hep-th/9904033 6 Apr 1999 Lectures given by D L ust 3 2 1 Recent Email LuestPhysikHUBerlinDe Email Ko ersPhysikHUBerlinDe Email MichaelHeraPhysikUniHalleDe ts together in Mtheory of the black holestates entropy in and the Hawking non radiation p erturbative and string nally sp ectrum a the brief application survey of of how all everything this to the computation of their BPS nature Polchinskis lutions gauge bative S and Udualitiesin their string unavoidable theory during the last years These lectures intend to give a p edagogical symmetry in Perturbative Institut f urPhysik Friedemann Bach Platz HalleSaale Germany general Developments Institut f urPhysik Invalidenstr Michael Haack at relativity xed MartinLutherUniversitatHalleWittenberg p oints y gauge Humboldt Universitatzu Berlin at of the the Saalburg observations theories Dualities They include p erturbative Tduality and non p ertur Boris Kors Tduality Abstract in demand for Dbranes an example of enhanced and that allow to view Dbranes as RR charged introduction into some of the developments Berlin Germany tendimensional String group Summer School in September y to Dieter L ust a review MTheory Theory sup ergravity of classical y a solitonic discussion From so

Contents

Introduction

Perturbative string theory

TDuality

Nonp erturbative dualities

Sduality

Uduality

Stringstringduality

Duality to elevendimensional sup ergravity and Mtheory

Tduality

Closed b osonic string theory

Compactication on a circle

D

Compactication on a torus T

Heterotic string theory

Type I IA and I IB sup erstring theory

Op en strings

Non p erturbative phenomena

Solitons in eld theory

Black holes

Magnetic monop oles

BPS states

Solitons in string theory

Extended charges as sources of tensor elds

Solutions of the sup ergravity eld equations pbranes

Dbranes as pbranes

Black holes in string theory

Mtheory

A Compactication on T and Tduality

Introduction

These notes are a summary and a substantial extension of the material that D L ustpresented

in his lectures at the summer school at Saalburg in They are intended to give a basic

overview over non p erturbative eects and duality symmetries in string theory including recent

developments After a short review of the status of p erturbative string theory as it presented

itself b efore the second string revolution in and a brief summary of the recent progress

esp ecially concerning non p erturbative asp ects the main text falls into two pieces In chapter

we will go into some details of Tduality Afterwards chapter and chapter will fo cus on

some non p erturbative phenomena The text is however not meant as an introduction to string

theory but rather relies on some basic knowledge see eg the lectures given at this school by O

Lechtenfeld or The references we give are never intended to b e exhaustive but only to

display the material that is essentially needed to justify our arguments and calculations

Perturbative string theory

Before string theory was only dened via its p erturbative expansion As the string moves

in time it sweeps out a two dimensional worldsheet which is embedded via its co ordinates in

a Minkowski target space M

X M

This worldsheet describ es after a Wick rotation in the time variable a Riemann surface

p ossibly with b oundary Propagators or general Greens functions of scattering pro cesses can

b e expanded in the dierent top ologies of Riemannian surfaces which corresp onds to an expansion

in the string coupling constant g see g The reason for this is that all string diagrams can S

+ + + ...

Figure String p erturbation expansion

b e built out of the fundamental splitting resp ectively joining vertex g This vertex comes

along with a factor g which is given by the vacuum exp ectation value VEV of a scalar eld

S

the so called dilaton

hi

g e

S

As there is no p otential for the dilaton in string p erturbation theory its VEV is an arbitrary

g s

Figure Fundamental string interaction vertex

parameter which can b e freely chosen Only if it is small the ab ove expansion in Riemann

surfaces makes sense Statements ab out the strong coupling regime on the other hand require

some knowledge ab out non p erturbative characteristics of string theory such as duality relations

combining weakly coupled string theories with strongly coupled ones First quantizing the string

amounts to quantizing the embedding co ordinates regarded as elds of a two dimensional con

formal eld theory living on the world sheet This is a two dimensional analog of p oint particle

quantum mechanics A sensible second quantized string eld theory is very dicult to achieve

and will not b e discussed here any further In the p erturbative regime there exist ve consistent

Type Gauge group of sup ercharges N

Heterotic E E

Heterotic SO

I includes op en strings SO

I IA nonchiral

I IB chiral

Table The ve consistent sup erstring theories in d

ten dimensional sup erstring theories see table Type I IA and I IB at rst sight do not contain

any op en strings In fact they do however app ear if one introduces the non p erturbative ob jects

called Dbranes which are hyperplanes on which op en strings can end Thus from the world sheet

viewp oint a Dbrane manifests itself by cutting a hole into the surface and imp osing Dirichlet

b oundary conditions These ob jects will b e studied in more detail b elow To get a string theory

in lower dimensional spacetime such as in the phenomenologically most interesting case d

one has to compactify the additional space dimensions There are several metho ds to construct

fourdimensional string theories and in fact there are many dierent ways to get rid of the extra

dimensions A priori each compactication gives rise to a dierent string vacuum with dierent

particle content gauge group and couplings This huge vacuum degeneracy in four dimensions is

known as the vacuum problem But despite of the large number of dierent known vacua it has

not yet b een p ossible to nd a compactication yielding in its low energy approximation precisely

the standard mo del of particle physics

TDuality

Tduality or target space duality denotes the equivalence of two string theories compactied

on dierent background spaces Both theories can in fact b e considered as one and the same

string theory as they contain exactly the same physics The equivalence transformation can thus

b e considered as some kind of transformation of variables in which the theory is describ ed Nev

ertheless we will always use the usual terminology sp eaking of dierent theories when we actually

mean dierent equivalent formulations of the same physical theory Tduality is a p erturbative

symmetry in the sense that the Tduality transformation maps the weak coupling region of one

theory to the weak coupling regime of another theory Thus it can b e tested in p erturbation

theory eg by comparing the p erturbative string sp ectra Examples of Tdualities are

p

0

Tdual

R

p

Het on S with radius R Het on S with radius

D

0

R

p

0

Tdual

R

p

I IA on S with radius I IB on S with radius R

D

0

R

These are sp ecial cases of the so called mirror symmetry As we will see in chapter Tduality

transformations for closed strings exchange the winding number around some circle with the

corresp onding discrete momentum quantum number Thus it is clear that this symmetry

relation has no counterpart in ordinary p oint particle eld theory as the ability of closed strings

to wind around the compactied dimension is essential

Nonp erturbative dualities

At strong coupling the higher top ologies of the expansion g b ecome large and the series

expansion do es not make sense anymore Non p erturbative eects dominantly contribute to the

scattering pro cesses Their contributions b ehave like

g g

S

S

A e or A e

The second exp onential with g g is the typical non p erturbative suppression factor in

S YM

gauge eld theoretic amplitudes involving solitons like magnetic monop oles or instanton eects

Solitons also play a role in general relativity in the form of black holes In general solitons are non

trivial solutions of the eld equations which have a nite action integral Their energy is lo calized

in space and they have prop erties similar to p oint particles Clearly it is of some interest to ask

what kind of solitonic ob jects app ear in string theory giving rise to the b ehavior of eq The

answer is that the string solitons are extended pspatialdimensional at ob jects called pbranes

ie p dimensional hypersurfaces in spacetime The sp ecial values p therefore

give p oint particles strings and membranes resp ectively Such ob jects can indeed b e found as

classical solutions of the eective low energy eld theories derived from the various sup erstring

theories see section It was however Polchinskis achievement to realize that some of them

namely those which do not arise in the universal sector have an alternative description as hy

p erplanes on which op en strings can end As these ob jects are necessarily contained in type

I IA and I IB string theory it is apparent that these theories have to contain op en strings Unlike

in type I theory the op en strings just have to start and end on the pbranes and are not allowed

to move freely in the whole of spacetime It is obvious that the b oundary conditions of the op en

strings have changed from Neumann to Dirichlet ones in the space dimensions transvers to the

branes That is why these pbranes are called Dbranes in contrast to the pbranes from the

universal sector which are sometimes also called NSbranes Note that precisely the Dbranes

are resp onsible for contributions to scattering amplitudes that are suppressed by the rst type of

suppression factor

This new insight into the nature of the non p erturbative degrees of freedom in string theory is

a fundamental ingredient of the recently conjectured non p erturbative duality symmetries Like

Tduality these dualities are supp osed to establish an equivalence of two seemingly dierent

full string theories but in their case the duality transformations map the weak coupling regions

of one theory to the strong coupling regions of the other one and vice versa Thus they eg

exchange elementary excitations and the solitonic pbranes Several dierent kinds of such non

p erturbative dualities have to b e distinguished

Sduality

By Sduality we mean a selfduality which maps the weak coupling regime of one string theory

to the strong coupling region of the same theory The existence of such a strongweak coupling

duality in string theory was rst conjectured in in the context of the compactication of the

heterotic string to four dimensions After some time accumulating evidence for the Sduality of

the heterotic string compactied on T was found found More recently it was realized

that also the type I IB sup erstring in ten dimensions is Sdual to itself and another In b oth

cases the transformation acts via an element of SL Z on a complex scalar whose imaginary

parts VEV is related to the coupling constant of the string theory see In the heterotic

theory it is given by a ie where is the dilaton anda the scalar which is equivalent

to the antisymmetric tensor B in four dimensions For the type I IB theory we have instead

a ie where now a is the second scalar present in the ten dimensional sp ectrum coming

from the RR sector Both theories are invariant under the Sduality transformation

a b

a b

with SL Z

c d

c d

combined with a transformation of either the four dimensional gauge b osons mixing F and its

dual

F F

a b

c d

F F

or the two antisymmetric tensors

B B

a b

0 0

c d

B B

in the heterotic or type I IB case resp ectively The case a a d b c shows

that the Sduality transformations comprise the inversion of the coupling constant In the rst

example the theory is additionally invariant under a Tduality group see b elow

Uduality

The Uduality group see eg for a review of a given string theory is the group which

comprises T and Sduality and embeds them into a generally larger group with new symmetry

d

generators The main example is the system type I IAI IB in d on T It will b e seen in

section that for d type I IA and type I IB have the same mo duli space and are Tdual to

each other in the sense that type I IA at large compactication radius is equivalent to type I IB at

small radius and vice versa The two tendimensional theories are dierent limits of a single space

of compactied theories which will in the following sometimes b e called the mo duli space of type

I I theory meaning all compactications of type I IA and I IB As indicated ab ove the theory is

invariant under Tduality which relates compactications of type I IA I IB to those of type I IB

I IA for a Tduality transformation in an o dd number of directions and to those of type I IA

I IB for a transformation in an even number Furthermore it inherits the Sduality of the type

I IB in ten dimensions All these transformations act however only on the scalars of the NSNS

sector ie the KaluzaKlein scalars coming from the metric and antisymmetric tensor including

the scalars coming from the ten dimensional RR scalar and RR antisymmetric tensor of I IB It

has however b een conjectured that there is a much larger symmetry group called Uduality

group which contains the S and Tduality group SL Z  SO D D Z as its subgroup but

transforms all scalars into each other including those coming from the RR sector In particular

the Uduality groups of type I I string theory on a ddimensional torus are

d

Uduality SL Z SL Z SO Z E Z E Z E Z E Z

group SL Z

where E denotes a noncompact version of the exceptional group E for n see eg

n

nn

and G for any group G means the lo op group of G ie the group of mappings from the

circle S into G

Stringstringduality

This duality some asp ects of the stringstring duality are reviewed in relates two dierent

string theories in a way that the p erturbative expansions get mixed up The p erturbative regime

of the one theory is equivalent to the non p erturbative regime of the other one The elementary

excitations on one side are mapp ed to the solitonic ob jects on the other side and vice versa

Examples are

4

The pro duct  is noncommutative as the Sduality transformation also acts non trivially on the antisym

metric tensor

Het on T I IA on K

Het with gauge group SO in d I in d

Duality to elevendimensional sup ergravity and Mtheory

Let us now consider the type I IA sup erstring At weak coupling it is the known tendimensional

theory However if we increase the coupling a new eleventh dimension op ens up Or more

precisely stated the eective Lagrangian of the tendimensional type I IA sup ergravity p erfectly

agrees with that of the elevendimensional sup ergravity compactifed on a circle of radius R if

the following identication of the type I IA string coupling and R is made

g R

S

The KaluzaKlein states of the elevendimensional theory get masses prop ortional to R and

they are mapp ed by the duality transformation to the Dbranes of the type I IA sup erstring

theory Something analogous happ ens for the heterotic string where the strong coupling limit is

dual to elevendimensional theory compactied on an interval S Z

All these dierent dualities have now led to the conjecture that all sup erstring theories are

connected to each other via duality transformations in dierent dimensions This suggests that

there is only one underlying unique fundamental theory and the dierent string vacua are just

dierent weak coupling regions in the mo duli space of this fundamental theory called Mtheory

see g From the type I IA string theory we have learned that this Mtheory is supp osed

to b e an elevendimensional theory whose low energy eective Lagrangian should coincide with

that of elvendimensional sup ergravity However a fundamental formulation of Mtheory is still

lacking We will come back to Mtheory in chapter and give more convincing arguments in favour of the ab ove claims

11-dim. SUGRA IIA

I M IIB

Het Het’

Figure Mtheory

Tduality

We are now going to have a closer lo ok at the p erturbative dualities namely Tduality We

will rst consider the most simple case the b osonic string on a circle resp ectively Ddimensional

torus and afterwards generalize to the sup erstring During the investigation of the type I string

we will see that it is unavoidable to introduce Dbranes ie hyperplanes on which the op en string

ends That is b ecause Tduality changes the b oundary conditions of op en strings from Neumann

to Dirichlet

Closed b osonic string theory

The principal eects of Tduality in closed b osonic string theory can b e studied within the context

of

Compactication on a circle

The string action of the b osonic string moving in a at background is given in the conformal

gauge

Z

S d d X X

The resulting equation of motion is simply the two dimensional wave equation

X

leading to the usual decomp osition

X X X

L R

are arbitrary functions of their arguments resp ectively and X where X

L R

just constrained to ob ey certain b oundary conditions which dep end on the background the string

is moving in Besides the tendimensional Minkowski space also a spacetime with one or several

dimensions compactied on a circle or higher dimensional torus has the prop erty of b eing

Ricci at which is required for the background of any consistent string theory We rst assume

that only one co ordinate is compact namely

X X R

and therefore have to implement in our solution the p erio dicity condition

X X mR

in the compact direction The general solution is given by

r r

X

il

X x p i e

R R R Rl

l

l

r r

X

il

p i e X x

L Ll L L

l

l

where we have used

p

R

p p

p m n

R

R

p

R

p p

n m p

L

R

p

for m n Z The canonical momentum is p p p nR The solution in has

L R

to b e supplemented with the usual solution of the wave equation in the non compact directions

x ie replace p and p in by the same continous p and b oth x and x by

L R L R

A string state in uncompactied dimensions is characterized by sp ecifying its momentum

and oscillations Analogously the states of the compactied string theory dep end on the two

quantum numbers m and n denoting the discrete momentum and winding of the string in the

th

dimension its momentum in the non compact dimensions and the oscillations internal and

external ones The winding is obviously a typical string eect with no analog in eld theory

The mass of the p erturbative states is given by M M M with

L R

p p p N M

L

L L

M p p p N

R

R R

where N and N denote b oth the internal and the external oscillations Tduality in this case

L R

refers to the symmetry of the mass sp ectrum under the Z transformation

p

R

p

R

m n

As the transformation just maps the p erturbative mass sp ectrum into and onto itself the T

duality is from the target space p oint of view a p erturbative symmetry Concerning the two

dimensional world sheet p oint of view however there is an exchange of the elementary excitations

momentum states with the solitonic ones winding states

to minus to itself and p It is obvious from and that the transformation maps p

R L

itself If the whole theory is supp osed to b e invariant under Tduality this should b e esp ecially

the case for all interactions ie the interactions of states in one theory should b e the same as

those of the dual states in the other theory Therefore the vertex op erators should also b e

which are X and exp ip X invariant They contain however phase factors like exp ip

R R L L

only invariant if we demand

X X

L L

X X ie and x x

R R Ri Ri R R

Now it is p ossible to show that this change of the signs of the rightmoving leaves all the

Ri

correlation functions invariant and therefore is a symmetry of p erturbative closed string theory

It should b e emphasized that Tduality thus is a spacetime parity op eration on the right moving

degrees of freedom only which will b ecome imp ortant in the context of type I I string theory

The mo duli space of a theory which dep ends on one or more parameters is dened as the

range of the parameters leading to distinct physics In our case the relevant parameter is the

radius of the compactication circle But whereas in eld theory every radius R R leads to

dierent physics the situation in string theory is dierent Here Tduality relates small with large

radii and there is a smallest resp biggest radius namely the xed p oint of the transformation

p

Thus the mo duli space is M fR R g resp M fR R g which ie R

x x x

can b e expressed in a more formal way as

M fR R Z g

This is a general feature of Tduality that the mo duli space in string theory can b e obtained

from the one in eld theory by mo dding out a discrete symmetry group namely the Tduality

group In string theory these parameters or mo duli describing dierent vacuum congurations

are typically given by vacuum exp ectation values of massless scalar elds In the case of circle

compactication the relevant scalar is

ji ji

R L

whose VEV corresp onds to the radius of the circle to b e more precise it really corresp onds to

the dierence of the radius and R The state ji denotes the vacuum state without winding

x

or internal momentum

D

Compactication on a torus T

We now want to generalize the results of the previous discussion to the case of higher dimensional

D

torus compactications on T for convenience we set throughout this section A torus

D D

can b e dened by identifying p oints of R which lie in a D dimensional lattice

D D D

T R

D D

where the lattice can b e sp ecied by giving D linear indep endent vectors in R namely

p

R e i D with e Ie

i i i

r

D

X

D D

m R e m Z L

i i i

i

p

D i

We also need the notion of the dual lattice given by the D vectors R e which is

i

D

dened as the lattice of vectors which have integer scalar pro ducts with all the elements of

In particular the basis vectors satisfy

D D

X X

j

a i a a j

e e e e

i b b i a

i

i a

i

The imp ortance of the dual lattice lies in the fact that the canonical momenta in the com

D D

pactied dimensions have to lie in for compactications on in order to ensure the single

i i

which is the generator of translations in the internal directions Fur valuedness of exp iX

D

thermore the metric G of is given by

ij

r r

p p

D D

X X

i j ij a a

e e G R e R e G

i j ij

a a i j

R R

i j

a a

In generalization of we now have D massless scalars

j

ij i

ji i j D j i

L

R

These elds corresp ond to the mo duli of the D dimensional torus compactication Their VEVs

can b e regarded as the internal comp onents of the metric ie and the antisymmetric tensor

eld B yielding D D resp ectively D D degrees of freedom Therefore we have to

ij

generalize to contain the antisymmetric tensor The action for the internal degrees of freedom

of a string moving in a background sp ecied by G and B then takes the form

ij ij

Z Z

i j i j

G X X B X X R d d S d d L

ij ij

i D

The elds X G and B are all given via their comp onents referring to the basis of

ij ij

namely referring to fe g One could as well take all comp onents referring to the standard basis

i

of R In this case we will take the indices to b e a b in contrast to i j From with

i j a b we can deduce the canonical momentum

Z

L

a ab a ab a

p B X X p B L d

b b b

X

a

5

Remember that the canonical momenta are in general not the same as the kinematical momenta denoted by

i

p

6

The second term did not show up in the previous section b ecause there is no antisymmetric tensor in one

(2)

dimension The third term do es not play any role for our purp oses is the dilaton and R the two dimensional

world sheet curvature scalar We shall return to this background eld action the linear mo del later on

D

As already mentioned has to b e an element of We also have

Z

a a a a

p p p d X

L R

a a a

and p p L compare eg and therefore we get

L R

ab a a a

B L L p

b

LR

D D

X X

p p

n m

k k

k a j a

e e B G

k j k j

R R

k j

k jk

where we have expressed the antisymmetric tensor via its comp onents referring to the basis of

the dual lattice ie

p p

k a j b ab

e B e B

k j

R R

k j

and we have used and to rewrite

p

D D

X X

a j a a

p

L e m R e m G

k k k k j

k

R

j

k jk

D a

are in general no elements of but they are the momenta which enter the analog The p

LR

of That is why the sp ectrum dep ends on b oth the shap e of the torus and the VEV of the

antisymmetric tensor eld

p p M N N

L R L R

It is obvious from this equation that the sp ectrum is invariant under sep erate rotations SO D

L

and SO D of the vectors p and p In order to determine the classical mo duli space it is

R L R

convenient to lo ok at the lattice which consists of all vectors p p and for which we

D D L R

choose a Lorentzian scalar pro duct ie p p p p p p p p Using and

L R L R L L R R

the antisymmetry of B it is easy to verify that we have

ij

D

X

m n n m Z p p p p

i i L R L R

i i

i

That means the inner pro duct is indep endent of the background elds G and B It can b e

ij ij

p

a a

R and B In this case calculated by taking for example G ie e

i ij ij ij

i i

it is obvious that is even and selfdual ie the length of any element is even clear from

D D

and the lattice is equal to its dual But as the scalar pro duct is indep endent of the

background elds the same holds for the selfduality and the eveness of the lattice Dierent

values of the D parameters lead to dierent such Lorentzian lattices and actually all

D D

of them can b e obtained by choosing the correct values for the background elds It is known

that it is p ossible to generate all dierent even and selfdual Lorentzian lattices via an SO D D

p

a a

R and rotation of a reference lattice eg the sp ecial one considered ab ove e

i

i i

B We have seen however in a hint that not all of them yield a dierent string theory

ij

In fact one can identify the classical mo duli space to b e

SO D D

M

class

SO D SO D

which is a D dimensional manifold

Like in the case of the circle compactication sp ecial p oints in the mo duli space are equivalent

b ecause of stringy eects while they were not equivalent classically Again the equivalent p oints

can b e mapp ed to each other via the op eration of a discrete Tduality group and it is obvious from

D D

the representation T S S that this group embraces Z coming from the Tduality

D

groups of the S factors which make up the torus Actually the whole Tduality group of T is

bigger namely

SO D D Z

Tduality

which consists of all SO D D matrices with integer entries Roughly sp eaking the Tduality

group is generated by the Tduality transformations of the dierent circles basis changes for the

dening lattice of the torus and integer shifts in B Instead of showing this in general we include

ij

an extensive treatment of the example D in the app endix We also demonstrate the feature

of gauge symmetry enhancement at xed p oints of the duality group on the mo duli space there

Heterotic string theory

The internal b osonic part of the heterotic string action on a Ddimensional torus including the

coupling to a background gauge eld V a D A and a background

aA

antisymmetric tensor is

Z

a b A C

S d d G B X X G B X X

ab ab AC AC

L L

a A

V X X

aA

L L

The background gauge eld is called a Wilson line and is a pure gauge conguration in the Cartan

subalgebra of the gauge group which yields however a non trivial parallel transp ort around non

trivial paths in space time The p otential for the gauge b osons not in the Cartan subalgebra

has for torus compactications no at directions so that it is not p ossible for them to obtain a

VEV If all the Wilson line mo duli are zero the gauge group is unbroken ie E E or SO

but for non zero values of the background gauge eld only the subgroup commuting with the

Wilson line remains a gauge symmetry ie it is broken to U for generic values Thus the

Wilson line parameters are further D mo duli sp ecifying the vacuum conguration which is

now characterized by D D background elds

It is obvious from that only the left moving part of the internal action has changed

compared to the b osonic case That is b ecause the heterotic string is a hybrid construction of a

left moving b osonic and a right moving fermionic string which are compactied on two dierent

tori The compactication torus for the left moving sector is the pro duct of that one for the

right moving sector times the one dened through the dual of the ro ot lattice of E E or

SO Thus it is clear that the right momenta p from do not change but the left ones

R

do Nevertheless it is again p ossible to show that the resulting vectors p p form an even self

L R

dual Lorentzian lattice with signature D D As b efore all of those can b e generated via

SO D D transformations of a reference lattice and again the sp ectrum is invariant under

individual SO D SO D rotations of the left and right momenta Thus the classical

mo duli space is

SO D D

M

class

SO D SO D

which is a D D dimensional manifold as one would have guessed Like in the b osonic case

the Tduality group is the maximal discrete subgroup of the numerator of the classical mo duli

space namely

SO D D Z

Tduality

7

The expression Wilson line refers to b oth the gauge conguration and the trace of the pathordered pro duct

H

P exp of the line integral of the vector p otential along a closed line tr dX A

and the quantum mo duli space is the coset space obtained from the classical one by mo dding out

this Tduality group

Type I IA and I IB sup erstring theory

The world sheet action in a at space time background for type I IA and I IB is given by

Z

S d d X X

R L

R L

The world sheet fermions and where we have introduced the notation

R

are right resp ectively left moving Both of them can either ob ey p erio dic Ramond R or

L

antiperio dic NeveuSchwarz NS b oundary conditions The zero mo des of in the R sector

LR

lead to a vacuum degeneracy such that the ground states transform according to one of the

irreducible spinor representations of SO or distinguished by their chirality The lowest

s c

lying state in the NS sector is a tachyon and the massless ground state transforms as a vector

under SO In b oth the left and right moving sp ectrum the GSO pro jection keeps only

v

one of the irreducible spinor representations in the massless R sector and skips the tachyon in

the NS sector As the sp ectrum of the closed string is derived by tensoring the left and the

right sp ectra and as the two choices or for each individual sector are physically equivalent

s c

diering only by a space time parity transformation there are two distinguished string theories

dep ending on whether the chirality of the fermions are the same in b oth sectors or not In the

rst case we end up with the chiral type I IB theory whose massless particle content is equal

to the one of N type I IB sup ergravity in ten dimensions namely

v s v s

The other alternative leads to the nonchiral type I IA theory with the same massless sp ectrum as

N type I IA sup ergravity in ten dimensions Space time fermions are

v s v c

made by tensoring states from the left moving R sector with those of the right moving NS sector

or vice versa leading for type I IA to two gravitinos and dilatinos of opp osite chirality and for

type I IB to two gravitinos and dilatinos of the same chirality Bosons are obtained in the NSNS

I IA

v c s v c c s s

I IB

v s s v s s s s

Table Fermionic massless sp ectrum

and the RR sector The NSNS sector is universal and leads for b oth theories to the same states

where the o ccuring states are in turn the dilaton the antisymmetric tensor

v v

and the graviton common to all string theories The RR sector is dierent for b oth theories It

is shown in table the sup erscript at the form in the I IB sp ectrum reminds at the fact

that its eld strength is self dual

I IA

s c v

A A

M

MNP

I IB

s s

0

A A

MN MNPQ

Table Massless RR states

Let us now have a lo ok at the new features coming up in the context of Tduality in type II

sup erstrings Consider rst the case of one compact dimension We have seen in the b osonic

case that Tduality reverses the sign of the compactied co ordinate in the right moving sector

This remains true in the type I I string theory Conformal invariance requires then also as

X should b e invariant the sup ercurrent

R

R

R R

This however switches the chirality on the right moving side ie which had to b e ex

s c

p ected as we have already seen in the b osonic case that Tduality is a spacetime parity op eration

on just one side of the worldsheet Thus the relative chirality b etween the left and right moving

sectors is changed and a Tduality transformation in one direction switches b etween type I IA

and I IB

p

0

R

p

I IB on S I IA on S

0

R

This remains true in higher dimensional torus compactications if we Tdualize in an o dd number

of directions On the other hand if we Tdualize in an even number of directions the relative

chirality of the left and the right moving sectors is not changed and in this case Tduality is a

selfduality compare the discussion in section

Op en strings

While type I string theory contains op en sup erstrings most of the generic features we shall b e

interested in already app ear with purely b osonic op en strings for which we choose a parametriza

tion For them there are two kinds of b oundary conditions p ossible The Poincare

invariant Neumann b oundary conditions mean that there is no momentum owing o the edges

of the string

X j

The ends of the string can however move freely in space time Dirichlet b oundary conditions on

the other hand break dimensional Poincareinvariance They are given by

X j ie X j c

Now the ends of the op en string are xed at p osition c That b oth endp oints are xed at the

same p osition b ecomes clear in but in fact the endp oints of all op en strings not charged

under a non ab elian gauge group with Dirichlet b oundary conditions in direction are xed to

the same value as can b e deduced by considering a graviton exchange of two op en strings

It is of course p ossible to choose dierent b oundary conditions in dierent directions In case

the op en string ob eys Neumann b oundary conditions in the p directions X p

i

and Dirichlet ones in the remaining X i p its end p oints can just move within the

pdimensional plane in space transversal to the directions in which Dirichlet b oundary conditions

are valid see g This plane is called Dpbrane Introduced in this manner Dbranes are just

rigid ob jects in space time dened via the b oundary conditions of op en strings It will b e seen

in the next chapter that they are in fact dynamical degrees of freedom with a tension T g

p S

Thus they only seem to b e rigid at weak coupling but b ecome dynamical in the strong coupling

regime hinting at the fact that they have to b e identied with the semi solitons already an

nounced in the introduction section

Let us now investigate the role of Dbranes for Tduality of op en strings Supp ose we start

with op en strings ob eying Neumann b oundary conditions Now we compactify one co ordinate

on a circle of radius R say X but keeping Neumann b oundary conditions The center of mass

momentum in this direction takes only the discrete values p nR like for the closed strings

In contrast to the closed string case however there is no analog of a winding state for the op en

string as its winding is top ologically always trivial The solutions of the equations of motion for X1 ,.., Xp-1

Xp

Xp+1 ,..., X25

Figure Op en strings moving on a Dpbrane

the compactied left and right moving co ordinates are

r

X

c x

il

p i e X

l R

l

l

r

X

x c

il

X p i e

L l

l

l

n

We see that the compactied co ordinate moves with momentum p on S

R

R

X X X x p osc

L R

As the radius is taken to zero only the n mo de survives and the op en string seems to move

only in spacetime dimensions but nevertheless still vibrates in or rather in the trans

verse ones This is similar to an op en string whose endp oints are xed at a hyperplane with

dimensions

This fact can b e b etter understo o d if one p erforms a Tduality transformation in the X

direction and introduces the Tdual co ordinate X X X This choice

L R

is motivated by the fact that Tduality is a one sided spacetime parity transformation on the

rightmoving co ordinate see eq and the comments in the corresp onding paragraph This

co ordinate on the Tdualized circle S with R R now takes the form

D

R

D

X X X c p osc

L R

q

0

25 25 8 25

p the additional factors of compared to stem p It is clear from with m that p

L R

2

from the dierent parametrization of the op en string world sheet

n

c osc

R

c nR osc

D

Thus the b oundary conditions have changed for the dual co ordinate from Neumann to Dirich

let ones ie the end p oints of the string are xed to the values X c resp ectively

X c mo d R Another way of saying this is that the op en string end p oints

D

are constrained to move within Dbranes which are sep erated from each other by a multiple

of R and are therefore identied as we have compactied the dual co ordinate on a circle of

D

radius R see g Like for the closed string Tduality exchanges winding and momentum D

π π ^

0 2 RD 4 RD X25

Figure c mo d R

D

quantum numbers for the op en string Before Tdualizing the winding number has b een zero

After the transformation the center of mass momentum is zero as can b e seen from ie not

only the end p oints but also the center of mass of the op en string is constrained to move in a

spatial dimensional hyperplane On the other hand as the string end p oints are now xed

in the compactied dimension it makes sense to talk ab out winding states Actually these are

states for which n is nonzero in b ecause X and X nR are identied for any n Z

D

Let us summarize this p oint Before Tdualizing the op en string ends move within a Dbrane

wrapp ed around the compact dimension which is an elegant way to say that they can move freely

in space time In the dual description however the Dbrane has changed to a Dbrane This

i

is a general feature Tdualizing in a certain compact direction X turns a Dpbrane which is

i

wrapp ed around this circle ie the op en strings ob ey Neumann b oundary conditions in the X

i

direction into a Dp brane The inverse is also true If the Dpbrane is xed in the X

i

direction b efore Tdualizing ie the op en strings ob ey Dirichlet b oundary conditions along X

th

dimension it is turned into a Dp brane which is wrapp ed around the compact i

This fact is also crucial for the incorp oration of Dbranes and op en strings in type I I string

theory We will see in section that Dpbranes with p even only app ear in the type I IA theory

and those with p o dd exist in type I IB see eg table As was explained ab ove Tduality in an

o dd number of directions switches from I IA to I IB theory and thus it is necessary for consistency

that Tduality in one direction changes the value of p by plus or minus one

Next we consider the massless sp ectrum of the b osonic op en string In the dual picture these

massless states are given by states without winding This can b e seen from the mass formula

n

M p N R N

D

9

This is true for generic values of R but as in the closed string case extra massless states app ear for the self

D

p

0

dual radius R R

D

Thus the massless states are given by N n

ji ji

where the rst state is a U gauge b oson and the second one a scalar whose VEV describ es the

X p osition of the Dbrane in the dual space This easily generalizes to the case of a Dpbrane

i

Then ji gives again a massless U gauge b oson and the VEVs of ji are the p

co ordinates of the Dpbrane in the space directions transversal to it As the op en string has no

momentum in these directions the same holds for the U gauge b osons Thus it has b ecome

conventional to sp eak of the gauge theory living on the world volume of the Dpbrane which

amounts to a dimensionally reduced gauge theory

So far we have only considered op en strings charged under a U gauge group We now want

to generalize the results to non ab elian gauge groups To do so let us consider orientable op en

strings whose end p oints carry charges under a non ab elian gauge group Consistency require

ments restrict the choice to U n for orientable strings and SO n resp ectively U S pn for non

orientable ones To b e more precise one end transforms under the fundamental representation

n of U n and the other one under its complex conjugate n The ground state wavefunction is

thus not only sp ecied by the center of mass momentum of the string but additionally by the

charges of the end p oints giving rise to a basis jk ij i Chan Paton basis Generally the op en

string states can b e characterized by their charges with resp ect to the generators of the Cartan

n

subalgebra U of U n which can b e taken as the n dierent n n matrices with just one

entry on the diagonal The states jk ij i of the Chan Paton basis are now those states which

th th

carry charge resp ectively under the i resp ectively j U generator Obviously the

whole op en string transforms as a bifundamental under the tensor pro duct n n which is just

the adjoint of U n From this p oint of view it seems more appropriate to take as basis for the

P

n

a a

ground state wavefunction the combinations jk ai jk ij i The factors are the

ij ij

ij

matrix elements of the U n generators and are called Chan Paton factors The jk ai transform

under the adjoint of U n if the i index transforms with n and the j index with n The usual

vector at the massless level jk ai is now a U n gauge b oson

The new feature in toroidal compactication of the op en string with gauge group U n is the

th

p ossible app earance of Wilson lines see fo otnote If we compactify the dimension on a

circle with radius R a p ossible background eld with non trivial line integral along this circle is

iX R iX R

n

A diag i diag e e

n

R

If or all equal to another constant value for i m and and all pairwise

i j

nm

dierent for j m n the gauge group is broken to U n U m U compare

our discussion of the Wilson lines of the heterotic string in section Thus the play the role

i

of Higgs elds Another imp ortant characteristic of the Wilson line is that it changes the value

of momentum p where the change dep ends on the Chan Paton quantum numbers of the state

Therefore we use the Chan Paton basis in the following In particular we have for a ground state

jl R ij i

l

j i

p

ij

R R

To see this consider rst the case of a U gauge theory and a p oint particle with charge q

Under a gauge transformation with the background gauge eld vanishes and one knows from

ordinary quantum mechanics that the wavefunction of the particle picks up a phase

Z

exp iq dx A

x

where x is a reference p oint Thus it is no longer p erio dic under X X R but gets

a phase expiq As the wavefunction in the momentum representation is a plane wave this

non p erio dicity is equivalent to a shift in the canonical momentum

l q

p

R R

Let us now turn to string theory The background gauge eld is an element of the Cartan

n

subalgebra ie of the subgroup U of U n The string states with Chan Paton quantum

th th

numbers jij i have charges under the i j U factor and are neutral with resp ect

to the others Thus follows immediately from If we insert into we get

l

j i

M N

R

from which we see that for generic values the only massless states are the diagonal ones i j

n

giving a gauge group U for l and N If some of the s are equal we get additional

massless states enhancing the gauge symmetry and conrming our discussion ab ove

Now we want to interpret the situation in the dual picture From and we see that

the dual co ordinate for a string with Chan Paton lab els jij i is given by

j i

R osc X c l

D

ij

If we set c R we get

i D

X R

i D

ij

X l R R

D j D

ij

Thus we have n dierent Dbranes whose p ositions mo dulo R are given by R see g

D j D

From we see that generically only op en strings with b oth end p oints on one and the

n

same Dbrane yield massless gauge b osons gauge group U and strings which are stretched

b etween dierent Dbranes give massive states with masses M R Obviously the

i j D

masses decrease with smaller distances and vanish if the two Dbranes take up the same p osition

In this picture the coincidence of Dbrane p ositions leads to the encountered gauge symmetry

enhancement In particular if all Dbranes are stuck on top of each other the gauge group is

U n

A feature of Dbranes which we just mention for later use but without pro of is that they break

part of the sup ersymmetry see eg In the sp ecial situation ab ove all the n Dbranes

were parallel to each other In this case half of the sup ersymmetries are broken leading to N

in D for type I I string theory To break even more sup ersymmetries one needs more general

congurations of intersecting Dbranes Two orthogonal Dbranes for example break altogether

of sup ersymmetry giving N in D if the number d of dimensions in which just

one of the two branes is extended but not b oth is or examples are a Dbrane together

with a Dbrane or two completely transverse Dbranes If d is however or it is always

even all of the sup ersymmetry is broken In situations with rotated Dbranes it is also p ossible

to get N sup ersymmetry in D ie of the SUSY generators are preserved For some

further details the reader is referred to

Non p erturbative phenomena

In the previous chapters interactions in string p erturbation theory were dened by a summation

over all conformally inequivalent world sheets each weighted with a factor of the string coupling

constant according to the top ology of the resp ective Riemann surface In this chapter we are going

to lo ok for explicit non p erturbative states in string theory to get a more complete description of θ θ θ π

0 1 RD 2RDD... n R 2 RD

Figure Dbrane conguration in the presence of a Wilson line

the sp ectrum and p ossible interactions One strategy to nd states of non p erturbative nature is

to lo ok for non trivial solutions of the classical equations of motion of the low energy approxima

tion to string theory We will nd these equations to have brane solutions which dep end only on

a subset of the co ordinates of spacetime such that the sources of the elds are multidimensional

membranes They are solitonic in the sense that their masses are prop ortional to inverse p owers

of the string coupling M g while the states of the p erturbative sp ectrum have masses

sol S

They are then called pbranes if they prop ortional to the string coupling constant M g

p er

S

extend into p spatial dimensions thus have p dimensional worldvolume Further we shall

demonstrate that also the Dbranes discussed earlier as hyperplanes in spacetime on which op en

strings might end with Dirichlet b oundary conditions display characteristic features of solitonic

states By an indirect way of reasoning we shall argue that they are in fact dynamical ob jects

ie they interact with op en strings thereby couple to gravity and gauge elds and uctuate in

shap e and p osition Their p erturbative degrees of freedom are op en strings ending on the brane

which describ e the uctuations of the Dbrane by their p erturbative sp ectrum Also we shall nd

their mass sp ectrum to b e of intermediate range M g which indicates that they are in

S

D

b etween elementary p erturbative states and solitons and therefore caused them b eing adressed as

halfsolitonic The three types of states p erturbative solitonic and halfsolitonic are mixed by

various conjectured S T and U dualities from the previous chapters They leave the sp ectrum

invariant but transform the mo duli thereby in some cases interchanging the p erturbative and

non p erturbative regimes of the theory

Solitons in eld theory

To motivate later discussions and interpretations we rst give an introduction into eld theoret

ical solitons in particular the black holes of Einstein gravity and the tHo oftPolyakov monop ole

of non ab elian gauge theories Such solitons are dened to b e non trivial solutions to the eld

equations with nite action Regarding the obvious symmetries of their sp ectrum we then p oint

out the idea of Sduality and how it is supp osed to b e realized in sup ersymmetric gauge theories

The bridge to the quantum theory will b e built by realizing the BPS nature of some of the classi

cal solutions after embedding these into sup ersymmetric theories which is assumed to guarantee

their existence in the sp ectrum of the quantum theory to o

Black holes

We start with discussing classical solutions to the Einstein equations that are not of p erturbative

nature as they involve large deviations from the Minkowski at spacetime The classical theory

of general relativity originates from the vacuum Einstein action

Z

p

g R S d x

E

of pure gravity where g is the determinant of the metric g and R the curvature scalar By the

usual metho ds one then nds the Einstein equation the equation of motion of the metric eld

without matter which in d simply demands the spacetime to b e Ricci at

R

Perturbative solutions to these equations are for instance given by gravitational waves freely

propagating small deviations from the at metric The most prominent among the nonp ertur

bative solutions are black holes and the prototype of such is the Schwarzschild metric

m m

dt d sin d dr r ds

S

r r

It is the unique stationary solution of the vacuum Einstein equations outside a star that has

collapsed to a pressureless uid of matter The parameter m is related to the fourdimensional

Newton constant G by m G mc m b eing the mass of the black hole which is derived by

N N

comparing the asymptotic large r b ehaviour of the g comp onent to the classical nonrelativistic

Newton law The Schwarzschild metric suers from two obvious divergencies First the event

horizon at r m which is only a co ordinate divergence and involves the change of the metric

signature in a way the interchange of radial and time directions while the

curvature and even R R stay nite When observed from the radial innity where the

metric tends to at Minkowski spacetime no geo desic can reach the horizon at nite values of

the ane parameter which can in case of timelike geo desics b e taken to b e its prop er time There

is also the physically more dramatic divergence at r where the square of the curvature tensor

R R diverges as an inverse p ower of the radial co ordinate Such divergencies cannot b e

removed by conformal transformations they imply that geo desics particle tra jectories cannot

b e completed into these p oints which is said to b e a generic feature of any gravitational collapse

Concerning the global causal structure the horizon signals the notion that all timelike or null

geo desics from inside r m are leading into the spacelike singularity therefore no matter can

escap e its destiny of falling into the singularity at the core which o ccurs even in nite prop er

time Generalizations of the Schwarzschild solution are given by the ReissnerNordstromsolution

to the vacuum action and F for charged black holes obtained by adding a Maxwell term

the Kerr solution that incorp orates also rotating black holes and is the most general form of any

stationary solution to the Einstein equations of the vacuum plus only electromagnetism These

metrics are related to more complicated global structures of space time and display dierent types

of curvature and co ordinate singularities As we shall later on refer to the charged black holes we

here give the solutions for the metric and the electromagnetic p otential of the ReissnerNordstrom

case

m q q m

dr r d sin d dt ds

RN

r r r r

q

A

r

Obviously q can b e interpreted as the total electric charge of the black hole Further analysis

shows that only in the case m q the curvature singularity at the radial origin is screened by

a horizon If m q the metric actually has two horizons

p

r m m q

which can b oth b e removed by analytic extension Dierent from the Schwarzschild case the

singularity itself is timelike and the causal structure more involved The case of an extremal

charged black hole is reached if m q which demands the horizons to coincide and the metric

simplies to

m m

dt dr r d sin d ds

ex

r r

An imp ortant prop erty of this situation is its vanishing surface gravity

r r

r

which is a measure of the lo cal accelaration that app ears to aect a particle near the horizon

as it is b eing watched from innity Note that the causal structure of spacetime of this solution

still is diering from the Schwarzschild solution though the metric might lo ok somewhat simi

lar Besides the dierences mentioned all black hole solutions have got the feature in common

that as long as the p ositive energy hypothesis holds any curvature singularity is shielded by a

horizon which viewed from spatial innity expressed in asymtotically at co ordinates cannot

b e reached by any particle any space or timelike geo desic in nite time

We shall now introduce concepts of particle physics and thermo dynamics into this classical

picture to explain the related problems of the loss of information and Hawking radiation As there

is no complete quantum theory available that allows a consistent treatment of the gravitational

interactions together with the strong and electroweak interactions we have to rely on semiclassical

metho ds and sometimes heuristic arguments Let us rst lo ok for the state equation of black hole

thermo dynamics Even the most general black hole metric the Kerr black hole do es not dep end

on the time and azimutal angle co ordinate such that these dene vector elds along which the

action is constant ie Killing vector elds

t

While in a curved background one cannot simply integrate the eld equations over some spacelike

region to get conserved quantities as usually done in eld theory on Minkowski at spacetime

the pro jections of the energy momentum tensor onto such Killing vector elds are conserved

Thus to any Killing vector eld one can asso ciate a conserved charge by integrating its covariant

derivative over the b oundary of some spacelike region V

I Z Z

d D d D D d R Q V

G G G

V V V

In the presence of a black hole the volume integral is conveniently split o into a surface integral

over the horizon H and a volume integral over the space time ST outside

Z I

d D D d D Q V

G G

ST H

The two Killing vector elds and in particular pro duce conserved charges that contain b esides

other terms the total energy mass m and the angular momentum J of the Kerr black hole

Using the explicit expressions one can derive the mass formula of Smarr

m A J q

H H

where A is the area of the horizon the constant angular velocity at the horizon J the

H

total angular momentum the electric p otential at the horizon We notice that the extremal

H

ReissnerNordstrom solution with vanishing and has mass equal to its electric p otential

H

energy m q This will b e identied to b e the BPS mass b ound later on By analyzing

H

the dimensionality of the various quantities cf Euler theorem on homogeneous functions one

further nds the dierential state equation to b e

dA dJ dq dm

H H

which is the so called rst law of black hole mechanics Interpreting this as a thermo dynamical

relation suggests to dene the black hole Hawking temp erature T and its Bekenstein

H

Hawking entropy S A It is in fact clear that the black hole must carry entropy anyway

BH

as throwing matter into it would otherwise destroy such On the other hand no such pro cess

could b e watched from the asymptotically at region The Hawking temp erature we have recov

ered here is also known from other arguments which explains the seemingly arbitrary choice of

numerical co ecients The second law of thermo dynamics now translates into the statement that

the horizon of an isolated black hole has nondecreasing area One should susp ect that this black

hole entropy can b e computed by a state counting pro cedure as usual in quantum statistics This

task has never b een solved in the context of QM or QFT as the quantum degrees of freedom of

the black hole are still unknown We shall later address this question from the p oint of view of

string theory

We now briey turn to Hawking radiation which refers to the fact that black holes

can radiate particles exhibiting a thermal sp ectrum The formalism of quantum eld theory on

curved spacetime allows under certain circumstances as are met when black holes are formed

non unitary changes of the Fock space basis From this it follows that the vacuum b efore the

collapse may lo ok like a many particle state afterwards Along these lines one can deduce the

claimed statement that black holes in fact radiate In a more heuristic manner it can b e gured

that after paircreation of a particle and an antiparticle by vacuum uctuations for instance of

the photon eld near the horizon one of the two particles is drawn inside the horizon while the

other one app ears to b e radiated away from the black hole By a semiclassical analysis of a eld

propagating along a tra jectory close to the future event horizon Hawking could demonstrate that

the sp ectrum of this radiation is approximately that of a black b o dy thermal radiation at the

Hawking temp erature

n

e

The black hole obviously can thermalize its entropy and energy by this mechanism A very sp ecial

case is the extremal ReissnerNordstrom black hole with so that no Hawking radiation

is emitted and it app ears to b e stable Although this analysis do es not take into account the

eect of decreasing the mass of the black hole in the pro cess back reaction it is convenient to

cite the riddle of loss of information when throwing matter into the hole that later on radiates

away thermalized The formerly pure quantum state of such matter has lost its coherence

A prop er unitary description of the whole phenomenon is exp ected to clarify the question but it

still remains to b e found Apparently it should have to include a consistent quantum treatment

of the gravitational eld an outstanding challenge so far Later on we shall reveal some of the

attempts that have b een undertaken in string theory recently

Magnetic monop oles

The second type of solitonic ob jects we shall discuss are the magnetic monop oles of gauge the

ories We will pro ceed very much like in the previous section by lo oking for classical solutions

to the eld equations which have nite energy densities and thus have p ossibly relevant con

tributions to the quantum theoretical path integral The magnitude of these contributions will

dep end on the coupling and is in general non p erturbative While we could not give a rigorous

answer to the question if black holes are stable states of the unknown quantum theory this

issue can b e addressed in gauge theory at least in some sup ersymmetric extension and leads to

the notion of so called BPS saturated states We now start with the most simple setting classical

electro dynamics and its Maxwell equations

The idea of magnetic monop oles traces back to Dirac who gured out what happ ened when

symmetrizing the Maxwell equations

E j rE r B

e e

t

B rB r E

t

by generalizing the electromagnetic duality

E B B E

of the vacuum equations to a symmetry of the full classical electromagnetic theory This is

p ossible by introducing magnetic currents j into the equations The corresp onding particles are

m

called magnetic monop oles Such a single stationary and p ointlike monop ole of magnetic charge

g at the origin generates a singular magnetic eld

r

B g

r

which itself is created by a vector p otential that cannot b e dened globally In fact it is singular

at least on a half line the so called Dirac string More mathematically sp eaking the Bianchi

identity of the eld strength tensor is no longer satised and thus it cannot b e exact globally

anymore The presence of such a Dirac string although classically meaningless can b e prob ed by

a BohmAharonov exp eriment Moving an electron around this string and demanding its wave

function to b e single valued imp oses the famous Dirac quantization condition on the pro duct of

the two charges

g e n

where n is some arbitrary integer It oers an explanation for charge quantization even if only

by p ostulating ob jects yet unobserved Another way of deriving it is nding that the vector

p otentials dened on dierent parts of a sphere around the magnetic charge can only b e matched

together on the whole sphere up to gauge transformations

A A

in the overlapping regions where is only dened mo dulo g while the transition function

exp ie should b e dened uniquely For n the Dirac condition tells us that one of the two

charges is obviously larger than and therefore naiv p erturbation theory as an expansion in this

parameter imp ossible The apparent symmetries of this generalized setting of electromagnetism

allow an electromagnetic duality interchanging the electric and magnetic elds like in and

simultaneously the two currents j j and thereby linking the p erturbative and non p ertur

e m

bative regimes This might b e thought of as the most simple case of Sduality

We shall pursue this idea further by insp ecting classical YangMills gauge theory which will

then b e embedded in its sup ersymmetric extension The challenge to nd an exact solution to a

phenomenologically interesting classical eld theory that exhibits prop erties of a particle ie has

lo cal supp ort in space and is constant in time was solved by the tHo oftPolyakov solution

of the SU YangMillsHiggs or GeorgiGlashow mo del Such solutions are called solitons and

they carry prop erties of monop oles The basic idea is to embed the electromagnetic U gauge

group in a larger simple group which is sp ontaneously broken to single out the electromagnetic

U as unbroken symmetry but now with charge quantization necessarily implicit The mo del

is constructed from the Lagrangian

a a

tr F D D V L

a a

where the gauge eld A and the scalar Higgs eld take their values in the adjoint repre

sentation of the SU gauge group The p otential of the Higgs eld is assumed to create a

nonvanishing exp ectation value alike the double well p otential The derivation of the equations

of motion and the energy momentum tensor is straightforward the former are

a abc b c

D F e D

V

a

D D

a

and the latter comes out in the symmetrized version

a a a a

F F D D L

From its time comp onent one can read o the energy density and realizes that the classical

vacuum has a constant Higgs eld with vacuum exp ectation value given by the minimum of the

p otential Cho osing

a a

V v

a a

this is h i v The constant Higgs eld leaves only a smaller gauge group U of rotations

around the direction of its exp ectation value intact and the manifold of degenerate vacua the

coset space is top ologically equivalent to a sphere

M SU U S

vac

Any solution to the eld equations that wants to have nite energy must lo cally lo ok like a vacuum

eld conguration at spatial innity Therefore at large r any Higgs eld has to obtain the vacuum

a a

value v This relation denes the sphere M in the isospin space whose p oints are

vac

connected by gauge transformations of the coset group rotations in that space Thereby the

asymptotic value of the Higgs eld induces a mapping of the spatial innity S onto the set

of degenerate vacua S which can b e characterized by its integer winding number n and is an

element of the second homotopy class of S

S Z

The constant Higgs eld b elongs to n but we shall also nd solutions to the eld equations

with nite energy that have nonvanishing winding number We construct the concrete form

of such a vector p otential leading to a monop ole solution by demanding the energy density to

b e decreasing faster than r when r to obtain a nite total energy Consequently the

covariant derivative of the Higgs eld has to decrease faster than r and from

a abc b c

e A faster than r

one can nd the most general expression for the gauge p otential

abc b c a a

A A

ev v

with some arbitrary smo oth A The corresp onding eld strength p oints into the direction of the

Higgs eld

a

a def d e f

F A A

v ev

Integrating the eld equations over the transverse space we then nd the magnetic charge of the

monop ole b eing prop ortional to the winding number n of the Higgs eld around the sphere at

radial innity

Z Z

n

ij k ij k abc a b c

g F dx dx

j k i j k i

ev e

S S

which explains its b eing called a top ological charge in contrast to the electric No ether charge

The Dirac quantization condition is thus repro duced up to a factor which is due to the fact

that the fermions we could p ossibly add into the mo del were half integer charged as they were to

lie in the fundamental representation of the SU That two seemingly very dierent arguments

repro duced the same quantization condition can b e traced back to the top ological statement that

SU U U

The lefthandside of the equation gives the range of winding numbers for the Higgs whereas the

righthandside is the p ossible number of windings of the electromagnetic gauge eld around the

Dirac string In the latter case we had to plug in the source term by hand to sp oil the Bianchi

identity in the former case this is achieved by the winding of the Higgs eld automatically

The general solution for the electromagnetic p otential allows a very sp ecial and symmetric

choice On the one hand the eld congurations can neither b e gauge invariant under the

full SU nor b e invariant under the SO of spatial rotations b ecause of the sp ontaneous

symmetry breaking and the non trivial b ehaviour of the Higgs eld at radial innity On the

other hand any spatial rotation can b e accompanied by an appropriate gauge transformation to

cancel the variation of the elds as the angular dep endence of the Higgs eld at radial innity is

lo cally pure gauge Such we can have solutions which are invariant under simultaneous rotations

and gauge transformations The arbitrariness that remains can b e summarized in the according

ansatz

a

r

a

H v er

er

j

r

a a

K v er A

i ij

er

a

A

a i

where r is the radial unit vector in the isospin space andr in the Minkowski space To render

charge and mass nite one has to imp ose b oundary conditions on H r and K r that allow

nite results for the integrated energymomentum tensor as well as for the integration of the eld

equation of the electromagnetic eld Dening mass and charge by such integrals over spatial

regions in the usual manner one can derive the BPS BogomolnyPrasadSommereld b ound on

the lowest p ossible mass of a monop ole

M v g

m

with equality if and only if V Note as an aside that extremal ReissnerNordstrom

black holes are indeed BPS saturated We shall later see that from the p oint of view of the

sup ersymmetric extension of this mo del the states that satisfy the equality are very sp ecial In

this case the equations of motion can b e translated into the dierential Bogomolny equations

dK x dH x

x K xH x x H x K x

dx dx

which are solved according to the relevant b oundary conditions by

x

H x x cothx K x

sinhx

Plotting the two functions one immediately realizes how they generalize the well known onedi

mensional top ologically charged kink solution Substituting back one nds that these mono

p oles satisfy the BPS mass b ound with equality holding as well as the Dirac charge quantization

condition with minimal winding number top ological charge eg so that we are tempted to

think of them as b eing the elementary magnetic charges stable p er denition if charge conser

vation holds It was also noticed that the same YangMillsHiggs mo del can have solutions that

carry b oth electric and magnetic charges If one also allows for the top ological term

e

L L F F

to b e present where the star marks the Ho dge dual their magnetic charge can b e related to the

angle the imaginary part of the gauge coupling This term is prop ortional to the instanton

number it violates parity and lo cally it can b e written as a total derivative Globally it might

lead to an additional violation of the Bianchi identity and an extra b oundary term in the eld

equation mo difying the electric charge of the state Together we have then got two top ological

charges the winding n of the gauge eld and the winding n of the Higgs eld enough to

e m

have dyonic states carrying b oth types of charges The Dirac condition and the BPS b ound are

mo died to b e

n q

m

n

e

e

p

M v q g

dyon

where n is related to the magnetic charge g of the dyon by the former Dirac condition while

m

q is the electric charge of the dyon now no longer b eing quantized in integer units of e The

ab ove discussion was a little sketchy and the situation app ears to have b ecome more complicated

than b efore introducing dyons We shall return to it from another p oint of view after lo oking at

sup ersymmetric extensions of the mo del But now already we can address the issue of duality

that had b een conjectured by Montonen and Olive They observed that in the mo del we

considered not only the dyonic charges saturate the BPS b ound but also the p erturbative degrees

of freedom ie the massive gauge b osons the remaining massless photon and also the Higgs

eld Also they found that monop oles do not excert any force onto each other by a cancellation

of attraction and repulsion mediated by the Higgs and gauge b oson exchange resp ectively This

lead to the assumption that there might b e some kind of mapping from the p erturbatively light

elds onto the non p erturbative dyon sp ectrum acting in some unknown way on the mo duli space

spanned by the vacuum exp ectation value of the Higgs and the couplings which could b e gured

to b e a symmetry of the theory a duality Montonen and Olive conjectured that there could exist

an entire charge lattice of states which were to b e p ermuted by duality transformations For the

sake of charge conservation these have to map the lattice onto itself and they therefore found

the most natural form of the duality to b e an SL Z group acting on the p oint lattice in the

complex plane Thus the duality has b een called S duality as well as its generalization in string

theory already mentioned in previous chapters A couple of unanswered questions remained

among them the problems that quantum corrections should b e exp ected to mo dify the classical

p otential and sp ectrum considerably and further that it was not p ossible to match the dyons

and the p erturbative elds together into multiplets of the Lorentz group The most imp ortant

progress in these issues was achieved in sup ersymmetric mo dels

BPS states

The success in identifying dualities in sup ersymmetric gauge theories has b een very impres

sive over the last couple of years In gauge theories with one sup ersymmetry N Seib erg

discovered dualities b etween the large distance IR b ehaviour of electric and magnetic versions

of the same theory The most prominent example of all is the treatment of N extended

sup ersymmetric mo dels by Seib erg and Witten that allowed to compute the sp ectrum of the

theory with gauge group SU exactly and identied the condensation of monop oles as a mech

anism of quark connement The solution of the theory also involved a duality when the

gauge symmetry had b een broken sp ontaneously to leave only a single gauge b oson massless ie

there exists a version of Sduality on the Coulomb branch of the mo duli space Much earlier it

had already b een found that the amount of sup ersymmetry necessary to implement the Sduality

in its full conjectured extent is even larger and that only N sup ersymmetric YangMills

SYM theory would allow the dyons and the gauge b osons to sit in the same representations of

the Lorentz group So let us briey discuss the way top ological charges are introduced in

sup ersymmetric gauge theories and stress the particular role that is b eing played by states that

saturate the BPS b ound

The most general algebra of N sup ercharges that can b e constructed in d dimensions is

given by two comp onent Weyl formalism

j

ij i

g P fQ Q

j

ij i

g U fQ Q

j

ij i

g V fQ Q

where i j lab el the dierent sup ercharges the spinor comp onents and C is

ij j i ij j i

the charge conjugation matrix P the momentum op erator U U and V V b eing

i

called central charges The Q are the generators of the sup ersymmetry and their conjugation

is dened by

y

i i

Q C Q

Applied to the elds of a given theory they generate the minimal eld content of the sup er

symmetrized version of that theory They carry a representation of the Lorentz group SO

as well as of some internal symmetry that acts on the sup ercharges In the absence of central

charges the algebra simplies and we can easily nd its representation in terms of a physical

ie sup ersymmetric multiplet of states In the case of a massless multiplet P we can

choose the Lorentz frame in which P E E and lo ok for a representation of the little

group SO that leaves P invariant which can afterwards b e promoted to a representation of

the full Lorentz algebra by application of the b o ost op erators Sup ersymmetry transformations

and b o osts commute Thus we obtain the simplied algebra

j j j

i ij i i

fQ Q g E fQ Q g fQ Q g

Obviously the upp er comp onents anticommute and by the usual p ositive norm argument they

have to annihilate the physical Hilb ert space thus they are trivially represented In other words

Massless states leave half of the sup ersymmetry unbroken The lower comp onents form a Clif

ford algebra and act as raising and lowering op erators of the helicity op erator the generator of

rotations around the direction of P They create an antisymmetric tensor representation of the

internal U N symmetry from a given state of lowest helicity s

j

i i

p

Q jP si Q Q jP si jP si

E

E

The series terminates after application of N raising op erators which in the case of N gives

a sp ectrum consisting only of two kinds of elds We shall only b e interested in multiplets that

contain no higher spin than s The p ossibilities we are then left with are two complex scalars

and a single fermion chiral multiplet or a fermion with a vector vector multiplet while in

N one could have a vector with a complex scalar and two fermions vector multiplet or two

fermions with two complex scalars hypermultiplet counting always onshell degrees of freedom

after eliminating auxiliary elds Multiplets with s that involve more massless elds can

always b e reduced to tensor pro ducts of these irreducible representations Such multiplets can

b e thought of as the minimal eld content a sup ersymmetric theory has to contain

We do not intend to lo ok at the case of massive elds explicitly as the situation gets more

complicated and we shall not b e interested in the massive elds with vanishing central charges in

the later discussion One gets aware at once that the matrix of the sup ercharge anticommutator

has no vanishing eigenvalues anymore and all sup ercharges have to b e represented non trivially

The shortest p ossible multiplets get very much longer than the massless multiplets ab ove their

internal symmetry is found to b e U S pN A general classication can b e found in the standard

literature a very recent review of this and some of the following is

We now also allow central charges in the sup eralgebra which implies N then lo ok for

the short multiplets as b efore and will nd them to b e BPS saturated Therefore we take massive

states with rest frame momentum P m whose sup eralgebra is

j

ij i

g m fQ Q

j

i ij

fQ Q g U

j

ij i

g V fQ Q

By using the mentioned internal symmetry one can redene the sup ercharges in a way that

combines the former ones into chiral comp onents and allows to diagonalize the right hand side

of the anticommutator with eigenvalues Z no summation

i

j

i y ij

fQ Q g m Z

i

j

i y ij

fQ Q g m Z

i

i i y

By the p ositive deniteness of the op erator Q Q again no summation we immediately get

the BPS b ound on the eigenvalues of the central charge matrix Z m Also we can identify the

i

short or BPS multiplets that corresp ond to the massless multiplets in the case of vanishing central

charges They are created when all the eigenvalues saturate the b ound Z m Generally the

i

multiplets contain less and less states when more and more eigenvalues satisfy this relation as

more sup ercharges get represented trivially then These states are of great imp ortance as they

are b elieved to b e not aected at least not to o much by renormalization when the classical

sup ersymmetric theory is taken to b e the bare Lagrangian of a quantum eld theory One then

generally exp ects that the pro cedure of renormalization might change some or all parameters

of the theory oating into some xed p oint of the renormalization group and thus sp oiling all

the results of classical analysis which we dealt with so far On the other hand if nothing very

dramatic happ ens and all parameters vary suciently smo othly the number of physical elds

should remain unchanged More precisely stated if charge and mass are co ecients of relevant

or marginal op erators in a Lagrangian that is at the foundation of a quantum theory for BPS

states their values should after renormalization still coincide if sup ersymmetry is present and the

number of elds unchanged In contrast to the co ecients of irrelevant op erators the renormal

ized values of such parameters are not determined by renormalization alone but have to b e xed

by an exp eriment A natural value is assumed to b e of the order of the quantum corrections

but in principle any value can b e obtained by adjusting the bare parameters order by order in

p erturbation theory as desired In sup ersymmetric eld theories there often o ccurs a cancellation

of p erturbative or even all quantum corrections which then renders the protected parameter to

b e a free mo dulus of the theory exactly determined by its bare value At least the p otential

of any sup ersymmetric eld theory is not aected by p erturbative quantum corections This

is also the reason why we can imagine consistently to vary the string coupling from strong to

weak by changing its bare value which is the only free mo dulus of string theory Because of such

reasoning one hop es that once the BPS states are identied in the classical approximation of some

sup ersymmetric theory ie the lowest order of its p erturbative expansion they should exist also

in the non p erturbative large coupling sector of the mo duli space Establishing duality relations

b etween p erturbative and non p erturbative sectors of theories very much relies on comparing

the BPS sp ectra of b oth theories in the resp ective domain There is in fact a case in which the

duality argument can b e made rigorous This is the maximal N sup ersymmetric extension of

YangMills theory which we already mentioned earlier to b e the only SYM theory that provides

the correct multiplets to have full MontonenOlive Sduality implemented This theory is

also conformally invariant and has all its functions vanishing exactly so that BPS states in

deed remain unaected by renormalization even if non p erturbative eects are taken into account

We now return to our previous sub ject of monop oles and dyons in the GeorgiGlashow mo del

and relate central charges of the sup eralgebra of its N sup ersymmetric extension to top olog

ical charges of solitonic solutions of the eld equations The Lagrangian of the SYM theory

that includes the former mo del

X

a a a a a

F F i D D D L

a i i i i

i

g Tr ig Tr

i j ij i j

a

contains a single N vector multiplet a vector gauge b oson with eld strength F two

a a a

fermions a scalar and a pseudoscalar in the adjoint representation of the gauge

i

group If one computes the variations of the sup ersymmetry transformations in terms of the

elds and keeps track of all b oundary terms one nds that the sup ersymmetry only holds up to

the b oundary terms

Z

ij k a i a

F U d x F

aj k i

a

Z

a i ij k a

V d x F F

i aj k

a

These are the generalizations of the top ological charges given by the winding of Higgs and gauge

eld of the former mo del If non vanishing they demand to implement central charges U and V

into the sup eralgebra

j j

ij i ij i

g V fQ Q g U fQ Q

Thus the numerical values of the central charges are equal to the top ological charges in the

SYM theory which in the non sup ersymmetric case we identied with the electric and magnetic

charges of the classical monop ole solutions BPS monop oles are then sp ecial solutions that

also saturate the BPS b ound and which are annihilated by one half of the sup ercharges In

other words the presence of a BPS monop ole somewhere in the universe can b e thought of as

a top ological non trivial mo dication of the vacuum that imp oses certain b oundary conditions

on the elds at innity and breaks half of the sup ersymmetry by these conditions The other

half of the sup eralgebra acts on the monop ole state by creation of fermionic solutions to the

eld equations Dirac equations in this background The space of these fermionic zeromo des is

parametrized by the mo duli of the monop ole solution its p osition and charges ie the number

of fermionic zeromo des in the monop ole background is related to the dimension of the mo duli

space of the monop ole All the states saturating the BPS mass b ound we have b een discussing

so far ReissnerNordstrom extremal black holes and PrasadSommereld monop oles can b e

embedded in sup ersymmetric theories to yield BPS states While we only deduced their prop erties

and existence from classical analysis the sup ersymmetry is assumed to protect them against

renormalization by quantum eects so that they remain BPS multiplets In this sense classical

arguments are extrap olated towards quantum exactness Reviews of the whole material can also

b e found in

Solitons in string theory

In analogy to solutions to classical eld equations of the previous chapter we shall now discuss

solitonic BPS ob jects in string theory In the following sections we shall nd two similar but

not completely identical types of candidates for such states solitonic pbranes and Dirichlet

branes We would like to p oint out some of their dierences and similarities We will start with

an introduction into the way how the calculus of dierential forms allows to deduce the p ossible

spatial dimension of charged ob jects in string theory on the grounds of the ranks of the tensor

elds o ccurring in the low energy eective action the sup ergravity theories in ten dimensions

These ob jects will then rst b e established classically as pbranes solutions of the sup ergravity

eld equations that only dep end on a subset of the co ordinates They are interpreted as higher

dimensional monop oles and black holes as they are sources for the generalized electromagnetic

tensor elds as well as for the gravitational metric eld that are extended in spatial directions also

Integrating the dual electromagnetic eld strengths over the space transverse to the worldvolume

of the branes then allows to introduce the notion of non p erturbative top ological charges into

the theory The second type of states the Dbranes have b een reviewed as defects of spacetime

which have p erturbative op en string world sheets ending on them By indirect arguing many

indications have b een found that they are intermediate states b etween p erturbative excitations

and solitonic pbranes that share many prop erties of the latter In addition to their b eing classical

solutions of the eective sup ergravity theory they further provide us with a prescription of how to

handle their p erturbative degrees of freedom via the op en strings ending on them which by the

continuity of BPS states can b e traced back into the strongly coupled non p erturbative regime

This is the key to a p erturbative quantum description of strongly coupled string phenomena We

shall nd an example of such metho ds in the treatment of black holes

Extended charges as sources of tensor elds

In this introductory section we rst explain the necessary mathematics to generalize classical

electromagnetism to higher dimensions ie the calculus of dierential forms Taking

the entries of the eld strength tensor F of the usual Maxwell theory as co ecients of the

form

F dA A dx dx F dx dx

corresp ondingly as a form j j dx we can rewrite the inhomogeneous and the current j

e

part of Maxwells equations

F d F j F

e

and represent the electric charge inside some spatial region by the integral of the eld equation

over the transverse space M

Z Z

e F j

e

M M

using Stokes theorem The homogeneous part of Maxwells equations b ecomes the Bianchi

identity dF The dualized set of equations including the magnetic current is then given by

adding ad ho c a term to the eld strength that is not closed but its Ho dge dual is thus sp oiling

the Bianchi identity but not mo difying the electric charge denition

F F dA

d F j dF d j

e g

The magnetic charge b ecomes

Z Z

F j g

g

M

M

The gauge freedom corresp onds to adding an exact form to the form p otential

A A d

All these statements can easily b e generalized to dierential forms of higher degree

p

p

dx dx p otential A A

p

p p p

gauge freedom A A d

p p

eld strength F p dA

Z

dp

electric charge e j

e

M

Z

p

j magnetic charge g

g

M

One nds that in general j is a p form if the electric charge is an ob ject extending into

g

p p space dimensions while the dual magnetic charge lives in p d p dimensions In

e m e

the classical d Maxwell theory we had of course pe p From this generalized setting

m

the Dirac quantization condition for the pro duct of the two charges can b e retained unchanged

as in the dualized Maxwell theory In a BohmAharonov exp eriment one would have to move

extended ob jects around extended singularities and the magnetic p otential integrated over non

trivial cycles around its singularities leads to a magnetic charge quantized by exactly the same

formula in integer units of the inverse electric charge times

Thus we have found a rule that allows to deduce from the rank of some antisymmetric tensor

eld rewritten in the language of a dierential form of same degree the spatial dimension of the

corresp onding charge that is the source of the generalized electromagnetic eld corresp onding to

that tensor eld As well this charge is quantized in analogy to Diracs condition We are now

in the p osition to discuss the dimensionality of the charges we do exp ect in the various string

mo dels by insp ecting their b osonic eld contents summarized in table Some comments are

necessary which will b ecome clear only in the next sections to follow The universal NSNS sector

is common to b oth heterotic the I IA and the I IB mo del Its form p otential couples to the

resp ective fundamental string itself ie the world sheet co ordinates of the string couple to the

space time antisymmetric tensor as usual in the mo del approach The solitonic ob ject of this

sector is the NSbrane solitonic in the sense of a pbrane as found in the universal b osonic

sector of d sup ergravity It is the magnetic dual of the string and couples magnetically

to the NSNS form eld strength All other charged states listed in the table come from the

Table Forms and charges in various string mo dels

Mo del Potential Field Strength p p

e m

Universal NSNS sector

Heterotic I IA I IB B F fundamental string NSbrane

RR sector

I IA A F DParticle

A F

I IB A F DInstanton

A F DString

selfdual A F

Mtheory d A F Membrane

RR sector they are assumed to b e realized as Dbranes These are string solitons which can

p erturbatively b e describ ed by op en strings with Dirichlet b oundary conditions and have an in

terpretation as pbranes in the low energy approximation by sup ergravity The meaning of these

statements will b e explored in the following sections All these states are related by several kinds

of conjectured duality transformations that also connect the p erturbative and non p erturbative

degrees of freedom Particularly the fundamental string and the NSbrane of the universal I IB

sector are related to the Dbrane the so called DString and the Dbrane of the RR sector of

the I IB mo del via Sduality The branes of the I IB are selfdual referring to the selfdual eld

strength tensor F F of that theory and the brane is apparently an ob ject lo cal in

space and time and therefore usually adressed as an instanton All these states can b e identied

as states arising in some particular compactication of states known from d Mtheory which

we shall turn to in the nal chapter

Solutions of the sup ergravity eld equations pbranes

The next task will b e to derive solutions to the low energy eld equations of string theory

These describ e the various elds whose sources are multidimensional ob jects that

carry mass and charges corresp onding to the various eld strengths We use the eld equation of

the eective theory sup ergravity in d dimensions to nd solitonic states that display nite

action Their supp ort in space has to b e lo calized and all elds must decrease fast enough at

innity but p ossibly with non trivial winding Thereby we always fo cus on the b osonic degrees of

freedom taking the fermionic elds then given automatically by sup ersymmetry The massless

b osonic elds of the string sp ectra are the graviton G the antisymmetric tensor eld B with

p

form eld strength and the dilaton from the universal sector as well as the various A

p

p otentials or their eld strengths F from the I IA and I IB RR sectors Their eective action

is dictated by the according sup ergravity theory

Z

p

X

E a E

n

E

d x F S r r e F R G G

n e

n

n

here written in the so called Einstein frame and all top ological ChernSimons terms b eing dropp ed

In the universal sector we have a n while in the Ramond sectors a n and

n runs over the relevant form degrees When commenting on the n case we shall eventually

ignore the problem that no consistent action for the selfdual form is known so far The action

can b e rephrased into the mo del or string frame by a Weyl rescaling

E E

G G e G

which results in

Z

p

G R G r r e S d x F F

e

X

n

F F

n

n

n

the sum now only running over the Ramond elds of course The latter form can also directly

b e derived as a low energy eective action of string theory using the formalism of world sheet

mo dels An imp ortant thing to notice is that the dilaton has obtained a universal coupling to

all elds of the universal sector but decouples from the elds of the RR sector Thus we exp ect

to get dierent dep endences of masses and charges on the string coupling hexp i for branes

originating from tensor elds of the dierent sectors The masses of the branes coupling to NSNS

while those from the RR sector will interpolate elds should b e exp ected to b ehave like g

S

To avoid certain diculties and for the sake of brevity we now take the b etween g and g

S

S

somewhat simpler type of Einstein action

Z

p

E E a d

n

E

S G R r r e F F d x

e n

n

keeping only a single tensor eld and derive the various equations of motion

E

S R

n

a E

n

S F F e F G

n

n nd

a

n

e F r

a

a

e F

n

The rst one is a generalized Einstein equation the third one the vacuum Maxwell equation

and the last one describ es a KleinGordon eld coupling somehow to electromagnetism Later

on we shall have to add a source term into the action to get non trivial solutions This will

lead to additional delta function like singular sources on the right hand side of the equations

of motion To make an ansatz we now split the co ordinates into the x p on a

p n dimensional hyperplane which is taken to contain the charged ob ject and the

transverse spatial directions y M p d We then demand usual Poincare P p

M

invariance in the rst p co ordinates the worldvolume of the brane and isotropy SO d p

invariance in the rest of space to b e satised by the solution All the elds necessarily have to

b e indep endent of the internal x co ordinates b ecause of translation invariance We also need

to determine the amount of sup ersymmetry that is left unbroken by the ansatz Therefore we

have to lo ok for the decomp osition of the tendimensional Poincare invariant sup ercharges into

p and d p dimensional spinors themselves invariant under the demanded spacetime

symmetries For d it is found that the tendimensional chirality condition

implies that also

p p p p

where is the resp ective chirality op erator and an arbitrary invariant

D

D D

spinor in the D dimensional subspaces By insp ecting the eigenvalues of these matrices one

nds that for the cases we consider one half of the sup ersymmetry is broken by the brane ansatz

which indicates that we might b e dealing with a BPS state In fact it is p ossible to go the other

way round and construct the solutions we shall uncover by just their prop erty of breaking exactly

one half of the sup ersymmetry Splitting o the metric according to the ansatz we can always

write it

Ar B r M

ds e dx dx e dy dy

M

E

p

M

y y is the radial distance where b oth contractions of indices only involve at metrics and r

M

in the space orthogonal to the brane The ansatz for the metric obviously resp ects Lorentz and

translation invariance on the brane and also rotation invariance in the transverse directions

For the eld strength tensor there are two dierent choices one can think of as we have the

two options to construct the electric charge from the eld strength tensor itself or the magnetic

charge from its dual such that one has the two options

C r e

e F

M

n

M

n

M

y

m

g F

M M M

n M M

n

n

r

The form degrees are related by n d n as b oth eld strengths are Ho dge dual in the d

dimensional spacetime This completes the ansatz It of course remains to b e veried that

charge and mass of these states take nite values We now omit lots of technical details which

can b e found for instance in Finally one can reduce the three undened functions to a

single one and a couple of parameters related by

d

Ar

d

e H r

d

B r

d

e H r

C r

p

H r e

a

r

e H r

The remaining function H r is harmonic in the transverse space ie ob eys the Laplace equation

N M

H r

NM

Only if we add a source on the right hand side of this equation we get non trivial solutions for

H r as integrable globally harmonic functions necessarily vanish Adding a source that only

has supp ort on the volume of the brane a p dimensional charged current now corresp onds

to turning the Laplace equation into a Poisson equation An explicit form for such a current

will b e discussed later The solution for H r with a brane at the transverse origin can then b e

written

d

H r r

the parameters b eing given by

p d

a

d

for electric solutions

for magnetic solutions

d d p

In the magnetic case the integration constant is related to the magnetic charge parameter

p

g

d

while in the electric case it is xed by the charge parameter of the source terms in the action

In analogy to the denition of charges in electromagnetism we can integrate the eld strength

p form over a surface that encapsulates the pbrane and obtain the charge of an ob ject

that has a p dimensional worldvolume This charge is the source of the metric eld in the

transverse directions as well as of the tensor eld Only in the case of a brane we know the

prop er quantum theory of the desired ob ject which is simply the fundamental string Taking

this example one can add a source term into the action which has only supp ort on a two

dimensional submanifold of spacetime and whose action there is given by the mo del action of

string theory

Z

p p

T

X X G X X B S d hh hR

All the analysis can b e carried out with the only exception that delta function source terms

app ear on the right hand side of the eld equations This is interpreted in the following way

The fundamental microscopic string is the source of a macroscopic eld conguration the brane

solution of the low energy approximation to string theory The elds that app ear in the world

sheet theory as the massless mo des of the string are now in the worldvolume theory organized

in sup ergravity multiplets and their source is the twodimensional string again Its electric charge

under the tensor eld is given by integrating the eld equation

Z

p

r

p

e F r T e

ST

M

The mass of this brane is dened by the integral over the time comp onent of the energy momen

tum tensor which can also b e calculated from the mo del source and for the given solution is

found to saturate the BPS b ound it is equal to the charge of the brane Despite from this case it

is an op en question what material branes are made of in general We can interpret the brane

as the solution coming from the string and the NSbrane as its d dimensional mag

netic dual For the rest of the pbrane solutions one has no elementary particles at hand Only

by the discovery of Polchinski it b ecame plausible that the desired ob jects are related to Dbranes

Let us now p oint out an asp ect of duality in the brane picture The electric and magnetic

pbrane solutions we have found corresp ond to a particle like state which is the source of the

n

electric eld strength tensor F as well as a solitonic ob ject the source for the magnetic dual

dn

eld F This setting allows a notion of duality in the sense that starting from the dual

tensor in the original action and splitting co ordinates accordingly would have interchanged the

role of the two charges In other words calling the n dimensional state the elmentary and the

d n dimensional the solitonic one is a matter of convention as long as we do not decide which

eld strength is to b e called fundamental Thus the NSbrane might b e as fundamental as the

string itself though we cannot say very much concerning its quantum theory which is supp osed

to b e given by a quantization of its co ordinates in the spirit of the Polyakov action approach to

string theory From computing its top ological charge by using the magnetic dual tensor eld

one can deduce the generalized Dirac quantization condition from a BohmAharonov exp eriment

T T n

where T is the tension of the brane

We can summarize that we found solutions to classical low energy eective string theory that

are extended and carry mass and charge under the various tensor elds Their particular values

saturate the BPS b ound and this allows us to strongly b elieve in the existence of these states also

in the unknown quantum theory and further exp ect them not to b e renormalized in a way that

would sp oil their b eing BPS saturated The severe and in general unsolved problem remains

how to give a description of the quantum theory involving branes of higher dimension

We next apply the solutions for the metric and the dilaton to various values for d and p to

get some examples which we of course choose from the string sp ectra The expressions for the

eld strengths are obtained as easy For the fundamental string in d having a worldvolume

of dimension and corresp ondingly n the metric and the dilaton eld read

ds dx dy

E M

r r

e

r

an expression that app ears to display singularities not completely unfamiliar from those found for

the black hole solutions in general relativity In fact there are black brane solutions that have

horizons shielding their singularities They are in many resp ects similar to higher dimensional

black holes and we shall eventually return to these examples when discussing black holes in

string theory For the NSbrane with n we get instead

ds dx dy

E M

r r

e

r

Comparing the solutions for the string and the NSbrane the two regions of spacetime the

brane and the transverse space app ear exchanged Also the coupling has b een inverted as

supp osed for the electricmagnetic duality of fundamental p erturbative states and monop oles

Particularly for the selfdual brane of the Ramond sector with n we get

ds dx dy

E M

r r

e

where there is no distinction b etween the two types of states electric or magnetic Finally the

Dbrane metric diers only in the dep endence of the dilaton from its Sdual the NSbrane

e

r

as the duality transformation exchanges p erturbative and non p erturbative states In this sense

the dualities of string theory are manifest in the sup ergravity brane solutions Sduality is an

involution of the I IB sup ergravity that can map the exp onential of the dilaton to its inverse and

leave the metric in the Einstein frame invariant Thus the Dbrane gets mapp ed to the ab ove

NSbrane solution while the fundamental string gets mapp ed to an ob ject with

e

r

and otherwise unchanged metric which is just the Dbrane solution In a more general anal

ysis also Tduality can b e recovered After p erforming a dimensional reduction of I IA and I IB

sup ergravity on a circle one nds a unique sup ergravity in d that again allows an involution

of its sup eralgebra and eld content in general exchanging the elds and charges that originate

from I IA and I IB Decompactifying afterwards one recognizes that this entire transformation

realizes the Tduality transformation which is known from the p erturbative string theory and

which maps o dd dimensional I IB branes into even dimensional I IA branes and vice versa This

has to b e discussed in the string frame where any Dpbrane has

dx dy ds

M

d d

r r

p

e

d

r

Compactifying along the brane worldvolume now leads to a brane wrapp ed around the circle

and the rank p of the tensor eld coupling to the brane is eectively reduced by one The

involution of the ninedimensional sup ergravity next is of such nature that a tensor eld of this

kind is mapp ed to another one that already in the tendimensional theory had the lower rank

p so that after decompactifying we end up with a brane of one dimension less On the other

hand starting with a compactication in a direction transvers to the brane the rank of the

tensor eld is unchanged at rst and it is then b eing mapp ed to a tensor eld that had originally

higher rank but lost an index during the compactication By decompactifying the additional

dimension op ens up and the brane gains an extra dimension in the end This discussion is so far

limited to the case of Dbranes and has to b e mo died for NSbranes We shall b e discussing the

role of the elevendimensional sup eralgebra in Mtheory later on which reveals the particular

imp ortance of central charges for the existence of corresp onding branes and should thereby make

the ab ove explanations also more transparent

Dbranes as pbranes

Another type of extended geometrical ob jects in string theory are the Dbranes we discussed ear

lier These have formerly b een introduced as xed hyperplanes in spacetime where op en strings

can end on The necessity of their existence had already b een demonstrated by Tduality

when Polchinski found an interpretation that allowed to view them as dynamical charged

ob jects that uctuate in shap e and p osition and couple to the RR elds of the string world volume

theory This induced a tremendous amount of work on Dbranes and related sub jects which left

very little doubt ab out the statement that Dbranes are some sort of non p erturbative states of

string theory relatives of the solitonic pbranes from the previous section They have a p erturba

tive description by op en strings ending on them and their BPS nature conserves generic features

of this in the non p erturbative regime There are numerous reviews of Dbrane physics only to

mention so that we restrict ourselves to illustrate Polchinskis original computation

of the tension and charge by regarding a scattering pro cess of strings emitted from and absorb ed

by Dbranes

We rst show why the pbranes of the NSNS sector cannot b e the sources of the RR elds Let

us recall the world sheet origin of the various elds that o ccur in the low energy eective string

actions In general the states are created by mo de op erators of the co ordinate and spin elds

sub ject to the constraints of sup erconformal invariance imp osed by the sup er Virasoro op erators

The space time elds are then given by the p olarizations of such states for a massless state in

the universal NSNS sector eg

j k i jG B k i G B

NSNS NSNS

and their equations of motion express the restriction imp osed by the constraints For these NSNS

elds the equations of motion can similarly b e determined by the mo del approach which con

sists in taking the action with the co ordinates coupling to the spacetime elds as a quantum

eld theory of the co ordinates in the usual sense The spacetime elds are coupling constants of

this theory and conformal invariance implies the vanishing of all the functions These can b e

computed by standard metho ds in terms of the bare values of the spacetime elds Demanding

them to vanish yields the equations of motion order by order in the mo del lo op expansion

parameter The onelo op result repro duces the equations known from the b osonic sector of

d sup ergravity

The elds of the RR sector on the other hand originate from the tensor pro duct of the space

time spinor elds s ands of the left and right moving sectors Thus they are p olarizations H

a a ab

which after expanding into gamma matrices lo ok like

n

X

i

ab

T

n

s j k i H k i s jH

b RR RR

n n

a

n

n

To b e more precise the H s are the eld strengths of the RR elds Their equations of motion

are derived by exploring the Dirac equations that come along with sup er Virasoro constraints

as well as chirality conditions while a corresp onding mo del necessarily involving a coupling

to the spin vertex is not known After rewriting the Lorentz tensors into dierential forms the

equations can b e summarized by simply demanding the forms to b e harmonic

dH d H

which allows to introduce p otentials As we have seen earlier the degree of the p otential de

termines the dimension of the brane which it naturally couples to Lo oking at the RR form

with form p otential everything app ears to b e quite similar to the NSNS form eld strength

except that the latter couples to the co ordinates of the world sheet while the former couples to

the spin eld But this crucial fact prohibits to interpret the fundamental closed string as the

source of the RR elds Assume a scattering pro cess of an incoming and outgoing closed string

with a vertex op erator of the RR eld inserted Because of the RR elds coupling only

via their eld strength to the world sheet this amplitude always carries a p ower of the external

momentum While the diagram itself is to b e interpreted as the coupling of the world volume

tensor eld to the macroscopic string its zero momentum limit is the charge of the string under

that eld which is vanishing Therefore we are forced to conclude that the pbrane solutions of

the sup ergravity eld equations which had fundamental strings as electric or magnetic sources

cannot b e the states that carry the RR charges There must b e dierent elementary ob jects in

string theory as RR charged elds These may then have a low energy description as pbranes

that are charged with resp ect to the RR eld strengths Dbranes are of course thought to b e

the right choice

The observation of Polchinski now was the following One computed the onelo op scattering

amplitude of an op en string in the vacuum but with Dirichlet b oundary conditions at its ends

D

signaled by putting L instead of L This can alternatively b e seen as two Dbranes of any

type I I mo del exchanging a closed string at tree level

D

L L t t L

ji jD i hje hD je

lo op tree

The length of the closed string is parametrized by t l while the op en string circling

around the cylinder has the length t The result for this matrix element included a contribution

or

Figure Op en string onelo op or closed string treelevel diagram

of the RR elds of the closed string to the scattering amplitude indicating a coupling of the RR

elds to the brane One then compares this to the same tree level amplitude in the sup ergravity

approximation to low energy type I I string theory with a Dirichlet brane eective action added

as a source of the elds and a WessZumino coupling of the RR eld to the co ordinates of the

brane Comparing the leading order contributions that come from the dilaton graviton and

RR elds separately allowed to deduce the charge density and tension of the brane In fact all

contributions cancel no force rule but apparently the Dbranes feel a repulsion via RR elds

We rst lo ok at the string calculation of the op en string onelo op vacuum diagram In general

one has to compute the path integral

Z Z

p

D h D X D D

L R

Z h h X X exp d d

vac

Vol

Cyl

L L R R

The elds have to ob ey the appropriate b oundary conditions Dirichlet at the ends and p erio dicity

or antiperio dicity around the cylinder While we shall compute Z as the onelo op zerop oint

vac

function of the op en Dirichlet string we could have alternatively called it the treelevel two

p oint function of two b oundary states in closed string theory What causes all the problems

in computing such integrals is in particular the integration measure which has to b e divided

by the volume Vol of the lo cal symmetry group of sup erconformal transformations and sup er

Weyl rescalings A mathematically more rigorous treatment can b e found in we indicate

the outcome here First we split the constant zeromo des of the co ordinate elds from the

integration which gives a prefactor equal to the innite volume V of the Dbrane only

p

as constant shifts transverse to the brane are prohibited Remember that the centre of mass

of the string also has to move on a hyperplane Then we can formally p erform the Gaussian

integrations

p

p

hh Z Det Det Det

vac

h

apply the old ln Det Tr ln trick as well as

Z

dt

tO t

e e ln O

t

which holds for any op erator O with sp ectrum in the right half plane of p ositive real part nally

getting

Z

0

dt

t k M

e ln Z V Tr

vac p

t

Here the information ab out the oscillator sp ectrum has b een translated into the degeneracy of

the mass levels which are b eing summed over We have included a factor of for the p ossible

orientations of the string and omitted the second term in demanding a dierent kind of

regularization for the integral later on In unoriented type I theory the factor is missing but

on the other hand one is forced by the orientifold pro jection only to consider pairs of branes

at mirror p ositions which in the end brings back the appropriate factor All the functional

determinants and traces involved have to b e p erformed on the right spaces and sup erspaces such

that all the diculties in ob eying constraints and b oundary conditions have only b een hidden

away so far The functional trace over the string sp ectrum consists of an integral over momenta

k on the brane as well as a sum over all oscillator levels of spin and co ordinate elds

Z Z

p

0

dt d k

t k M

Tr e ln Z V

osc vac p

p

t

Z

0

dt

p

t M

V e t Tr

p osc

t

The mass M of a state is given by the formula and esp ecially dep ends on the separation r of

the branes The knowledge of the discrete mass sp ectrum of the theory that is gained from the

oscillator expansion of the elds sub ject to the sup er Virasoro constraints now allows to actually

p erform the sum over oscillators The pro cedure for the spin elds is a little tedious as all allowed

p erio dicity conditions for going around the cylinder NS and R combinations or spin structures

have to b e regarded This can b e found in the standard literature eg chapter of and

has b een explicitly went through in The result can b e written most easily using Jacobi

and Dedekind functions

Z

it

X

j

0

V dt

p

p

s

r t s

ln Z e t

vac

it

t

s

The sum in the integrand is actually vanishing by some identity of functions but we can split o

the two contributions that cancel each other and by their resp ective p erio dicity condition identify

the RR s and NSNS s contributions separately This implies that we reinterpret

the op en string lo op amplitude in terms of the treelevel closed string contributions The p oles

of the amplitude arise from the UV region ie small t and such we expand the integrand around

t getting the leading contribution

it

X

j

s

s t

t o e

it

s

We dene the propagator

Z Z

ipx d

e d p

d

x t

dt t e x

d

d

p

and write the nal result

p

0 p

r

r o e ln Z V

vac p

p

We can summarize the discussion in the following way The force b etween Dbranes due to the ex

change of dilaton graviton and RR eld vanishes by a cancellation of the attractive gravitational

force and the repulsive electric force This is completely analogous to the b ehaviour of magnetic

monop oles in eld theory which do not excert any force on each other by a cancellation of Higgs

and vector b oson exchange In particular the contribution of the RR eld to the amplitude do es

not vanish Therefore in the eective eld theory the Dbrane somehow couples to the RR elds

and thus has to carry a corresp onding charge

The leading contribution of the ab ove string amplitude can also b e computed in the low

energy approximation to string theory which corresp onds to widely separated Dbranes large r

or eectively small We now demonstrate how this is done in the eld theory that is given by

the eective type I I action for the bulk an appropriate world volume action for the brane and

a coupling term This will allow to deduce the macroscopic charge density and brane tension

p

T To decouple the dilaton and graviton propagators one can rewrite the eective action of type

p

I I sup ergravity in the Einstein frame

Z

p

II E

S G R d e dB d x

e

X

p p

dC e

p

p

Note the abuse of notation that mixes forms and functions in the integrand and leads to some

changes of signs compared to earlier notations but helps very much to keep notations short In

a similar manner as was sketched ab ove an eective action for op en strings in the presence of a

Dbrane can b e extracted from a prop er mo del

Z

p

D p p

S T d e det G B F

p

e

p

M

where G and B are the pullbacks of the metric and the antisymmetric NSNS tensor to the

ij ij

Dbrane world volume and F p dA is the eld strength of the U gauge p otential

A that couples to the b oundary of the op en string The world volume coupling constant T

p

denes the string tension The ab ove action is taken as the eective action of the Dbrane itself

by noticing that the p erturbative op en strings ending on the brane eectively carry its degrees

of freedom as they are the only p erturbative manifestation of Dbranes in string theory Thus the

low energy theory of Dbranes is the p erturbation theory of op en strings with Dirichlet b oundary

conditions Note that the dep endence of the Dbrane action on the dilaton diers from the type

II eective action by a factor exp p which in the mo del frame b oils down to the

dierence b etween the dilaton dep endence exp and exp Thus the eective string

tensions of a Dbrane and a pbrane dier by hexp i g This fact leads to the name half

S

solitons for Dbranes Further we add the natural electric coupling of the RR eld to the brane

volume the pullback of the RR eld onto the worldvolume of the brane

Z

WZ p

p

S d C

p p

e p

p

M

where the coupling constant is the charge density Finally one substitutes the identity

p

Z

p p

d x x x a

k

into the Dbrane action where a is the co ordinate of the Dbrane in the transverse space in order

to integrate out the world sheet co ordinates static gauge Altogether we have got a eld theory

with an extended Dbrane source for the NSNS elds and the RR tensor whose propagation in

the bulk of spacetime is governed by type I I sup ergravity This theory might b e ill dened as

a quantum eld theory but one can still extract the classical approximation by computing the

tree level contributions to the twopoint function of dilaton graviton and RR eld Therefore we

naively expand the functions of the elds to get all terms that are linear sources and quadratic

propagators in the elds

Z

p

p II D WZ

GR dC d S S S d x

e e e

p

X

p

p

p p p

T x a G C x a

p i p i

i

By dropping all coupling and higher terms we restrict ourselves to those which are relevant

for the treelevel twopoint functions We only did not explicitly expand the metric eld of the

pure gravity EinsteinHilb ert part around a at background and avoid a discussion of diculties

concerning gauge xing of the gravitational action etc In this form one can immediately read o

the propagators and source terms and quite easily compute the three contributions to the total

tree level amplitude The steps are displayed in and the result reads

r T ln Z V

vac p

p

p p

which is compatible with the string result if

p

T

p p

holds The RR charge density of the Dbrane is equal to its tension a version of the BPS

mass b ound The mo died Dirac quantization condition can b e deduced from a BohmAharonov

exp eriment as usual The pro duct of the charge of the brane of p spatial dimensions with the

charge of its d p p dimensional dual has to b e an invisible phase factor

n

T T

p p

p p

This relation is satised by the ab ove Dbrane charge densities with n thus Dbranes are

states with minimal RR charge and can b e called elementary from this p oint of view

We have only recalled the rst of very many considerations that all lead to the conclusions

that Dbranes are the RR charged BPS states that were missing in the non p erturbative part

of the string sp ectrum b efore Polchinskis discovery Let us collect the facts again Dbranes

break one half of the sup ersymmetry by the same arguments as for pbranes They carry charge

density equal to their tension and they satisfy the minimal version of the Dirac charge quantiza

tion condition Their low energy limit is a sup ergravity pbrane solution and their p erturbative

degrees of freedom are the light mo des of op en strings ending on them

Black holes in string theory

In this chapter we shall review the application of the ideas concerning Dbranes their interactions

and duality relations to the entropy computation and information loss dilemma of black holes

There have b een several mo dels in various dimensions suggested after the rst treatment in d

We shall refer to and d a more complete discussion of the whole material can for

instance b e found in The basic metho d consists in taking congurations of Dbranes and NS

branes which have a low energy description in terms of sup ergravity solutions pbranes whose

elds esp ecially their metric can b e written explicitly as in chapter These are then b eing

compactied by a KaluzaKlein pro cedure down to say d dimensions which corresp onds

to choosing a spacetime background vacuum of appropriate top ology most simply the direct

pro duct of a sixdimensional torus T with fourdimensional at Minkowski space While this

background preserves N sup ersymmetry in d which is further reduced to N by the

presence of the branes there have also b een N compactications on CalabiYau folds b een

considered which introduce a dep endence of the black hole entropy on the top ological prop erties

of the CalabiYau manifold If this is done in a skilful manner the resulting fourdimensional

metric can b e tuned to exactly resemble one of the metrics of black holes we know from general

relativity Particularly generalizations of ReissnerNordstromsolutions can b e obtained this way

Of course we get a lot more elds by the sup ersymmetry which are in general supp osed not to

mo dify the conclusions severely and are omitted in the following We then intend to count the

degeneracy of the resulting state of a couple of Dbranes NSbranes and strings with xed

values of macroscopic energy and charges whose logarithm simply is the entropy of the resultant

fourdimensional black hole But as we do not know the non p erturbative degrees of freedom

of a Dbrane we have to make sure that we can change the coupling towards the p erturbative

regime of the string mo duli space without lo osing control ab out the states we are lo oking at

The rst thing to notice is then that we shall have to take BPS states which means extremal

ReissnerNordstromblack hole solutions They can b e assumed to exist in the non p erturbative

as well as in the p erturbative sp ectra The second p oint is that the classical pbrane solutions

also include a dep endence of the dilaton eld on the radial co ordinate of the transverse space

which is of the kind that it generically tends to blow up at the horizon r thus preventing

any small coupling treatment at the p osition of the brane no matter what the bare coupling is

chosen The classical value of the dilaton will coincide with its quantum exp ectation value as

the p otential go es unrenormalized Thus the brane conguration we choose also has to take care

to have regular dilaton at the horizon

We now sketch how the mo del is constructed in detail and how the state counting pro ceeds

following the concrete steps of One uses N Dbranes N parallel Dbranes and N

parallel NSbranes of the I IA string theory which are living in a spacetime of top ology R T

all b eing wrapp ed around the torus Strictly sp eaking we have not shown that such many brane

states with several intersecting Dbranes and NSbranes exist and are stable In fact the no

force condition allows to consider rather arbitrary congurations which can also b e managed to

b e BPS saturated The metric in the non compact four dimensions then has to b e tuned to b e the

ReissnerNordstromclassical solution of general relativity Also one adds op en strings carrying

quantized purely left moving momentum N R in one of the compact directions The Dbranes

have to wrap around all the directions of the torus while the Dbranes are taken to intersect the

NSbranes only in a onedimensional subspace of their resp ective worldvolumes in a way that

all branes have one compact direction in common The string momentum is supp osed to ow

exactly in that particular direction The low energy solution for the eld strengths is then given

similar to the RR form eld strength originates from the Dbrane sources the form

eld strength from the Dbranes and the NSNS form eld strength from the NSbranes

These we shall not need again but we have to make sure that the dilaton is regular Dening

the resp ective harmonic functions according to by

N q

p

p

h

p

r

we can write the solution for the dilaton

e h h h

which proves the regularity of the dilaton at the p osition r of the branes as well as in the

asymptotically at region r Thus we can take the coupling to b e small everywhere by

choosing its nite value at the horizon extremely small The unique dep endence of the dierent

harmonic functions on the radial co ordinate through r results from the fact that they all have

a threedimensional uncompactied transverse space and r is the appropriate Greens function

of the Laplacian The values for the charge parameters q in d are in fact given by applying

p

the dimensional reduction prescription to the charges in ten uncompactied dimensions The

tendimensional elds are decomp osed according to the lower dimensional Lorentz group and then

taken to b e indep endent of the higher compact dimensions The higher dimensions can thus in the

case of a torus compactication trivially b e integrated out which for the mass of a pdimensional

Dbrane wrapp ed around the torus gives

R R R

p

p

p p

m

D

g

S

This mass is then inserted in Newtons formula for the gravitational p otential and compared to

the classical large radius b ehaviour of the g comp onent of the metric deduced in

q

p

g

r

Thus we can express the parameters in the harmonic functions in geometric quantities and New

tons constant A similar charge parameter has to b e dened for the discrete momentum of the

op en strings

N q

k

r

The solution for the metric from is the low energy approximation for a single brane The

rules for sup erp osing several such branes harmonic function rule lead to the string frame line

element

p

K

p p

dt H dt dx H H dx dx dx ds

RN

H H H H

r

H H

p p

dx dx dx dx

H

H H H H

where the H and K denote the tendimensional ancestors of the h and k b efore dimensional

p p

reduction After reducing to d and switching to the Einstein frame we can get back to the

desired ReissnerNordstrommetric by a prop er choice of the numbers of branes and the radii of

the compactication The thermo dynamic BekensteinHawking entropy can then b e found by

computing the area of the horizon

p

A

N N N N S

BH

G

N

and thermo dynamical prop erties can b e discussed This is the p oint of view of the low energy

approximation through sup ergravity and its pbrane solutions

We now turn to string theory by regarding the quantum degrees of freedom of the state that

consits of the branes and strings we have put together to match the ReissnerNordstrommetric

As we have managed to get a solution with a regular dilaton at the horizon we feel free to

change the coupling constant from the non p erturbative to the p erturbative regime and discuss

p erturbations of the brane state which are small uctuations of the Dbranes p ositions and

shap e that can b e describ ed by weakly coupled op en strings ending on the Dbranes Counting

the degeneracy of the black hole state in four dimensions now means counting all p ossible con

gurations of strings attached to a given set of intersecting branes that leave the macroscopic

value of energy mass and charges invariant ie one has to count the number of states of the

op en strings stretching b etween the Dbranes and the Dbranes that carry the momentum

N R along the compact direction common to all the branes Thus the degeneracy of the string

sp ectrum will b e resp onsible for the statistical entropy of the black hole Again in other words

The quantum degrees of freedom of a macroscopic fourdimensional black hole are op en strings

travelling in the internal compact space carrying some given amount of energy and momentum

The details of the degeneracy calculated in a prop er way are given in we shall b e content

to use only a brief and heuristic treatment First one has to notice that all the Dbranes are cut

into half innite branes at the intersections with the NSbranes thus the number of Dbranes

is multiplied by the number of NSbranes present N N N This conguration is depicted

in gure with arrows indicating the momentum ow For a general deduction how branes

can end on branes see Take next into account the dierent p ossibilities of b oundary

conditions an op en string can have here ND NN DN DD or equivalently the combinations of

branes which it can end on Counting the two orientations this together gives

N N N dierent ways to attach an op en string to the Dbranes Further one has to observe

that the massless excitation level of these strings contains two fermionic and two b osonic onshell

degrees of freedom Thus we got a gas of N N N N fermions and the same number N of

F B

b osons moving in a twodimensional spacetime and carrying momentum N R and corresp onding

energy The state density dN N N of such systems is known from conformal eld theory to

B F

grow exp onentially with energy for high excitation levels according to

r

N N N

B F

dN N N exp

B F

The logarithm of this is the macroscopic entropy

p

S ln dN N N N N N N

B F

which p erfectly matches the BekensteinHawking result The key ingredient to this remarkable

result can b e seen in the sp ecial prop erty of string theory or twodimensional conformal eld

theory to have exp onentially growing state density Thus we can say that the sp ecic input

of string theory into this derivation app ears crucial for the correct magnitude of the quantum

degeneracy of a macroscopic black hole state of general relativity

The metho ds we have sketched ab ove have also b een applied to calculate the entropy of non

extremal black holes These can b e constructed most easily from the extremal mo del by D6

D2 D6 NS5

D2 D6 NS5

D2

N/R

Figure The internal brane geometry of a fourdimensional black hole

adding right moving momentum N R carried by additional op en strings not to b e confused

R

with the left and right moving sectors of a single closed string in the compact direction common

to all the branes Naively the ab ove state counting pro cedure can b e rep eated and the left and

right moving strings at small coupling contribute indep endently to the degeneracy

S ln d N N N d N N N

R B F R L B F L

The result is again in p erfect agreement with the BekensteinHawking area law In this picture

Hawking radiation is gured to arise from a recombination of left and right moving op en strings

forming a closed string that leaves the brane and moves freely in the bulk This enables to

compute the sp ectrum of the radiation by an evaluation of the treelevel partition function

which allows by standard metho ds of statistical physics to obtain the exp ectation values of the

o ccupation numbers of string oscillator levels This is in fact the sp ectral density of the closed

string radiation from the black hole Indeed one nds a black b o dy sp ectrum not surprisingly

for a set of free harmonic oscillators and one can identify the Hawking temp erature in terms of

the charges and momentum quantum numbers As this is a completely unitary computation in

a fully quantum theory there is a priori no way left for any information coherence to dissipate

away Any pure state will stay to b e one forever On the other hand as has b een p ointed out

already by the authors of the success of the naive application of the pro cedure that was

employed to compute the entropy of the extremal black hole is not clear It very essentially

relies on the opp ortunity to change from the strong to the weak coupling regime Starting from

a situation with many Dbranes present we would assume to have relevant if not dominating

corrections due to interactions near the horizon as the large number of branes leads to an increase

of the dilaton there These corrections are suppressed by going to the very small coupling regime

The continuity of the former Dbrane conguration and in particular the existence of all of the

degenerate quantum states in b oth regimes was guaranteed by their BPS nature in the extremal

case This argument do es no longer hold for non extremal black holes that are clearly non BPS

They should b e aected by renormalization in presumably very dierent manners for dierent

values of the string coupling constant For instance in there have b een arguments given

why op en string lo op corrections might not change the results of the small coupling computation

but a generally excepted answer is not available

Mtheory

In this nal chapter we shall hardly b e able to give anything more than a little taste of what

Mtheory is meant to b e while its nal version surely is still under construction anyway The

large amount of symmetry that is found in the sp ectrum and the interactions of string theory

has lend a lot of heuristic evidence to the very tempting conjecture there might b e a unique

theory of gravity and sup ersymmetric gauge theory at the heart of it This is assumed to incor

p orate all degrees of freedom encountered in string theory in a single type of theoretical mo del

the p erturbative string excitations as well as all the branes we have found so far Also it has

to repro duce the low energy theory of string theory the two types of d sup ergravity in

a consistent manner An astonishing fact is that these requirements app ear to b e met quite

naturally if one takes this Mtheory to b e living in an elevendimensional spacetime that after

compactication of a single spatial direction yields in dierent limits the string theories we like

This explains them all to b e connected by duality transformations as they originate from the

same mother theory The only free parameter in the pro cedure app ears to b e the radius of the

compact additional dimension while string theory in the critical dimension also has a single free

parameter its coupling constant Let us start by briey indicating some motivating evidence for

the eleventh dimension

The maximal dimension in which any sup ergravity theory could b e dened is d Higher

dimensions necessarily lead to spin elds in the theory which one do es not know how to deal

with consistently This theory is thus naturally assumed to b e the low energy eective theory

that approximates Mtheory There is a unique sup ergravity in d b eing scale invariant ie

it do es not have any free parameters This makes us b elieve that it itself do es not come from

a compactied even higher dimensional and even more unknown theory as the compactication

should include some scale parameters according to the geometry As it is also non chiral in a

dimensional KaluzaKlein type reduction by a compactication in one direction it leads to type

I IA sup ergravity The eld content of its b osonic sector contains only a form p otential A

plus the metric eld The eective action follows

Z

p

d

g R dA d x S

e

As earlier we did not write fermionic elds and also left out top ological ChernSimon terms A

KaluzaKlein reduction to d dimensions is p erformed by putting the eleventh co ordinate x

on a circle of radius R The elds will then b e decomp osed with resp ect to the tendimensional

Lorentz group and reveal the eld content of d type I IA according to

A B A A

ij ij ij k

g g A g g

i i ij

where indices i j only run from to To make this more precise we can write the elevendimen

sional line element and p otential

i

e ds dx dx A ds e

i

A fA B g

ij k ij

The tendimensional elds then are taken to b e indep endent of the eleventh internal co ordinate

for consistency reasons This assures that any solution derived in the lower dimensional theory is

also a solution of the original one The eleventh dimension can now b e integrated over trivially

and yields a prefactor in front of the action relating the tendimensional gravitational coupling

constant to the elvendimensional by

R

The string coupling constant g hexp i in the tendimensional I IA theory is also related to

S

the radius R of the compactication by

R g

S

thus the string coupling constant is revealed to b e given by the radius of the additional dimension

of Mtheory This implies that the p erturbative regime of string theory do es not know anything

ab out the eleventh dimension as the small coupling sector corresp onds to the small radius region

Vice versa the non p erturbative regime of string theory should imply a decompactication of the

eleventh dimension Mtheory do es by this mechanism automatically include the non p erturba

tive eects of string theory Also the other degrees of freedom of the d sup ergravity

theory match together with their ancestors in eleven dimensions For the fermionic elds this

check is as easy as the one we made ab ove for the b osonic elds while it also app ears to hold as

it b etter has to in the non p erturbative sector of the sp ectrum

In the non p erturbative part of its sp ectrum the elevendimensional sup ergravity has by the

sake of its form eld strength an electric membrane Mbrane as well as a solitonic Mbrane

of the same nature as the pbranes we discussed earlier From these the states of the type I IA

string sp ectrum can b e constructed by a prop er choice of the co ordinates which are b eing com

pactied The string itself can b e seen to b e a membrane wrapp ed around the compact circle the

string Dbrane is an uncompactied Mbrane and similarly the Dbrane and the NSbrane

descend from the Mbrane A less obvious case is the Dbrane which can b e traced back to

a KaluzaKlein monop ole of Mtheory The resp ective string tensions can b e computed on

b oth sides of this corresp ondence as a test They are related by the minimal Dirac quantization

conditions and one pair of brane tensions can b e xed by hand But afterwards the relation

b etween the two coupling constants has to hold when comparing the rest of the tensions and in

fact it do es

A more systematic way to see how the elds and non p erturbative states of low energy string

theory emerge from d is to analyze the N sup ersymmetry algebra We

shall indicate how a couple of the features of string theory including the dierent string mo dels

their sp ectra and the duality transformations that relate them are understo o d to b e descendants

of the sup ersymmetry in Mtheory This reasoning gives a very comprehensive and systematic

treatment of separate issues in string theory So let us lo ok at the most general sup eralgebra in

eleven dimensions A given Ma jorana spinor sup er charge Q has real comp onents such

that fQ Q g is a symplectic S p matrix with indep endent entries Under

the Lorentz subgroup SO of S p it can b e decomp osed into irreducible representations

which are a vector the momentum a second rank tensor and a fth rank tensor The last two

are the p ossible central terms in the algebra which thus reads

C Z C Z fQ Q g C P

The terms are not central in the sense of central charges which we introduced in chapter

Those were only allowed for extended sup ersymmetry while the central terms here break the

Lorentz invariance If we take the tensor elds in the Lagrangian to have supp ort only in a par

ticular subspace the invariance is restored on the transverse co ordinates which is a hyperplane

worldvolume of a pbrane The presence of the charges then mo dies the eld equations of the

RR tensor elds by adding sources which are top ological in the sense that they are lo cally exact

forms They are vanishing unless the branes wrap around non trivial cycles in spacetime for

instance

Z

Z q dx dx

M

which is zero if the brane volume M a cycle is contractible By similar arguments as

we used in chapter one can now deduce that there can b e Mbranes and Mbranes as

presumably elementary and solitonic BPS states In fact one also susp ects dual Mbranes and

Mbranes to exist The Mbrane was the rst to b e established as a solution to the eld equa

tions in a similar manner as we followed in chapter There have also b een extensive

researches for its quantum theory the quantization of its co ordinates Also the Mbrane

is thought to b e rather well understo o d in terms of its worldvolume action One can for

instance give explicit formulas for the metrics and eld strengths and discuss the singularity

structures the masses and the charges which veries them to saturate the BPS mass b ound

The latter dual branes are still somewhat mysterious and are b eing under observation

The easiest way to get an impression what might b e happ ening when these Mtheory states

or b etter say d sup ergravity states are b eing compactied down to the critical string

dimension is again to lo ok at the dimensional reduction of the sup eralgebra

fQ Q g C P C P C Z C Z

Z Z C C

where the indices now run from to only We notice that the central terms in d originate

from the central terms of d as well as from the eleventh entry of the momentum All the

central terms we need as charges of the tensor elds in the d I IA sup ergravity are present

such that all the brane solutions we have constructed and conjectured in the previous chapters

could have b een foreseen from this simple analysis In particular we recognize our earlier state

ment that the states carrying KaluzaKlein momentum P in the compact direction from the

tendimensional p oint of view lo ok like Dbranes charged under the scalar central term of the

I IA sup er algebra The relation to I IB is a little more subtle It was however found that a torus

compactication of Mtheory leads to I IB compacied on a circle where the SL Z acting on

the complex mo dulus of the torus is exactly mapp ed to the I IB selfduality group that acts on the

complex combinations of the NSNS and RR scalars and forms Along similar lines one can

pro ceed further to recover more dualities of string theory By combining chiral comp onents of the

sup ercharges and after p erforming a chirality ip on one half of the comp onents one gets from

I IA to a chiral type I IB sup eralgebra of only say left handed sup ercharges This can b e related to

the Tduality of the two type I I string theories in d Also one can truncate the I IA theory

down to an N theory by a one sided parity op eration keeping only invariant sup ercharges

This then yields the sup eralgebra of the heterotic theory with all its central terms It also reveals

that these central terms involve only the sum P Z of momentum and top ological winding

charge therefore it is unaected by exchanging the two In this way the heterotic Tduality arises

very naturally from symmetries of the truncated d sup eralgebra which are invisible from

ten dimensions A large part of the web of string dualities has thus b een observed emerging from

the sup eralgebra of the elvendimensional ancestor

After having returned to the p erturbative Tduality we started with we like to stop our

journey at this p oint leaving all the more advanced topics to more sp ecialized reviews a couple

of which we have cited ab ove The most prominent omissions we have left out surely include the

developments initiated by which allowed the study of gauge theories via the Dbrane world

volume eective eld theories or via Mbranes alternatively Neither did we explore the ideas

concerning the more realistic non extremal and non sup ersymmetric black holes in detail and

completely omitted a discussion of the Maldacena conjecture of the duality b etween I IB string

theory on anti de Sitter space and an ordinary conformal eld theory on its b oundary

A Compactication on T and Tduality

In order to illustrate the statements of section we will now fo cus on the compactication of

a b osonic string on a twodimensional torus We have then four background elds three coming

from the metric G and one from the antisymmetric tensor B B spanning the classical

ij ij ij

mo duli space

SL R SO R SL R

H j H j M

T U class

SO SO U U

L R

T U

Here we have introduced the two complex mo duli

p

G detG

U U iU i H

G G

p

B i T T iT detG H

where U is the complex structure mo dulus describing the form of the torus and T is the Kahler

p

mo dulus det G gives the volume of the torus That means we represent the two dimensional

lattice in the complex plane It is p ossible to express the metric in terms of U and T as

T

U U U

G

U

U

One can also write p and p in terms of the mo duli

L R

jn U n T m U m j p

L

T U

n U n T m U m p

R

T U

where n n are the momentum numbers and m m the winding numbers The sp ectrum

is given in It can b e shown as p ointed out at the end of section that its symmetry

group is

I II

SO Z SL Z SL Z Z Z

Tduality U T

We will demonstrate that this is indeed a group of transformations under which the sp ectrum is

invariant To b e honest we will just fo cus on the p p part of eq The number

L R

op erators

X

j

i

N G B G B G

ij

LR

LRn

LRn

n

can b e shown to b e manifestly invariant under Tduality b ecause of the non trivial trans

formation of the oscillators which comp ensates the transformation of the metric That is

precisely the entire Tduality group relies on the general result stated at the end of section

which can also b e found in In fact the symmetry group contains one further element namely

symmetry under the worldsheet parity transformation implying B B The rst

SL Z in reects the fact that the target space is a two dimensional torus whose com

U

plex structure mo dulus always has an SL Z symmetry Not all values of U lead to dierent

complex structures but only those in the fundamental region M H S L Z see g the

thick lines of the b oundary b elong to the mo duli space the thin ones not The transformation

aU b

a b

with SL Z U

c d

cU d

10

This can b e checked after a straightforward but tedious calculation with the help of

r r r

r r r

r r r

Figure Dierent basis vectors can dene the same lattice

do es not change the complex structure of the torus and just amounts to another choice of basis

vectors for the same lattice see g This symmetry of the sp ectrum is classical in the

sense that it do es not need any sp ecial features of the string but just dep ends on a symmetry

of the target space The second SL Z is stringy from nature Its generators are as usual

T

T T which is a shift in B and T T which for B amounts to an inversion

of the torus volume Invariance under the rst transformation can b e understo o d from If

i j

B is constant the second term is a total derivative namely B X X and thus its

ij ij

contribution top ological An integer shift in B ie in general a shift by an antisymmetric matrix

with integer entries amounts to a shift of the action by an integer multiple of and therefore do es not change the path integral

U

1/2+ 3 /2 i

-1 -1/2 1/2 1

Figure Mo duli space of the complex structure mo dulus of a torus

If one considers the sp ecial case of a background with B G ie a lattice with

p

basis fR iR g leading to G R G R and det G R R we have U iR R and

p p

T iR R The element T T U U acts on the lattice according to R R

p p

R and R

I

The rst Z exchanges the complex structure and Kahler mo duli U T and is a two di

mensional example of mirror symmetry It is related to the Tduality of one of the circles making

up the torus This b ecomes clear if one lo oks again at the sp ecial case of U iR R and

p p

T iR R It translates to R R and R R The corresp onding Tduality for

I

the second circle is achieved by a comp osition of this Z transformation and twice the SL Z

U

transformation U U T T ie R R altogether this amounts to T U and

II

U T The second Z is given by T U T U which translates into B B and

G G It is an easy exercise to verify explicitly that the ensembles of all dierent values

p resp ectively p from are sep erately unchanged under the ab ove transformations

L R

and therefore the target space p erturbative sp ectrum is invariant Of course like in the one

dimensional case the single states are in general not invariant and winding and momentum num

b ers mix under the transformations To b e more precise the winding and momentum numbers

of the transformed states tilded quantities can b e expressed via the old ones according to table

Obviously if n m n m take all values of Z the same is true for n m n m The

Transformation n n m m

U n n m m

U

U U n n n m m m

m m n n T

T

T T n m m n m m

T U n m m n

U U T T n n m m

Table Transformation of winding and momentum numbers

invariance of the mass sp ectrum is of course only a necessary condition for the whole theory to

b e invariant under the Tduality group It is p ossible to show that the partition function is also

unchanged to all orders in p erturbation theory

Tduality is the remainder of a sp ontaneously broken gauge symmetry Only at sp ecial p oints

in the mo duli space it is partially or completely restored These p oints corresp ond to xed p oints

or higher dimensional xed manifolds of some of the symmetry transformations of the Tduality

group To illustrate this fact take the example of the circle compactication at the self

dual radius R where the gauge group is SU SU see b elow It can b e shown in this

x

case that there are nine massless scalars b esides the dilaton which transform as under

SU SU The comp onent can furthermore b e identied as the mo dulus R for the

radius of the compactication circle or to b e more precise for its dierence from the self dual

radius Moving away from the self dual radius amounts to giving an exp ectation value to R

and thereby breaking the gauge symmetry to the generic group U U However rotating

by around the axis in one of the SU s changes the sign of the comp onent This shows

that decreasing the value of the radius from the self dual value is gauge equivalent to increasing

it This fact survives the breaking of the gauge group in the form of Tduality

The gauge symmetry enhancement at sp ecial lo ci in the mo duli space happ ens of course also

I

in our two dimensional example The Z transformation ie T U has the xed line T U

In the sp ecial case of B G this amounts to choosing the self dual radius for R From

the exp erience with the circle compactication one therefore exp ects a symmetry enhancement

according to U U SU This actually happ ens for T U which can b e seen from

the formulas for the left and right momenta viewed as complex numbers namely

p

n T n T m T m p

L

T

p

p n T n T m T m

R

T

For B G we have T T iT and thus four additional massless vectors cf and

jp j N see table The last two columns are determined by the level matching condition

L L

jp j N If the oscillators carry indices of non compact dimensions the corresp onding vertex

R R

m m n n p p N N

L R L R

p

i

p

i

Table Additional SU gauge b osons

p p

op erators X exp ik X exp i X and X exp ik X exp i X represent

L R

four new gauge b osons which together with the generically for all values of the radius existing

gauge b osons X X exp ik X and X X exp ik X combine to the gauge elds

L R

of SU SU in the CartanWeyl basis This can b e checked directly by considering the

L R

currents

p p p p

p

cos X z exp i X z exp i X z j z

LR LR LR

p p p p

p

j z sin X z exp i X X z exp i z

LR LR LR

i

j z i X z

LR

One can verify that the algebra formed by their Laurent co ecients is given for the left resp

right moving currents sep erately by a so called level one SU KacMoody algebra

q

k l k lq k l

m i j j j

mn

mn m n

k

which reduces for the j elements to the usual SU Lie algebra the fact that we get a much

bigger innite algebra is of course due to the z dep endence of the currents in For more

details on this p oint see eg

X exp ik X is X exp ik X and X The U related to the vectors X

R L

enhanced at the self dual value for the radius R Since the Tduality transformation for this

circle is given by T U and U T see ab ove the xed p oint is T U Again

we have for B G four additional gauge b osons which enlarge the symmetry to an

SU SU namely

L R

m m n n p p N N

L R L R

p

p

We have used

p

n T m m p n

L

T

p

p n T m m n

R

T T

and T T If we satisfy b oth conditions U T and UT at the same time all the U s

are enhanced to give the gauge group SU This is obviously the case for T U i which

is also a xed p oint of the SL Z and SL Z transformations U U and T T

T U

II

and of the Z transformation T T U U This is a generalization of the circle compact

ication in the sense that we have compactied on two orthogonal circles with self dual radii

D D

On Ddimensional tori it is p ossible to get in a similar way the gauge group SU SU

L R

To get more general gauge groups we need a non vanishing B Like b efore the right choice

for T and U is a value that is invariant under a subgroup of It is easy to verify that

p

is invariant under T U T T U U and T U i

T T U U In this case the restored symmetry group is SU SU as we

L R

get twelve additional massless gauge b osons whose left resp ectively right momenta make up the

ro ot lattice of SU ie we have chosen the compactication torus to b e dened by the lattice

dual to the ro ot lattice of SU The new states are summarized in table It is again p ossible

to show that the internal parts of the corresp onding vertex op erators together with those of the

generically present gauge b osons generate a level one SU SU KacMo o dy algebra

L R

m m n n p p N N

L R L R

p

i

p

p

i

p

p

i

p

i

p

p

i

p

p

i

Table Additional SU gauge b osons

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Censor Phys Rev D

PK Townsend DBranes from MBranes Phys Lett B

A Strominger Open pBranes Phys Lett B

JL Cardy Operator Content of twodimensional conformally invariant Theories Nucl Phys

B

G T Horowitz A Strominger Counting States of Near Extremal Black Holes Phys Rev

Lett

W Nahm Supersymmetries and their Representations Nucl Phys B

E Witten String Theory Dynamics in various Dimensions Nucl Phys B

PK Townsend The elevendimensional Supermembrane revisited J Math Phys

PK Townsend MTheory from its Superalgebra hepth

PK Townsend Four Lectures on MTheory hepth

PK Townsend P Brane Democracy hepth

JA de Azcarraga JP Gauntlett JM Izquierdo PK Townsend Topological Extensions

of the Superalgebra for Extended Objects Phys Rev Lett N

E Bergsho e E Sezgin PK Townsend Supermembranes and dimensional Supergravity

Phys Lett B

B de Wit M L uscher H Nicolai The Supermembrane is Unstable Nucl Phys B

I Bandos K Lechner A NurmagambetovP Pasti D Sorokin M Tonin Covariant action

for the SuperFiveBrane of MTheory Phys Lett B

A Hanany E Witten Type IIB Superstrings BPS Monopoles and Three Dimensional Gauge

Dynamics Nucl Phys B

E Witten Solutions of FourDimensional Gauge Theories via M Theory Nucl Phys B

JM Maldacena The Large N Limit of Superconformal Field Theories and Supergravity

Adv Theor Math Phys

E Witten Anti de Sitter Space and Holography Adv Theor Math Phys

SS Gubser I Klebanov AM Polyakov Gauge Theory Correlators from Non Critical String

Theory Phys Lett B