

The idea that black holes correspond to highly

degenerate states in string theory is quite old and

dates back to ’t Hooft (1990) and Susskind (1993).

In the following two sections we discuss such black

holes which are described by D-branes. For reviews

see Maldacena (1996), Das and Mathur (2001), and

David et al. (2002).

We have so far discussed the string-theoretic

branes in two different ways. In the first description,

branes are solutions of the low-energy equati ons of

motion – this is the setting in which branes provide

conventional descriptions of black holes. In the

second description, branes are certain states in the

quantum theory of superstrings. More specifically,

D-branes are described in terms of states of the

open-string field theory which lives on the branes.

The first description is necessarily approximate. On

the other hand, the second description is exact in

principle, although in practice one might not know

how to write down and analyze the string-field

theory in an exact fashion.

The description in terms of open-string field

theory should reduce to the descript ion in terms of

a classical solution when the charges and masses

become large. If black-hole thermodynamics has a

microscopic origin, D-branes should be highly

degenerate states in this limit and the entropy

should be given by the Boltzmann formula. Further-

more, Hawking radiation should be understood as

an ordinary decay process.

For a system of Q

parallel Dp branes, the mass

is Q

, while Newton’s gravitational constant

G  g

. Gravitational effects are controlled by

GM  g

. A semiclassical limit in closed-string

theory requires g

! 0, while a nontrivial gravita-

tional effect in this limit requires g

finite, which

implies one must have Q

1. Furthermore, when

 1 the typical curvatures are small compared

to the string scale and the sem iclassical string theory

reduces to classical supergravity. This is the limit in

which branes are well described as classical

solutions.

Similar considerations apply for brane systems with

multiple charges. For example, in the D1–D5–N

system the classical solution becomes a good

description when all the quantities g

, g

,and

N become large. (The relevant quantity which

comes with the momentum has g

rather than g

because the mass contribution from the momentum is

simply N/R without any inverse power of g

However, g

is the square of the coupling constant

of the open-string theory living on the brane – in fact,

eqn [18] shows this relation in the low-energy limit.

It is well known that in a U(Q

) gauge theory the real

coupling constant is g

ﬃﬃﬃﬃﬃﬃﬃ



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

.Thismeans

that the semiclassical limit corresponds to a strongly

coupled string-field theory which reduces to strongly

coupled gauge theory in the low-energy limit and the

picture of D-branes as a collection of open strings is

not very useful. In fact, known calculational methods

in gauge theory or open-string theory are not valid in

this regime.

Microscopic Entropy for Two-Charge Systems

The prospects are much better for extremal black

holes, which appear as BPS states in string theory.

This is because the spectra of BPS states do not

depend on the coupling. The degeneracy of such

states may therefore be calculated at weak coupling,

where techniques are well known and the result can

be extrapolated to strong coupling without change.

The simplest BPS state is the ground state of a set of

parallel D-branes of the same type. This state is indeed

128-fold degenerate, which would imply a micro-

scopic entropy. This entropy, however, is small and

therefore invisible in the corresponding classi cal

solution. Indeed, the classical solution shows that in

the extremal limit the horizon area is zero, leading to a

vanishing Bekenstein–Hawking entropy.

The next interesting class of states consists of

threshold bound states with two kinds of

charges. Consider, for example, the D1–D5 system

on T

 S

considered above with no momentum

along the D1’s. By known duality transformations,

this is equivalent to a fundamental IIB string which

is wound Q

times around the S

and with a net

momentum P = Q

=2Q

R (where R is the radius of

the S

), with four of the trans verse directions

compactified on a T

. For this system, it is easy to

count the number of states for given values of Q

and Q

at weak string coupling by simply enumer-

ating the perturbative oscillator states of the string.

For large values of Q

and Q

, we can alternatively

calculate this entropy by using a canonical ensemble

of eight massless bosons corresponding to the eight

transverse polarizations and their supersymmetric

partners – eight massless fermions – moving on the

string with some temperature T and a chemical

potential  for the total momentum.

Consider a noninteracting gas of f massless bosons

and f fermions living on a circle with circumference

L. The average number of left- and right-moving

particles with some energy e, denoted by 

, 

respectively, are



ðeÞ¼

e=T

 1

; i ¼ L; R ½19

Branes and Black Hole Statistical Mechanics 377

where the  sign refers to fermions and bosons,

respectively, and we have introduced left- and right-

moving temperatures T

, T

. The physical tempera-

ture is



½20

The extensive quantities, such as the energy E,

momentum P, and entropy S, then become the sum

of left- and right-moving pieces:

E ¼ E

þ E

; P ¼ P

þ P

; S ¼ S

þ S

½21

and the distribution function [19] leads to the

following thermodynamic relations:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Lf

fL

; i ¼ L; R ½22

Since the total mom entum P = P

þ P

= E

 E

nonzero, the lowest-energy state is clearly the one in

which all the particles move in the same direction,

for example, right moving. This is a BPS state and

corresponds to the extremal solution in supergravity.

Then E = E

= P = P

. This approach to the black

hole entropy was initiated by Das and Mathur

(1996) and Callan and Maldacena (199 6).

For our two-charge syst em, f = 8, P = 2Q

=L,

and L = 2RQ

. Using [22] we get

2-charge-II

micro

¼ 2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

½23

This is the microscopic entropy for the fundamental

string with momentum in the type II theory. By

duality, this is also the microscopic entropy of the

D1–D5 system. This is a large number which should

agree with the macroscopic entropy calculated from

the corresponding classical solution.

The discussion is almost identical for the funda-

mental heterotic string, except that now we have

24 right-moving bosons, eight left-moving bosons,

and eight left-moving fermions, and the BPS state

consists only of right movers. If n

denotes the

winding number and n

the quantized momentum

the extremal heterotic string entropy is

2-charge heterotic

micro

¼ 4

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

½24

The supergravity solution for the D1–D5

system may be obtained by substituting  = 0in

eqns [8]–[13]. In the extremal limit, the classical

Bekenstein–Hawking entropy vanishes as is clear

from the expression [15], in which N = 0. This

appears to be in contradiction with the fact that the

state has a large microscopic entropy.

The key point, however, is that the two-charge

solution has a singular horizon where the string

frame curvature is large. Consequently, low-energy

tree-level supergravity breaks down near the horizon

and higher-derivative terms (e.g., higher powers of

curvature) become important. This issue has been

best studied for the fundamental heterotic string

compactified on T

. This is dual to the D1–D5

system in type IIB theory compactified on K3  T

The classical supergravity solution is then a singular

black hole in four spacetime dimensions. In one of

the first papers on the string-theoretic understanding

of black hole thermodynamics, Sen (1995) showed

that, for large n

, n

, string-loop effects are small

near the horizon so that the only relevant correc-

tions are higher-derivative terms coming from

integrating out the massive modes of the string at

tree level. Furthermore, a robust scaling argument

shows that regardless of the detailed nature of the

derivative corrections, the macroscopic entropy

defined through the horizon area must be of the

form a

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

, where a is a pure number. Finally,

one can define a ‘‘stretched horizon’’ as the surface

where the curvature becomes of the order of the

string scale and the area of the stretched horizon

is indeed proportional to

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

. This result gives

a strong indication that string theory provides a

microscopic basis for black hole thermodynamics,

although the coefficient a cannot be determined

without more detailed knowledge of higher-

derivative terms.

Microscopic Entropy of Extremal Three-Charge

System

Brane bound states with three kinds of charge

provide examples of black holes whose extremal

limits have large horizons with curvatures much

smaller than the string scale. In this case, a

microscopic count of states in string theory should

exactly account for the Bekenstein–Hawking

formula, without corrections coming from

higher derivatives. This is indeed true, as first found

by Strominger and Vafa (1996). In the following, we

will outline how this calculation can be done in the

D1–D5–N system on K3  S

or T

 S

following

the treatment of Dijkgraaf et al. (1996).

D1 branes can be considered as ‘‘instanton

strings’’ in the six-dimensional supersymmetric

U(Q

) gauge theory of D5 branes (actually, these

should be called solitonic string s rather than

instantons, since the configurations are time

independent). The total instanton number is the

D1-brane charge Q

. The moduli space of

these instantons is then a blown-up version of the

378 Branes and Black Hole Statistical Mechanics

orbifold (T

)

=S(Q

)or(K3)

=S(Q

)

and is 4Q

dimensional. Since any instanton

configuration is independent of time x

and the S

direction x

, the collective coo rdinate dynamics is a

(1 þ 1)-dimensional field theory which lives in the

, x

) space. At low energies, this flows to a

conformal field theory with a central charge

c = 6Q

since there are 4Q

bosons each

contributing 1 to the central charge and an equal

number of fermions each contributing 1=2. The BPS

state with momentum N=R is a purely right- or left-

moving state in this conformal field theory which

has a conformal weight N. From general principles

of conformal invariance, the degeneracy of such

states for large N is given by Cardy’s formula

dðNÞe

2

ﬃﬃﬃﬃﬃﬃﬃﬃ

cN=6

½25

so that the microscopic entropy is

micro

3-charge

¼ log dðnÞ¼2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

cN=6

½26

Substituting the value of c = 6Q

, this is in exact

agreement with the Bekenstein–Hawking entropy of

the classical solution given in [15].

Nonextremal Black Holes and Hawking

Radiation

The BPS property of ground states of D-brane

systems enables us to compute the degeneracy of

microstates exact ly in the regime of parameters

where the state can be reliably described as a black

hole solution in the low-energy theory. How ever,

extremal black holes have vanishing temperature

and do not radiate. To understand the microscopic

origins of Hawking radiation, one has to go away

from extre mality. Such states are not supersym-

metric and an extrapolation of weak-coupling

calculations to strong coupling is not a priori

justified. Nevertheless, it turns out that for small

departures from extremality, weak-coupling results

still reproduce semiclassical answers for entropy,

temperature, and luminosity.

Near-Extremal Entropy

Nonextremal properties are best understood for the

D1–D5–N system on T

 S

. In the orbifold limit,

the conformal field theory which describes the low-

energy dynamics is equivalent to a gas of strings

which are wound around the S

and which can

oscillate along the T

. The total winding number is

k = Q

and may be achieved by sets of strings

which are multiply wound in various ways. As

argued below, entropically the most favored config-

uration is a single long string wound around Q

times. Thus, the thermodynamics may be analyzed

exactly along the lines of the fundamental string in

the previous section. The thermodynamic relations

are given by [22] with f = 4andL = 2RQ

. The

extremal state consists entirely of right movers and

E = E

= N=R. Substituting these values in [22]

yields the correct formula for the microscopic

entropy

3-charge

micro

¼ 2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

½27

The same expression follow s if f = 4Q

and

L = 2R corresponding to Q

singly wound

strings. However, for statistical methods to hold,

the entropy must be much larger than the number of

flavors. The ratio of the entropy to the number of

flavors is S=f 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

N=Q

for multiple singly

wound strings and is not guaranteed to be large

when all of Q

, Q

, N are large. On the other hand,

this ratio is S=f 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

for the long string.

This shows that the long string is always entropi-

cally favored.

A departure from the extremal state is achieved by

adding a left-moving momentum 2n=L as well as a

right-moving momentum 2n=L to the extremal

state, thus adding energy to the system but main-

taining the total momentum. For the long string, this

yields

¼ 2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

N þ n

; S

¼ 2

ﬃﬃﬃ

½28

For small departures from extremality, n  N, the

expressions for the total entropy and tem perature as

a function of the excess energy E = 2n=Q

agree exactly with the near-extremal Bekenstein–

Hawking entropy and the Hawking temperature of

the classical solution, as shown by Callan and

Maldacena (1996) and by Horowitz and

Strominger.

The necessity of the long string appears in another

important physical consideration. For statistical

mechanics to be valid, the specific heat of the system

has to be larger than unity. This implies that for

the case considered here the energy gap E must be

larger than 1=RQ

, which is precisely what the

long string yields.

Hawking Radiation

A nonextremal state described above is unstable,

since a left mover can annihilate a right mover into a

closed-string mode which may leave the brane

system and propagate to the asymptotic region.

The resulting closed-string state will be in a thermal

state whose temperature is the physical temperature

of the initial state. This process is the microscopic

Branes and Black Hole Statistical Mechanics 379

description of Hawking radiation. The decay rate is

related to the absorption cross section of the

corresponding mode by the principle of detailed

balance, encoded in eqn [4].

From the point of view of the classical solution,

the absorption cross section can be calculated by

solving the lin earized wave equ ation in the

background geometry and calculating the ratio of

the incident and reflec ted waves. It follows from

these calculations that at low energies, absorption

(and hence emission) are dom inated by massless

minimally coupled scalars. In fact, for any spheri-

cally symmetric black hole in any number of

dimensions, there is a general theorem which

ensures that the low-energy limit of this absorption

cross section is exactly equal to the horizon area.

In the microsc opic model for the three-charge

black hole, this absorption cross section may be

calculated by the usual rules of quantum mechanics.

In the long-string limit and in the approximation

that the modes on the long string form a dilute gas,

the result has been derived by Das and Mathur

(1996):

ð!Þ¼

2G

!=T

 1

ðe

!=2T

 1Þðe

!=2T

 1Þ

½29

where V isthevolumeoftheT

and T is the

physical temperature given by [20]. For a near-

extremal hole T

 T

,sothatT  2T

.Then

in the extreme low-energy limit !  T

,sothat

the corresponding Bose factor may be approx i-

mated as 1=(e

!=2T

 1)  2T

=!.Thecross

section [29] becomes

 ¼

4Q

ð2RÞV

¼ 4G

extremal

¼ A

½30

where G

is the five-dimensional Newton’s gravita-

tional constant. We have used the relation [22] with

L = 2RQ

and f = 4. The fact that in the near-

extremal limit S

is simply the extremal entropy and

the fact that the extremal entropy reproduces the

Bekenstein–Hawking formula has been used as well.

Thus, the microscopic cross section exactly reproduces

the semiclassical result at low energies. Even more

remarkably, the full cross section [29] agrees with the

semiclassical answer for the gray-body factor for

parameters which correspond to the dilute-gas regime,

as shown by Maldacena and Strominger.

It is rather surprising that the results for micro-

scopic absorption cross section calculated at wea k

coupling agree with the semiclassical answers, since

the relevant process involves states which are not

supersymmetric and therefore a naive extrapolation

to strong coupling is not a priori justified. There

are strong indications, however, that low-energy

nonrenormalization theorems are at work. This

agreement has been established not only for black

holes with finite-horizon areas, but also for other

systems with no horizons – most significantly, a set

of parallel 3-branes – and forms the basis for

Maldacena’s conjecture abou t AdS/CFT Correspon-

dence (see AdS/CFT Correspondence).

Effects of Higher-Derivative Terms

The classical low-energy limit of string theory is

supergravity. The effects of the massive modes of the

string as well as effect of string loops is to add terms to

the supergravity action which involve higher number

of spacetime derivatives, for example, terms containing

higher powers of the curvature. In the presence of such

terms, the Bekenstein–Hawking formula for black hole

entropy [2] receives corrections which can be calcu-

lated in a systematic fashion. It turns out that for a

class of extremal black holes, this corrected entropy as

computed in the modified supergravity is also in exact

agreement with a microscopic calculation.

One example of this agreement is provided by four-

dimensional extremal black holes in type IIA string

theory compactified on a Calabi–Yau manifold. These

are obtained by wrapping D4 branes on three different

4-cycles on the Calabi–Yau and having in addition a

number of D0 branes. Let p

, A = 1, ...,3 denote the

three D4 charges and q

denote the D0 charge. The

microscopic entropy of the BPS state can be computed

by embedding this in M-theory:

CYBlack hole

micro

¼ 2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jðC

ABC

þ c

½31

where C

ABC

is the intersection number of the

4-cycles and c

denotes the second Chern class of

the Calabi–Yau space. When all the charges p

are

large, the term involving c

is subdominant. In this

case, the result agrees with the Bekenstein–Hawking

entropy of the corresponding classical solution.

When the charges are not all large (so that the

second term is appreciable), the curvatures of the

supergravity solution become large at the horizon

and higher-derivative corrections to the action

cannot be ignored. In this particular case, it turns

out that these higher-derivative corrections are

string-loop corrections and can be computed using

general properties of N = 2 supersymmetry, so that

one can compute corrections to near-horizon

geometry. Furthermore, one has to now modify the

380 Branes and Black Hole Statistical Mechanics

expression for macroscopic entropy using the

formalism of Wald. Putting these together, it is

found that the macroscopic entropy following from

the modified supergravity is in exact agreement with

[31]. This subject is reviewed in Mohaupt (2000).

These methods have also been applied to the

problem of tw o-charge black holes in heterotic

string theory on T6 or, equivalently, type IIA on

K3  T

(Dabholkar 2004). Recall that in this case

the horizon of the usual supergravity solution is

singular. It has been found that leading-order

higher-derivative corrections smoothen out the

horizon into a AdS

 S

spacetime and the

modified expression for the macroscopic entropy is

again in exact agreement with the microscopic

answer [23].

Geometry of Microstates

A satisfactory solution of the information-loss

paradox requires a much more detailed understand-

ing of black holes in string theory. The discussion

above shows that black holes have microstates

which may be described well in the weak-coupling

regime. It is interesting to ask whether there is a

description of these microstates in the strong-

coupling regime in terms of the effective geometry

perceived by suitable probes. This questi on has been

answered for the two-charge system in great detail

(see Mathur (2004)). It turns out that the D1–D5

microstates can be described by perfectly smooth

metrics with no horizons, and they asymptote to

the standard two-charge metric discussed above.

The location of the erstwhile stretched horizon

marks the point where the different microstates

start differing from each other significantly. Since

each such geometry does not have a horizon, neither

does it have any entropy – this is consistent with

their identification with nondegenerate microstates.

Indeed, the number of such microstates correctly

accounts for the microscopic entropy. Whether a

similar picture holds for the three-charge system

remains to be seen in detail, although there are some

indications that this may be true. In this approach, it

is not yet fully understood how a horizon emerges

and why the entropy scales as the horizon area.

Outlook

One key feature of the understanding of black hole

statistical mechanics from the dynamics of branes is

the fact that a problem in gravity is mapped to a

problem in a theory without gravity, for example,

open-string field theory. In fact , the closed strings in

the bulk are already contained in the spectrum of the

open strings. This is a consequence of the basic

duality between open strings and closed strings.

Furthermore, the open-string theory lives in a lower-

dimensional spacetime. Thi s is a manifestation of

the holographic principle. As argued by Maldacena,

the prese nce of a horizon implies that the low-

energy limit retains all the modes of the closed

strings near the horizon, while it truncates the open-

string theory to a gauge theory. Open–closed duality

then reduces to gauge–string duality. This provides a

strong evidence that black holes obey the normal

laws of quantum mechanics and hence their time

evolution is unitary.

One of the most outstanding problems in the

subject is a proper understanding of neutral black

holes. Most of the quan titative results described

above depend on supersymmetry, which allows

extrapolation of weak-coupling answers to the

strong-coupling domain. Some of these results can

be extended to situations which have small depar-

tures from supersymmetry, for example, near-

extremal black holes. States corrresponding to

neutral black holes are, however, far from super-

symmetry and known calculational techniques fail.

There are good reasons to expect, however, that the

general philosophy – in particular the holographic

principle – is still valid. Final ly, so far string theory

has been able to attack problems of eternal black

holes. A satisfactory understanding of the informa-

tion-loss problem requires an understanding of the

dynamics of black hole formation and subsequent

evaporation. Unfortunately, very little is known

about this at the moment.

See also: AdS/CFT Correspondence; Black Hole

Mechanics; Supergravity; Superstring Theories.

Glossary

ADM (Arnowitt–Deser–Misner) mass – Mass of a gravita-

tional background which is asymptotically flat.

AdS

(anti-de Sitter space) – A space (or spacetime) with

constant negative curvature in n dimensions.

BPS state (Bogomolny–Prasad–Sommerfeld state) – In a

theory of extended supersymmetry, a state that is

invariant under a nontrivial subalgebra of the full

supersymmetry algebra. These states always carry

conserved charges, and supersymmetry determines the

mass exactly in terms of the charges.

Calabi–Yau space – Complex Kahler manifold with

vanishing first Chern class.

Compactify (n. compactification) – To consider a field or

string theory in a spacetime some of whose spatial

dimensions are compact.

Dirichlet boundary condition – The boundary condition

which fixes the value of a field on the boundary.

Branes and Black Hole Statistical Mechanics 381

Duality Equivalence of systems which appear to be

distinct. For string theories, such equivalences relate

string theories on different spacetimes as well as

theories with different coupling constants.

Einstein–Hilbert action – The standard action for gravity

which leads to Einstein’s equation,

S = (1=16G)

ﬃﬃﬃ

R, where R is the Ricci scalar,

g denotes the determinant of the metric, and G is

Newton’s gravitational constant.

Instanton – A classical solution of Euclidean field theory

with finite action.

Kaluza–Klein gauge field – In a compactified theory, the

gauge field which arises from the metric of the higher-

dimensional theory.

K3 – The unique Calabi–Yau manifold in four dimensions

having an SU(2) holonomy.

Loop levels – In a Feynman diagram expansion of a field

theory, terms which contribute in higher orders of the

Planck constant h.

Macroscopic entropy – Entropy associated with gravita-

tional backgrounds via the Bekenstein–Hawking for-

mula or its generalization.

Microscopic entropy – Entropy which follows from the

degeneracy of states of a system via Boltzmann’s

relation.

Minimally coupled scalar – A scalar field whose equation

of motion is the standard Klein–Gordon equation

where the derivatives are covariant derivatives.

Neveu–Schwarz/Neveu–Schwarz states – In type I and II

string theories, bosonic closed-string states whose left-

and right-moving parts are bosonic.

No-hair theorem – A theorem in general relativity which

states that black holes with nonsingular horizons are

uniquely characterized by their mass, angular

momenta, and charges which can couple to long-

range gauge fields.

Orbifold – A coset space M=G where G is a group of

discrete symmetries of a manifold M.IfG has a fixed

point, the space is singular.

p-Form – A fully antisymmetric p-index tensor.

Ramond–Ramond states – In type I and II string theories,

bosonic closed-string states whose left- and right-

moving parts are fermionic.

Reissner–Nordstrom black hole – Black hole solution of

general relativity with electric Maxwell charge.

– n-Dimensional sphere.

Supergravity – Supersymmetric extension of general

relativity.

Supersymmetry – A symmetry between bosons and

fermions.

Threshold bound state – A bound state which is margin-

ally bound, that is, the binding energy is zero.

Tree level – In a Feynman diagram expansion of a field

theory, terms which contribute to lowest order of the

Planck constant h.

U(N) – The group of N  N unitary matrices. If the

determinant is unity, the subgroup is called SU(N).

Further Reading

Callan CG and Maldacena M (1996) D-brane approach to black

hole quantum mechanics. Nuclear Physics B 472: 591

(arXiv:hep-th/9602043).

Dabholkar A (2004) Exact counting of black hole microstates,

arXiv:hep-th/0409148.

Das SR and Mathur SD (1996) Comparing decay rates for black

holes and D-branes. Nuclear Physics B 478: 561 (arXiv:hep-

th/9606185).

Das SR and Mathur SD (2001) The quantum physics of black

holes: results from string theory. Annual Review of Nuclear

and Particle Science 50: 153 (arXiv:gr-qc/0105063).

David JR, Mandal G, and Wadia SR (2002) Microscopic

formulation of black holes in string theory. Physics Reports

369: 549 (arXiv:hep-th/0203048).

Dijkgraaf R, Moore GW, and Verlinde E (1996) Elliptic genera of

symmetric products and second quantized strings. Commu-

nications in Mathematical Physics 185: 197 (arXiv:hep-th/

9608096).

’t Hooft G (1990) The black hole interpretation of string theory.

Nuclear Physics B 335: 138.

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

Press.

Maldacena JM (1996) Black holes in string theory, arXiv:hep-th/

9607235.

Maldacena J, Strominger A, and Witten E (1997) Black hole

entropy in M-theory. Journal of High Energy Physics

9712: 002 (arXiv:hep-th/9711053).

Mathur SD (2004) Where are the states of a black hole?,

arXiv:hep-th/0401115.

Mohaupt T (2000) Black hole entropy, special geometry

and strings. Fortschritte der Physik 49: 3 (arXiv:hep-th/

0007195).

Sen A (1995) Extremal black holes and elementary string states.

Modern Physics Letters A 10: 2081.

Strominger A and Vafa C (1996) Microscopic origin of the

Bekenstein–Hawking entropy. Physics Letters B 379: 99

(arXiv:hep-th/9601029).

Susskind L (1993) Some speculations about black hole entropy in

string theory, arXiv:hep-th/9309145.

Wald R (1994) Quantum Field Theory In Curved Space-Time and

Black Hole Thermodynamics. Chicago, IL: University of

Chicago Press.

382 Branes and Black Hole Statistical Mechanics

Breaking Water Waves

A Constantin, Trinity College, Dublin,

Republic of Ireland

Introduction

Watching the sea or a lake it is often possible to

trace a wave as it propagates on the water’s surface.

One can roughly distinguish two types of breaking

waves. All waves break while reaching the shore but

certain waves break far from the shore. In the first

case, the change in water depth or the presence of an

obstacle (e.g., a rock) seems to cause wave breaking,

while for certain waves within the second category,

these factors appear not to be essential. It is a matter

of observation that for many waves that break in the

open water a drastic increase in their slope near

breaking is noticeable. This leads us to the following

mathematical definition: the wave profile gradually

steepens as it propagates until it develops a point

where the slope is vertical and the wave is said to

have broken (Whitham 1980). Throughout this

article, we are concerned with wave breaking that

is not caused by a drastic change of the topography

of the bottom; for a disc ussion of wave breaking at

the beach we refer to Johnson (1997). The governing

equations for water waves (see the next section) are

too difficult to be dealt with in their full generality.

Therefore, to gain some insight, one has to find

simpler models that are more tractable mathemati-

cally. Investigating the properties of the model,

certain predictions can be made. The conclusions

reached will reflect reality only to some limited

extent. The value of a model depends on the number

and the degree of accuracy of physically useful

deductions that can be made from it – the ‘‘truth’’ of

the model is meaningless as all experiments contain

inaccuracies and effects other than those accounted

for (while deriving the model) cannot be totally

excluded. We intend to discuss the way in which a

recent model due to Camassa and Holm (1993) can

lead to a better understanding of breaking water

waves. Firstly we survey a few classical nonlinear

partial differential equations that model the propa-

gation of water waves over a flat bed (within the

confines of the linear theory one cannot cope with

the wave breaking phenomenon) and discuss their

relevance to the study of breaking waves. We then

analyze the breaking of waves within the context of

the Camassa–Holm equation: existence of breaking

waves, criteria that guarantee that a certain initial

shape develops into a breaking wave, specific

features of wave breaking (blow-up rate and blow-

up set for certain types of breaking waves). We

conclude the presentation with a discussion of the

way in which solutions to the Camassa–Holm

equation can be continued after wave breaking.

The Governing Equations

The water waves that one typically sees propagating

on the surface of the sea or on a lake are, as a matter

of common experience, approximately two dimen-

sional. That is, the motion is identical in any direction

parallel to the crest line. To describe these waves, it

suffices to consider a cross section of the flow that is

perpendicular to the crest line. Choose Cartesian

coordinates (x, y)withthey-axis pointing vertically

upwards and the x-axis being the direction of wave

propagation, while the origin lies at the mean water

level. Let (u(t, x, y), v(t, x, y)) be the velocity field of

the flow, let y =  d be the flat bed (for some fixed

d > 0), and let y = (t, x)bethewater’sfreesurface.

Homogeneity (constant density) is a physically reason-

able assumption for gravity waves (Johnson 1997),

and it implies the equation of mass conservation

þ v

¼ 0 ½1

The inviscid setting is realistic since experimental

evidence confirms that the length scales associated

with an adjustment of the velocity distribution due to

laminar viscosity or turbulent mixing are long com-

pared to typical wavelengths. Under the assumption of

inviscid flow the equation of motion is Euler’s equation

þ uu

þ vu

¼P

þ uv

þ vv

¼P

 g

½2

where P(t, x, y) denotes the pressure and g is the

gravitational constant of acceleration. The free

surface decouples the motion of the water from

that of the air so that (Johnson 1997) the dynamic

boundary condition

P ¼ P

on y ¼ ðt; xÞ½3

must hold if we neglect surface tension, where P

the (constant) atmospheric pressure. Moreover,

since the same particles always form the free surfa ce,

we have the kinematic boundary con dition

v ¼ 

þ u

on y ¼ ðt; xÞ½4

On the flat bed we have the kinematic boundary

condition

v ¼ 0ony ¼d ½5

Breaking Water Waves 383

expressing the fact that the flow is tangent to the

horizontal bed (or, equivalently, that water cannot

penetrate the rigid bed). The governing equations

for water waves are [1]–[5]. Other than the fact that

they are highly nonlinear, a main difficulty in

analyzing the governing equations lies in the fact

that we deal with a free boundary problem: the free

surface y = (t, x) is not specified a priori. In our

discussion, we suppose that initially (at time t = 0), a

disturbance of the flat surface of still water was

created and we analyze the subseque nt motion of

the water. The balance between the restoring gravity

force and the inertia of the system governs the

evolution of the mass of water and our primary

objective is the behavior of the free surface.

An important category of flows are those of zero

vorticity, characterized by the add itional assumption

¼ v

½6

The vorticity of a flow, ! = u

 v

, measures the local

spin or rotation of a fluid element. In flows for which

[6] holds the local whirl is completely absent and for

this reason such flows are called irrotational. Relation

[6] ensures the existence of a velocity potential, namely

a function (t, x, y) defined up to a constant via



¼ u;

¼ v

Notice that [1] ensures that  is a harmonic

function, that is, (@

þ @

) = 0. In this way, the

powerful methods of complex analysis become

available for the study of irrotational flows. Thus,

while most water flows are with vorticity, the study

of irrotational flows can be defended mathemati-

cally on grounds of beauty. Concerning the physical

relevance of irrotational water flows, experimental

evidence indicates that for waves entering a region

of still water the assumption of irrotational flow is

realistic (Johnson 1997). Moreover, as a conse-

quence of Kelvin’s circulation theorem (Acheson

1990), a water flow that is irrotational initially has

to be irrotational at all later times. It is thus

reasonable to consider that water motions starting

from rest will remain irrotational at later times.

Nonlinear Model Equations

Starting from the governing equations [1]–[6] one can

derive a variety of model equations using the non-

dimensionalization and scaling approach: a suitable

set of nondimensional variables is introduced, which,

after scaling, leads to the appearance of parameters.

The sizes and relative sizes of these parameters then

govern the type of phenomenon that is of interest. An

asymptotic expansion in one or several parameters

yields an equation that is usually of significance in

some region of space/time. The aim of this process is to

obtain a simpler model that can be used to gain some

understanding and to make some predictions for

specific physical processes. This scaling method yields

the Korteweg–de Vries (KdV) equation



þ 

þ 

xxx

¼ 0; t > 0; x 2 R ½7

as a model for the unidi rectional propagation of

shallow water waves over a flat bed ( Johnson 1997).

In [7] the function (t, x) represents the height of the

water’s free surface above the flat bed. We would

like to emphasize that the ‘‘shallow water’’ regime

does not refer to water of insignificant depth – it

indicates that the typical wavelength is much larger

than the typical depth (e.g., tidal waves are

considered to be shallow water waves although

they affect the motion of the deep sea). The KdV

model admits the solitary wave solutions



ðt; xÞ¼3c sech

ﬃﬃﬃ

ðx  ctÞ



; c 2 R ½8

For any fixed c > 0, the profile 

propagates without

change of form at constant speed c on the surface on

the water, that is, it represents a traveling wave. Since

the profiles [8] of the traveling waves drop rapidly to

the undisturbed water level  = 0 ahead and behind the

crest of the wave, 

are called solitary waves. Notice

that [8] shows that taller solitary waves travel faster.

They have other special properties: an initial profile

consisting of two solitary waves, with the taller

preceding the smaller one, evolves in such a way that

the taller wave catches up the other, there is a period of

complicated nonlinear interaction but eventually both

solitary waves emerge completely unscathed! This

special type of nonlinear interaction (the superposition

principle is not valid since KdV is a nonlinear

equation) in which solitary waves regain their form

upon collision occurs only for special equations, in

which case the solitary waves are called solitons. A

further interesting property of the KdV model, relevant

for the understanding of the interaction of solitons, is

the fact that it is completely integrable (McKean

1998): there is a transformation which converts the

equation into an infinite sequence of linear ordinary

differential equations which can be trivially integrated.

Moreover, the KdV-solitons 

are stable: an initial

profile that is close to the form of a soliton will evolve

into a wave that at any later times has a form close to

that of a soliton (Benjamin 1972). Despite all these

intriguing features of the KdV-model, for all initial

profiles x 7!(0, x) within the Sobolev space H

(R)of

square-integrable functions with a square-integrable

distributional derivative, eqn [7] has a unique solution

384 Breaking Water Waves