niko jokela- stable and unstable d-branes

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Pro gradu -tutkielma

Teoreettinen fysiikka

Stable and Unstable D-branes

Niko Jokela

2004

Ohjaaja: Dos. Esko Keski-Vakkuri

Tarkastajat: Prof. Keijo Kajantie, Dos. Esko Keski-Vakkuri

HELSINGIN YLIOPISTOFYSIKAALISTEN TIETEIDEN LAITOS

PL 6400014 Helsingin yliopisto



Acknowledgments

I would like to express my greatest appreciation to the following people, who havehelped and accompanied me during my time as a M.Sc. student at University of Helsinki:

First, I sincerely acknowledge my supervisor, Esko Keski-Vakkuri, for motivatingand encouraging me to study this interesting subject. His enthusiasm and patiencemade working with him a pleasure.

Secondly, I would like to thank Esko Keski-Vakkuri, Matti Jarvinen and BrunoCarneiro da Cunha for careful reading through my thesis and for many valuablecomments they made.

I would like to express my appreciation to many people. I owe many thanks toMatti Jarvinen and Kalle Kytola for a number of motivating and fruitful discussionswhich took place during studies. They are my first true teachers in the art of science.I am especially grateful to my family and friends for their love and support. Thisthesis is dedicated to them.

Finally, the financial support that I have received from Helsinki Institute of Physicsis greatly appreciated.



Contents

1 Introduction 1

2 Basic Properties of D-branes 62.1 Open String Boundaries and Chan-Paton Factors . . . . . . . . . . . 62.2 T-duality with Chan-Paton Factors and Wilson Lines . . . . . . . . . 92.3 D-brane Dynamics and Gauge Symmetry Breaking . . . . . . . . . . 112.4 Ramond-Ramond Charges . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 D-branes as BPS States . . . . . . . . . . . . . . . . . . . . . . . . . 162.6 Anti-D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Classical p-brane Solutions 193.1 Type IIA Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Type IIB Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Singly Charged Extremal and Black p-branes . . . . . . . . . . . . . . 213.4 Region of Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Comparing Black-branes with D-branes . . . . . . . . . . . . . . . . . 30

3.5.1 T-duality Revisited . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Unstable D-brane Configurations 344.1 Sen’s Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Stable and Unstable Non-BPS D-brane Configurations . . . . . . . . 364.3 Static Properties of Unstable D-branes . . . . . . . . . . . . . . . . . 37

4.3.1 Non-BPS D-brane as Tachyonic Kink on the Brane-Anti-branePair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.2 The BPS D-brane as Tachyonic Kink on the Non-BPS D-brane 404.3.3 Descent Relations . . . . . . . . . . . . . . . . . . . . . . . . . 414.3.4 Descent Relations Extended to Bosonic String Theory . . . . . 41

4.4 Effective Field Theory Analysis . . . . . . . . . . . . . . . . . . . . . 45

4.4.1 Homogeneous Rolling Tachyon . . . . . . . . . . . . . . . . . . 464.4.2 Rolling Tachyons Coupled to U(1) Gauge Fields . . . . . . . . 48

5 Brane Decay 505.1 Time-dependent Solution . . . . . . . . . . . . . . . . . . . . . . . . . 505.2 Conformal Field Theory Description . . . . . . . . . . . . . . . . . . 515.3 Boundary State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4 Closed String Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 555.5 Identification of Emitted Closed String Modes . . . . . . . . . . . . . 59

5.5.1 Closed String UV limit: s → ∞ (t → 0) . . . . . . . . . . . . 595.5.2 Closed String IR limit: s → 0 (t → ∞) . . . . . . . . . . . . . 60

5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 61



1 Introduction

“The world is held up by the trunks of three giant elephants.”“What holds up the elephants?”“The elephants are standing on the shell of an even larger tortoise.”“What does the tortoise stand on?”“It stands on the shell of yet an even larger tortoise.”“What does...”“Sorry, it’s tortoises from here on.”

- Unknown

The standard model of particle physics is an extraordinarily successful theoryof the “fundamental” constituents of Nature, i.e., quarks, electrons and so forth.General relativity is a quite well tested description of the large scale structure of theuniverse. Inflationary cosmology has met with success in describing the origins of the universe. All of these stories are, however, incomplete. The standard model canonly be a correct description of physics up to energies well below the Planck scale,and contains 19 parameters as input. General relativity with standard quantum fieldtheory provides no good understanding of the black hole information paradox andinflationary cosmology does not explain the cosmological constant. Whether there is

a single underlying fundamental theory of Nature remains to be seen. Certainly thereis room for improvement.

It is expected that a more fundamental theory will require extreme modificationsin one’s perception of the world. Einstein’s introduction of spacetime as a dynamicalquantity, the idea that space and time together form a dynamical four-dimensionalmanifold, is today a common notion. However, at the time of its inception therewas scant evidence for such ideas. What this example teaches us is that intuitive,seemingly obvious, notions of space are often naive or simply wrong. Such misunder-standing can occur because the circumstances under which phenomena are observedoften give the illusion that the world is simpler than otherwise suspected. For exam-ple, moving at speeds significantly lower than that of light makes it very difficult tounderstand the intertwining of time and the space dimensions. Mathematical expla-nations of sub-luminal phenomena do not require time and space to be put on equalfooting.

One of the attempts at understanding some of the deeper mysteries of both par-ticle physics and gravity is superstring theory, or simply string theory. This theorychallenges two notions of standard quantum field theory and general relativity. First,string theory is rooted in the notion that at a sufficiently short length scale the elemen-tary constituents of Nature are no longer described by the physics of zero-dimensionalobjects, commonly referred to as point particles, but rather by the physics of one-dimensional objects or strings. Second, it requires that the total number of spacetime

dimensions is 9+1. These are radical suggestions, but not entirely unbelievable giventhe historical perspective. Even further, as we will learn, giving up the restriction topoint particles actually implies that objects of all dimensionalities need to be included.

Prior to 1995, string theory consisted of 5 different perturbative variants: TypeIIA, Type IIB, Type I, heterotic E 8×E 8 and heterotic SO(32). These theories live in

1



ten dimensions, and compactifications to lower dimensions produce further variants.The connections between different theories, known as string dualities, unified theseseemingly different perturbative vacua, by smoothly interpolating between them, bothat 10D and at lower dimensions.

We now want to take a step in understanding some non-perturbative aspects of string theory. This is the understanding of some of the solitons of the theory. Solitonsare typically nontrivial localized solutions of a theory. They are usually very heavyat weak coupling and can often be understood classically. An indication of a solitonis very often a (generalized) gauge field which couples to it. For example, in elec-tromagnetism there are magnetic monopole solutions, which are (singular) classicalsolutions of the theory and couple to the (dual) gauge field. As will be explained insection 2, Type II string theories contain a very important set of massless fields, theRamond-Ramond (RR) fields. Those are generalized gauge fields which, as we willsee, couple to Dirichlet branes, or D-branes. D-branes are thus playing a fundamentalrole in the theory.

The discovery of D-branes is a remarkable milestone. A D-brane is described as a(hyper)surface (for our purposes this surface is flat) on which open strings can end.Its tension is proportional to 1/gs, where gs is the closed string coupling constant. Soat weak coupling it becomes a very heavy object. A ( p + 1)-dimensional brane, called

a D p-brane, will couple to a ( p + 1)-form (gauge potential).The historical road to finding Dirichlet branes was through the low-energy de-

scription of the Type II string theories, the Type II supergravities (SUGRA), andthis is what we will study in section 3. We construct singular solutions which arecharged under the RR fields and which describe branes of different dimensions. Thesesolutions are known as extremal black p-branes. These p-branes are now understoodas spacetime counterparts of D p-branes. The p-brane solutions are “black” since theyare generalizations of black holes ( p = 0), i.e., they have horizons and singularities,and “extremal” since they have the following property which is typical for solitons.There exists an inequality giving a lower bound for the mass of the p-brane which isproportional to the magnitude of its charge. Such a bound is called a Bogomol’nyi-Prasad-Sommerfeld (BPS) bound. If this bound is saturated, the soliton is calledextremal or BPS.

In order to “trust” these classical supergravity solutions, the curvature has tobe very small (compared to the length of a typical string). This can be overcome,e.g., by putting a lot of charge at the location of the brane, which makes it veryheavy and suggests that the solution actually describes multiple p-branes sitting ontop of each other. This description has its own drawbacks, e.g. the perturbativecalculations in string theory in the background of such solitonic solutions are heavilyinvolved. However, Polchinski suggested that this is just another picture of D-branes,hyperplanes to which open strings are attached. Since D-branes are heavy objects

which carry charge, they curve spacetime around them. For small curvatures we canignore stringy effects and the theory is well approximated by classical supergravity.The equations we get are just the classical Einstein-Maxwell-dilaton equations andso the geometries are in a sense analogous to charged, extended black holes.

The second part of this thesis explores the instability of both supersymmetric and

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non-supersymmetric constructions using D-branes. In the bosonic string theory, thespectrum contains unexpected particles, tachyons, which are particles with negativesquared mass. These particles were considered unphysical because they will lead to anenergy unbounded from below in the theory. Thus tachyons have to be removed fromthe theory. In the superstring theory the tachyons are removed from the spectrumwith the GSO projection. In the last couple of years, the tachyonic fields have drawnattention since Ashoke Sen posted his famous conjecture of tachyon condensation.Sen’s conjecture of tachyon condensation is summarized in section 4.

Unstable D-branes are defined at the maximum point of the tachyon potential,the unstable vacuum. With a small perturbation, tachyonic fields will spontaneouslydecay to the minimum point, the stable vacuum, where the kinetic term of openstring mode vanishes. We then reach the stable closed string vacuum where openstrings excitations disappear from the spectrum. During this process, the rolling of the tachyon, an unstable D p-brane will decay to a stable lower dimensional D-braneor directly to the closed string vacuum with no D-branes. The energy density betweenthe two vacua is conjectured to equal the tension of the unstable D p-brane. Evidencefor this conjecture has been found in many recent investigations. In particular we canask, what happens if we displace the tachyon from the maximum of its potential andlet it roll? One framework where to study the rolling of the tachyon is the low-energy

effective field theory on the unstable brane. In section 4 we review the basic featuresof this approach.

Another method to study tachyon condensation is to use second quantized for-malism of string theories, i.e., string field theories (SFT). As in point-like particlephysics, quantum mechanics is a first quantized formalism which can only computeon-shell processes. To deal with off-shell processes such as pair creation, we have touse the second quantized formalism, i.e., quantum field theory. The traditional fivetypes of superstring theories mentioned earlier, are formulated in the first quantizedform. We can use the first quantized string theories to study some aspects of tachyoncondensation, but the full tachyon condensation is an off-shell process. SFTs auto-matically choose the right set of field variables around the tachyon vacuum, thus theyoffer a more suitable and nicer framework to study the complete picture of tachyoncondensation. The full SFT approach is however beyond the scope of this thesis.

We then explore some aspects of time-dependent solutions of string theory onan unstable D-brane describing the tachyon rolling away from the maximum of itspotential. Due to the presence of higher derivative terms in the SFT action it is nota priori clear how to construct such solutions. In section 5 we argue that it is indeedpossible to construct a family of such solutions of the SFT equations of motion labeledby the initial position and velocity of the tachyon field. Furthermore, we explicitlywrite down the boundary conformal field theory (BCFT) action and the boundarystate for an unstable D p-brane associated with these solutions.

The analysis of the boundary state is a topic on its own, hence we just summarizesome essential properties relevant for the computation of scattering amplitudes of closed string modes from the decaying brane. We furthermore limit our considera-tions to bosonic string theory, where D p-branes for all values of p are unstable. Theassociated boundary state shows that if we push the system towards the minimum

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of the tachyon potential, its energy-momentum tensor evolves smoothly towards anasymptotic configuration. Instead, if we push the system in the other direction, inwhich the potential is unbounded from below, the energy-momentum tensor quicklyhits a singularity. This aspect is absent in the supersymmetric system.

In closed string theory amplitudes are evaluated through a perturbative sum overclosed Riemann surfaces of various topology. The genus of these surfaces correspondsto the number of closed string loops: a diagram associated with a sphere is a tree levelcontribution, a torus diagram corresponds to one-loop contribution, etc. A surfaceof genus g is weighted with a power g2g−2

sof the string coupling. In presence of one

or more D-branes this sum is extended to Riemann surfaces with boundaries, with achoice of boundary conditions for the worldsheet fields at each boundary. DifferentD-branes correspond to different boundary conditions. Each boundary added to thesurface is weighted with another factor of gs.

While D-branes are non-perturbative objects in closed string theory, for smallstring coupling constant gs they admit a simple description in perturbation theoryas “extra objects” inserted in the closed string theory. Most of the work on tachyoncondensation has been performed in the case with vanishing string coupling, gs = 0.In particular, as gs → 0 the tachyon becomes decoupled from the bulk closed stringmodes. In this limit, and at large distance scales, we then expect the effective field

theory on the brane to be a good approximation. Then, the endpoint of the rollingtachyon is a somewhat mysterious substance called “tachyon matter”. However, thecoupling to the bulk strings cannot really be neglected, as they are the natural modesinto which the initial energy of the unstable brane should decay. Consideration of thefull problem with gs = 0 would presumably eliminate the need for mysterious tachyonmatter.

As gs increases D-branes become lighter and more dynamical. In the remainderof section 5 we are interested in computing at tree level (g0

s) the emission of closedstring modes from decaying D-branes, in which interactions between emitted closedstrings may be neglected. The result for emitted energy is divergent depending on thenumber of dimensions transverse to the decaying unstable brane: for unstable D p-branes with p ≤ 2 there is a need to invoke a cutoff to get a finite result. If we cut off the energy at E ∼ 1/gs we see that we get an energy of the same order of magnitudeas the initial mass of the brane. Of course the emitted energy cannot be greater thanthe mass of the initial brane, so higher order corrections must cut it off so that thetotal emitted energy is finite. Energies of order E ∼ 1/gs are precisely the energieswhere we expect the tree level computation to become invalid, since at this energythe gravitational force between the emitted state and the initial D-brane become of order 1. In any case, unstable brane decay should presumably be a physically smoothprocess for gs finite. We furthermore approximate the average transverse velocity of emitted closed string massive modes and argue that the emitted closed string massive

modes slowly move apart from the initial unstable D-brane. It is proposed that thetachyon matter may be a possible description of the remainder of D-brane decay, i.e.,it may be identified with a collection of emitted closed string massive modes of highdensity.

Although we review several key results in describing the actual decay of unstable

4



D-branes, the approach is still in its infancy and many questions remain open. Weexpect the study of unstable branes to continue to be an active and fascinating areaof research.

5



2 Basic Properties of D-branes

Until recently, the formulation of string theory was only perturbative with Feynmandiagrams corresponding to surfaces. As is also the case for many quantum fieldtheories, this perturbation expansion is only asymptotic, so that the theory was atbest incomplete. There was also no a priori reason to assume that the string couplingconstant, which controls the expansion in terms of surfaces, was small. Furthermore,there seemed to be five independent perturbative critical string theories, that differeddramatically in their basic properties, such as worldsheet geometry, gauge groups andsupersymmetries.

However, we have witnessed dramatic developments bringing for the first timenon-perturbative questions into reach. Crucial in all this has been the concept of string duality. String duality is the statement that all five different perturbativestrings are related and are just expansions of one single unified theory around differentbackgrounds. Some of these duality symmetries are non-perturbative, in the sensethat the string coupling gs in one theory is related to the inverse string coupling1/gs in the dual theory. This means that calculations in one particular superstringtheory at large coupling constant can be translated to different calculations in anothersuperstring theory at small coupling constant. A key role in these dualities is played

by D-branes.In this section we study some basic properties of D-branes. For more detailedanalysis see, e.g., [1–4] and/or the books [5,6] and references therein.

2.1 Open String Boundaries and Chan-Paton Factors

One of the major advances of the second superstring revolution (1995-2002) was thestudy of the boundary conditions of open strings. Let us say that the endpoints of anopen string can propagate freely through all nine spatial dimensions. These are calledNeumann boundary conditions. However, the endpoints of an open string could alsobe fixed, and not free to move. These are called Dirichlet boundary conditions. The

endpoint of an open string could be fixed in some dimensions and not in others. Forinstance, it could be fixed in seven of the nine spatial dimensions, and free to move inthe other two. It would then be free to slide around on a two-dimensional plane. Thattwo-dimensional surface can then be considered a physical object called a Dirichletmembrane, or D-brane. If the endpoints of an open string have p Neumann boundaryconditions, the string therefore has 9 − p Dirichlet boundary conditions, and ends ona D p-brane1.

To parametrize the open string worldsheet, let the “spatial” coordinate run 0 ≤σ1 ≤ π. Consider fixing the boundary conditions at both ends, σ1 = 0, π, but sepa-rately for each X µ. We shortly comment on relaxing this condition in subsection 2.2.

1We could think of a string as simply a D1-brane, although we should remember that D-branesare defined in terms of strings ending on them, whereas strings are postulated from the beginningas the fundamental objects of the theory (F-strings). Furthermore, it is found that the tensions of F-string and D1-brane differ.

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=1,...,pα

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Bulk spacetime

Dp−brane

a=p+1,...,D−1

X0

Figure 1: A D p-brane in a D-dimensional spacetime. The worldvolume extends in

X 0

, . . . , X p

. This leaves X p+1

, . . . , X D−1

as transverse coordinates. Open stringendpoints are stuck on the D-brane, whereas closed strings can propagate in thebulk.

Neumann boundary conditions

∂ nX µ|σ1=0,π = 0 , (2.1)

where ∂ n is the derivative normal to the boundary, respect translational invarianceand are hence momentum conserving. Dirichlet conditions

δX µ|σ1=0,π = 0 , (2.2)

on the other hand, break translational invariance and describe spacetime defects. Astatic defect extending over p flat spatial dimensions is described by the boundaryconditions

∂ nX α=0,...,p = X a= p+1,...,9 = 0 , at σ1 = 0, π , (2.3)

which force open strings to move on a ( p + 1)-dimensional (worldvolume) hyperplanespanning the dimensions α = 0, . . . , p. Since open strings do not propagate in thebulk in Type II theory, their presence is intimately tied to the existence of the defect,which we will refer to as D p-brane, see figure 1.

The Dirichlet condition specifies a spacetime location on which a string ends.It is natural to think of open strings as always ending on D-branes, so for Neumannboundary conditions, we imagine the string ending on a D-brane that fills all of space,so the string can be located anywhere.

7



For simplicity, let us now consider the open bosonic string. Recall that bosonicstring theory is 1 + 25-dimensional and the mass-shell condition is

m2 = −kµkµ =1

α

∞

n=1

nN n − 1

, (2.4)

where N n is the number of excited oscillators in the nth mode, and −1 is the zeropoint energy of the 24 transverse physical bosons. Following the convention in theliterature, we refer to the level of the state by N = ∞

n=1

nN n. We will use the letterN to denote other things as well, but it should be clear from the context what itstands for.

We now consider multiple coincident D-branes2. In this case open strings havefreedom to end on different branes. We denote this by adding non-dynamical degreesof freedom at each of its two endpoints. A ground state of a generic open string state,which is defined to be annihilated by the lowering operators and to be an eigenstateof the center-of-mass momentum, now has the form

|k; ij , (2.5)

where the additional indices i and j incorporate the information that the string

stretches from D-brane j to D-brane i. Now let us consider a general non-oscillatingopen string state, which we denote by

|k; a . (2.6)

It can consist of a linear combination of states which start and end on different branes.A basis for a general state is then given by

|k; a =N

i,j=1

|k; ijλaij . (2.7)

The N ×N matrices λaij have to be Hermitian, to ensure that the amplitudes of stringinteractions are still unitary. If we choose the normalization Trλaλb = δab, they forma representation of a U (N ) gauge group. The matrices λa

ij are called the Chan-Paton (CP) factors of the string. Thus a U (N ) gauge group is associated with the endpointsof the string with i transforming according to the fundamental representation while

j transforming according to the complex conjugate representation3:

|i = U ii|i ; | j = | jU † jj . (2.8)

From (2.8) we see that matrices λaij transform according to the adjoint represen-

tation (N × N ) of U (N )

λaij → U iiλ

aijU † jj = (UλaU †)i j . (2.9)

2More precisely, the worldvolume of every coincident D-brane is the same subspace of the wholespacetime, that is, D-branes are located at the same point along transverse dimensions.

3Sometimes this is called antifundamental representation.

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Now let us consider N D25-branes, for simplicity. A D25-brane fills space, so thestring endpoint can be located anywhere. The lowest excited states of the string areobtained by exciting one of the n = 1 modes once, resulting to a massless state (2.4):

αµ−1|0; ijλij ; m2 = 0 . (2.10)

This state transforms as a vector under the spacetime Lorentz transformations andcan therefore be regarded as the vector boson of the U (N ) gauge symmetry on theD-brane.

2.2 T-duality with Chan-Paton Factors and Wilson Lines

We observed in the previous subsection that open strings satisfying Neumann bound-ary conditions in all the directions can be thought as being attached to a space-fillingbrane that is a D25-brane in the bosonic string4. We now want to consider the effectof compactification in the presence of Chan-Paton factors. The action of T-duality5

on an open string theory consists in transforming Neumann boundary conditions intoDirichlet ones. As a consequence, starting with a space-filling brane through a T-duality transformation we can obtain an arbitrary D p-brane. In this subsection wesee that a consistency between T-duality and the freedom to add background gauge

field requires the existence of D-branes. For the sake of simplicity we compactify justone coordinate X 25 ∼ X 25 + 2πR. Furthermore, we turn on a background U (N ) fieldwith constant, non-zero component A25,

A25(X µ) =1

2πRdiag(θ1, . . . , θN ) = −iU −1 ∂U

∂X 25(2.11)

U (X 25) = diag(eiX25θ1/2πR, . . . , eiX25θN /2πR) , (2.12)

which, as we will see, amounts to determine the location of D p-branes along compacti-fied dimension. Locally we could always gauge A25 away, but the gauge transformationis not periodic in the case of compact coordinate, thus we get nonzero Wilson lines

eiR 2πR0 A25dx25 = diag(eiθ1 , . . . , eiθN ) . (2.13)

Remember from (2.8) that a string with Chan-Paton state |ijλij has charge −1under U (1) j and +1 under U (1)i and is neutral under the remaining ones, so a paralleltransport around the compact coordinate transforms the open string state (2.7) as

|k; a =N

i,j=1

ei(θi−θj)|k; ijλaij . (2.14)

So a translation of 2πR acts both on the string and the gauge degrees of freedom thatare located at its endpoints. Requiring that this combined action leaves the stateinvariant:

e2πiRk25ei(θi−θj)|k,ij = |k,ij , (2.15)

4A D9-brane in the superstring.5For a review, see [7].

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we obtain a general result for the momentum of the state

k25 =n

R+

θ j − θi

2πR, (2.16)

where n/R is called the Kaluza-Klein momentum and is well-known from field theorywith compact dimensions. Open strings cannot wind around periodic dimension,hence they have no quantum number comparable to w, the winding number of thestring. So what is the interpretation of θ j − θi?

We can see this in the T-dual picture,

R ↔ R =α

R, n ↔ w , (2.17)

in the presence of the Wilson lines, where we get

X 25(π) − X 25(0) = −(2πn + θ j − θi)R ∼ −(θ j − θi)R . (2.18)

This means that the open string is stretching between two D p-branes whose coordi-nates are θiR

and θ jR. Comparing to equation (2.11) we see that

θiR = 2πα(A25)ii , θ jR = 2πα(A25) jj , (2.19)

so we can conclude that turning on U (N ) Wilson lines in a theory of open stringsalong a compactified direction corresponds, in the T-dual theory, to introducing N D p-branes located respectively at

X 25 = −2πα(A25)11 , . . . , −2πα(A25)NN . (2.20)

Note that if several coordinates X a = X 25, X 24, . . . , X p+1 are periodic and werewrite the periodic dimensions in terms of the dual coordinate, this whole procedureworks similarly. The open string endpoints are then confined to N ( p+1)-dimensional

hyperplanes. The Neumann conditions on the worldsheet, ∂ nX a

(σ1

, σ2

) = 0, havebecome Dirichlet conditions, δX a(σ1, σ2) = 0, for the dual coordinate. Since T-duality interchanges the Neumann and Dirichlet boundary conditions, a further T-duality in a direction tangent to a D p-brane reduces it to a ( p − 1)-brane, while aT-duality in an orthogonal direction turns it to a ( p + 1)-brane.

Note that simple T-duality explained above gives parallel D-branes all with thesame dimension. After having accepted that D-branes are natural objects to consider,it is natural to study more general configurations. Let us shortly comment on havingtwo D-branes, D p and D p’, each parallel to the coordinate axes, but taking p = p’.Now there are four possible different sets of open string boundary conditions for eachspatial coordinate X i, namely NN (Neumann at both ends), DD, ND and DN. But

what really matters is the number ν of ND plus DN coordinates; a T-duality canswitch NN and DD, or ND and DN, but ν is invariant6. The D p-D p’, p = p’, systemis discussed in detail in [8] and it is extended to D-branes at angles in [9].

6In Type II ν has to be even since, as we will see, we have p even or p odd in a given theory.

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2.3 D-brane Dynamics and Gauge Symmetry Breaking

Up to now we have treated a D p-brane as a pure geometrical hyperplane to whichopen strings are attached and we have completely disregarded the excitations of theattached open strings. It is natural to expect that this hyperplane is dynamical ratherthan rigid. For one thing, the theory naturally has a graviton, and it is difficult to seehow a perfectly rigid object could exist. Rather, we would expect the hyperplanes tofluctuate in shape and position as dynamical objects. This is actually what we foundout in equation (2.20) for a Wilson line, which was a constant gauge potential. From

point of view of the worldvolume, with ξα the p + 1 coordinates on the D-branes’worldvolume Σ( p+1), the scalar fields X a(ξα), a = p + 1, . . . , D − 1, vary as we movearound on the worldvolume of the D-brane and describe small fluctuations of theD-brane. This therefore embeds the brane into a variable place in the transversecoordinates. If we assume the D-brane to be approximately flat, and is close to thehypersurface X a = 0, a > p, we can take a static gauge choice X α = ξα. This issimply describing the specific shape to the brane as it is embedded in spacetime.

From now on, we focus on the case of superstrings and work in ten spacetimedimensions. As is explained in section 2.1, D-branes in Type II string theory canbe introduced as open string boundaries. Open strings are then constrained to liveon the D-branes, whereas the Type II closed strings can propagate in the bulk. Themassless open string modes on a D p-brane in Type IIA or IIB superstring theorydescribe a ( p + 1)-component gauge field Aα, 9 − p transverse scalar fields X a, anda set of massless fermionic gaugino fields. As discussed before, the scalar fields X a

describe small fluctuations of the D-brane around a flat hypersurface. If the D-branegeometry is sufficiently far from flat, it is useful to describe the D-brane configurationby a general embedding X µ(ξ), where X µ are ten functions giving a map from Σ( p+1)

into the spacetime manifold R9,1. Just as the Einstein equations which govern thegeometry of spacetime arise from the condition that the one-loop contribution tothe closed string beta function vanishes, a set of equations of motion for a generalD p-brane geometry and associated worldvolume gauge field can be derived from a

calculation of the one-loop open string beta function [10]. These equations of motionarise from an effective action, the classical Dirac-Born-Infeld (DBI) action, describinga single D p-brane:

S = −T p

d p+1ξe−φ

− det(Gαβ + Bαβ + 2παF αβ ) + S WZ + fermions , (2.21)

where G, B and φ are the pullbacks of the ten-dimensional metric, antisymmet-ric tensor and dilaton to the D-brane worldvolume, while F is the field strength of the worldvolume U (1) gauge field Aα. S WZ represents the Wess-Zumino (or Chern-Simons) terms, which we will discuss shortly after we have first learned that D-branescarry charges.

The action (2.21) can be verified by a perturbative string calculation [8], whichalso gives a precise expression for the brane tension

T p =1

gs

√α

1

(2π√

α) p∼ 1

gs, (2.22)

11



where gs = eφ is the closed string coupling constant, equal to the exponential of thevacuum expectation value of the dilaton. The proportionality (2.22) clearly indicatesthat the D p-branes, the solutions of (2.21), are non-perturbative configurations of string theory; however, they are of a non-standard type, since their tension scaleswith the inverse power of the coupling constant, while typical solitonic solutions7 arecharacterized instead by the inverse square of the coupling constant. At weak couplingthese solitonic solutions become massive, so we expect that they curve spacetime.Corresponding solutions are discussed in detail in section 3.

If we want to study dynamics of more than one D p-brane, we expect to encountermany difficulties along the way. To keep the discussion elementary, we just considerthe following simple example to illustrate the effect of gauge symmetry breaking.Consider two parallel D p-branes, separated by some distance Y = (Y p+1, . . . , Y 9)along the compact directions. Consider the open string state

|k,ijλij , i, j = 1, 2 . (2.23)

If the branes reside at the same point (|Y |2 = 0), we obtain an enhanced symmetryas new degrees of freedom arise from massless strings that can stretch between thetwo branes. If the Chan-Paton degrees of freedom are contracted with a Hermitian

matrix, these matrices transform in the adjoint representation of U (2). Therefore theeffective action (the analogous non-Abelian version of (2.21) describing several D p-branes) on the worldvolume of the brane is a supersymmetric U (2) gauge theory. Weobserve that the complete multiplet transforms in the adjoint representation. Thisimplies that the scalars which correspond in the string theory picture to the transversecoordinates of the brane become matrix valued.

Now we want to relax the condition of coincidence and move the branes apart. Thestrings |k,ii which start and end on the same brane remain massless, while strings|k,ij ,i = j which stretch between the branes obtain a mass m2 ∝ T |Y |2, where T is the string tension. Strings stretching between the branes are charged under theU (1) gauge fields living on each brane. The worldvolume vector bosons, defined in

(2.10), with charge (±1, 1) obtain a mass, breaking the U (2) gauge symmetry downto U (1) × U (1). The scalars and fermions in the (12)+(21) sectors obtain the samemass.

For definiteness, we take two D3-branes at the same point. The spectrum consiststhen of a vector multiplet of d = 4, N = 4 supersymmetry, consisting of a vector, sixscalars and their fermionic partners, all transforming in the adjoint representation of U (2). If we now separate the branes, two of the four vector multiplets become massive.Because of the conserved U (1) charge, the multiplets from the strings between thebranes consist of one massive vector, five scalars and their massive fermionic partnersgiving 24 states in all, which is exactly the same number of degrees of freedom as fora massless multiplet. If we repeat the above analysis for more than two branes, thesame mechanism applies. Generally, if from a stack of N branes we move away M branes, we break the U (N ) gauge theory down to a U (N − M ) × U (M ) gauge theory.

7Soliton is defined as time-independent, non-singular, localized solution of classical equation of motion with finite energy in a field theory.

12



D-brane scattering, the velocity-dependent forces between D-branes, has been thecase under great interest. This can be analyzed by calculating the semi-classical phaseshift for two moving external sources. Detailed study for two identical D p-branes inNeveu-Ramond-Schwarz formulation8 can be found in [11]. The same results can beobtained in light-cone boundary state formalism in [12–14], and can be furthermoreextended to different D-branes [15], non-vanishing worldvolume fields [16–18], orbifoldbackgrounds [19], Type I theory [20], and to study spin-dependent interactions [21,22].Furthermore, the question of what is the effective size of D-branes as measured byother D-branes, instead of elementary strings, has been studied in [11,15,23–27] andlength scales shorter than the string length [28] were found in cases where the couplingis weak, and the heavy D-branes move very slowly. Thus, using D-branes as probesreveals new aspects of their substructure, but the necessary methods is a topic onits own, and hence not covered here. Instead, we wish to conclude the discussion of D-branes’ thickness, the physical size of the Dirichlet p-branes in Type II theories,by noting that it is usually determined by standard particle physics methods, i.e.,by scattering off them massless string states, is a quantity of order string lengthls9 [33, 34] which increases with the energy of the probe, thus exhibiting the Regge

behavior [35–38].

2.4 Ramond-Ramond Charges

So far we have discussed only open strings. In this section we address the case of theclosed string sector of Type II superstrings. Both Type IIA and Type IIB superstringshave identical fields in the NSNS (Neveu-Schwartz-Neveu-Schwartz) sector, but theclosed string spectrum differs in the RR (Ramond-Ramond) sector. The key propertyof the D-brane is that it is a source of charge of the RR field.

The states of the closed string are given by the tensor product of the left- andright-moving worldsheet sector. At the massless level each sector contains a ten-dimensional vector and a ten-dimensional Majorana-Weyl spinor. This is depicted asfollows:

(|µNS ⊕ |aR)left ⊗ (|ν NS ⊕ |bR)right , (2.24)

where µ, ν = 0, . . . , 9 are vector indices and a, b = 1, . . . , 16 are spinor indices. Bosonicfields from the NS sector thus include a two-index tensor, which can be decomposedinto a symmetric traceless (graviton Gµν ), a trace (dilaton φ) and an antisymmet-ric (Bµν ) part. In addition, massless bosonic fields include a RR bispinor F ab. Thebispinor field F can be decomposed in a complete basis of all gamma matrix anti-

8Historically, it was also called the spinning string theory. In this formalism, the supersymmetryis on the two-dimensional worldsheet as it propagates through spacetime. A different formalism (GSformalism), developed by Michael Green and John Schwarz in 1981, has the supersymmetry on the10-dimensional spacetime itself. These two formalisms have been shown to be equivalent to each

other.9The D(-1)-brane, or Dirichlet instanton, defined by Dirichlet conditions in the time direction as

well as all spatial directions [29–32], is a special case exhibiting a point-like structure.

13



symmetric products

F ab =10

p=0

i p

p!F µ1...µp(Γµ1...µp)ab . (2.25)

Here Γµ1...µp ≡ 1 p!

Γ[µ1 · · · Γµp], where square brackets denote the alternating sum over

all permutations of the enclosed indices10.In view of the decomposition (2.25), the Ramond-Ramond massless fields are

a collection of antisymmetric Lorentz tensors, where the rank of these tensors is

constrained due to definite chirality projections:

F = F Γ11 = ±Γ11F . (2.26)

The choice of sign +1 , − 1 distinguishes the Type IIA and Type IIB models, respec-tively. It follows from (2.26) that only even- p (odd- p) terms are allowed in the TypeIIA (Type IIB) case. Furthermore the antisymmetric tensors obey the Hodge dualityrelations

F µ1...µp =µ1...µ10

(10 − p)!F µ10...µp+1 , or equivalently F ( p) = ∗F (10− p) , (2.27)

the generalization of the electromagnetic duality.The mass-shell conditions imply that the bispinor field obeys two Dirac equations

/ pF = F / p = 0 . (2.28)

After some algebra we find the Bianchi identity and free massless equation for anti-symmetric tensor field strength in momentum space:

p[µF ν 1...ν p] = pµF µν 2...ν p = 0 (2.29)

or equivalentlydF ( p) = d

∗F ( p) = 0 . (2.30)

Solving the Bianchi identity, we can express the p-form field strength as the exteriorderivative of a ( p − 1)-form potential

F µ1...µp =1

( p − 1)!∂ [µ1Aµ2...µp] , or F ( p) = dA( p−1) . (2.31)

Thus the Type IIA theory has a vector (Aµ) and a three-index tensor potential(Aµνρ), in addition to the non-propagating zero-form field strength (F (0)), while theType IIB theory has a zero-form (A), a two-form (Aµν ) and four-form (Aµνρσ) po-tential, the latter with self-dual field strength. From Hodge duality (2.27) it follows

that, the ( p + 1)-form “electric” potential can be traded for a (7 − p)-form “magnetic”potential, obtained by solving the Bianchi identity of the dual field strength.

10We use conventions, where ten-dimensional gamma matrices are purely imaginary and obey thealgebra Γµ,Γν = −2ηµν . The chirality operator is Γ11 = Γ0 . . .Γ9, Majorana spinors are real andthe Levi-Civita tensor density 01...9 = 1.

14



The natural objects coupling to a ( p+1)-form are p-branes, extended objects with p spatial dimensions. In string theory these are exactly the Dirichlet p-branes. Thuswe conclude that in order to RR fields of the closed strings to couple to the potentialspresent in the different Type II theories, the Type IIA theory must contain D p-branesfor p = 0, 2, 4, 6, 8 and Type IIB must contain D p-branes for p = −1, 1, 3, 5, 7, 9.

We found out that A( p+1) couples naturally to a D p-brane, that is an object witha ( p + 1)-dimensional worldvolume, the coupling taking the form

Σ(p+1)

A( p+1)

, (2.32)

where Σ( p+1) is the worldvolume of the D p-brane. The corresponding charge in tenspacetime dimensions would then be11

QE =1

2κ210

S 8−p

∗dA( p+1) , (2.33)

where S 8− p is a (hyper)sphere surrounding the D p-brane, in transverse space. Themagnetically dual object to the D p-brane would then be an object coupling to a(7

− p)-form, i.e., a D(6

− p)-brane with magnetic charge

QM =1

2κ210

S p+2

dA( p+1) . (2.34)

Note, however, that a D p-brane is charged not only under the RR ( p + 1)-form,but also under other lower-degree RR forms, if the worldvolume gauge bundle is non-trivial. We can see this by looking at the full set of Wess-Zumino terms mentionedin subsection 2.3, up to normalization,

S WZ ∼ Σ(p+1) p

A( p+1) ∧ Tr

eB+2παF

∧ G( p+1) , (2.35)

where the trace is taken in the N -dimensional representation of U (N ). The geometricpart of Wess-Zumino action, i.e., terms describing the coupling of a D p-brane to thebulk RR fields, can be written as an expansion in even powers of the curvature two-form,

G =

A( N )A(T ) = 1 − (4π2α)2

48[ p1(T ) − p1( N )] + · · · , (2.36)

where T and N are the tangent and normal bundles of the D-brane, A is theappropriately-normalized Dirac “roof genus”, and p1 is the first Pontrjagin class12.The next term in expansion (2.36) is an eight-form, and to our knowledge, it has not

been checked if it enters in the full effective action describing the dynamics of theD p-branes in curved backgrounds.

11Notice that in this normalization it is actually the charge density.12See [39] or [6] for definitions.

15



For instance, we may consider the Wess-Zumino couplings for a D3-brane

S WZ ∼ Σ(4)

A(4) +

Σ(4)

A(2) ∧ TrF +

Σ(4)

A(0) ∧ TrF 2 . (2.37)

Consider a worldvolume gauge bundle with non-zero first Chern class, i.e., TrF isnon-trivial on the D3-brane worldvolume. The above couplings imply that the D3-brane is charged under the RR 2-form A(2), or that we are dealing with a boundstate of a D3-brane and a D1-brane. In a sense, the system can be thought of as a

D3-brane with a D1-brane diluted in its worldvolume. More generally, a non-trivialTrF on a general D p-brane induces D( p−2)-brane charge, a non-trivial second Chernclass (or instanton number) TrF 2 induces non-trivial D( p − 4)-brane charge, etc [40].For the rest of this thesis we are interested in D p-branes that are charged only underRR ( p + 1)-form.

2.5 D-branes as BPS States

An important feature in theories with extended supersymmetry (SUSY), i.e., theorieswith N > 1 supercharges QA=1,..., N , is the fact that the energy spectrum is boundedfrom below. In these supersymmetric theories an important class of states with massM are those which saturate the BPS bound [41] which, for point-like states, is of theform

M ≥ |zi| for all i = 1, . . . , N /2 , (2.38)

where zi are the elements in the central charge matrix Z IJ . We can choose a basis inwhich Z IJ takes a standard form [2]:

Z IJ =

0 z1−z1 0

0 z2

−z2 0

. . .. . .

0 z N /2−z N /2 0

, (2.39)

where the zi can be chosen to be real and non-negative. To simplify the discussionwe have assumed that N is even, but one should keep in mind that when N is odd,there is a 1 × 1 zero block in the above normal form of Z IJ .

The bound (2.38) represents a lower bound on the mass of a state with respectto the central charges of the extended supersymmetry algebra. It is an unavoidable

consequence of having a unitary representation of the supersymmetry algebra. There-fore, provided that supersymmetry is not broken quantum mechanically, the boundwill be maintained.

Suppose that some of the zi saturate the bound (2.38): zi = M , for i = 1, . . . , k ≤ N /2. This implies that 2k of the generators act trivially and hence can be taken to

16



zero. The remaining 2 N − 2k SUSY generators obey a Clifford algebra whose uniqueirreducible representation has dimension 22 N−2k, and is called BPS representation .These massive multiplets are also known as short multiplets. Notice that the smallestrepresentation occurs when all central charges saturate the bound, in which case allthe generators can be taken to be zero. Then we are left only with 2 N states, just asis the case of a massless multiplet. These multiplets are called ultrashort .

For example, for N = 2 there is only one charge z = z1. If z < M the massivemultiplet contains 24 = 16 states, whereas if z = M the ultrashort multiplet containsonly 22 = 4 states. For

N = 4, compare to the discussion in subsection 2.3, there

are two charges zi. If both zi < M , the massive multiplet contains 28 = 256 states,whereas if both zi = M , the ultrashort multiplet contains only 24 = 16 states. Andfinally if exactly one of the zi = M , we are left with 26 = 64 states.

There are a number of important and interesting properties of BPS states for N = 2 [42], which is the case we are interested in Type II string theories:

• They break half the supersymmetry: half of the supercharges annihilates them,the other half does not.

• The force between such BPS states is zero.

• The spectrum of masses and charges of BPS states in a given theory is exact: itmay be computed at weak coupling, and there is a non-renormalization theoremwhich protects this spectrum from corrections, to all orders in perturbationtheory.

The BPS states described above can be realized in Type II string theories aspoint-like soliton solutions of the relevant effective supergravity theory. There are,however, BPS versions for extended objects (BPS p-branes). In the presence of theextended objects, supersymmetry algebra can acquire central charges that are notLorentz scalars. Their general form can be obtained from group theory, in which casewe find that they must be antisymmetric tensors. Such central charges have values

proportional to the charges of p-branes. Then the BPS condition would relate thesecharges with the p-brane tensions of the relevant p-branes. Such p-branes can beviewed as extended soliton solutions of the effective field theory.

Due to Polchinski [1] we are led to accept that D-branes describe non-perturbativeextended BPS states of the Type II string carrying non-trivial RR charge. We cansee this partial supersymmetry breaking by considering a scattering process with aclosed string scattering from a D-brane. Computing the amplitude of this processrequires a worldsheet with a boundary. Now the open string boundary conditions areonly invariant under D = 10, N = 1 supersymmetry13: the left-moving worldsheetcurrent for spacetime supersymmetry J α(z) flows into the boundary and the right-moving current J α(z) flows out of it. If we denote the conserved charges by Qα

and Qα, corresponding to the currents J α(z) and J α(z), respectively, only the sumof the charges Qα + Qα of left- and right-moving currents is conserved. Under T 9-duality this becomes Qα + (P 9Q)α, where P 9 denotes the action of T 9-duality on the

13Note that now there are no central charges.

17



spinorial supersymmetry charges. Therefore, if a closed string scatters from a D-brane, only half of the supersymmetry is conserved. The same happens for the caseof more than one T -dualized direction. Since a D-brane conserves exactly half of thesupersymmetry it is a BPS state.

D-branes are BPS states if their tensions [43] saturate the bound (2.38): theybreak half the supersymmetries and have zero force between each other [2].

2.6 Anti-D-branes

In analogy with particles and anti-particles in quantum field theory, every object insuperstring theory has the corresponding anti-object, with equal tension but oppositecharge. In particular, for every D p-brane there exists a corresponding anti-D p-branestate, denoted by D p -brane, such that when they are put together they can annihilateeach other into the vacuum. So D p -branes and D p-branes have the same tension butopposite charges under RR ( p + 1)-form. This implies that D p -branes are also BPSstates, preserving half of the supersymmetries of the vacuum, but they preserve thehalf broken by D p-branes, and vice versa.

The supersymmetric D-branes are always oriented. Even the spacetime-fillingD-branes are oriented; the spacetime-filling anti-D-branes are their mirror image.

Although we cannot “rotate” the D-brane inside the space - if it already fills thewhole space at the beginning - its volume-form still carries a sign. Bosonic D-branesare unoriented, since they are not carrying any corresponding conserved charge.

D p -branes are oppositely oriented with respect to D p-branes due to the oppositecharge. Imagine again particles and their anti-particles with non-zero charge, e.g.,electrons. The positron can be viewed as the electron that propagates backwards intime: the orientations of the worldline of an electron and of a positron (anti-electron)are opposite. This is a way to describe the anti-particles that can be generalized tohigher dimensional branes. An anti-brane is a brane whose volume-form (the sign) isinverted.

18



3 Classical p-brane Solutions

After having discussed D-branes from open string viewpoint, we now study what kindof source they provide for closed strings in the low-energy limit, α → 0, of superstringtheory, the supergravity. The Type IIA and Type IIB supergravity theories haveblack-brane solutions [44]. These are extended objects with horizons and singularities.Some of these solutions carry RR charges and in the extremal limit preserve half of the supersymmetries of spacetime. We already established to all orders in stringperturbation theory that D-branes are the basic sources of the RR fields. It is naturalto suppose that there is some kind of connection between BPS D p-branes and extremalblack p-branes. From historical point of view, p-branes were first thought to be theextended objects needed for string dualities before the D-brane idea came, but nowthey are realized as long-range spacetime description of the D-branes. Thus, the D-brane description of the black-branes in terms of boundary conditions of open stringsgives us a powerful tool to analyze the behavior of the black-branes. It has been usedto calculate the black hole entropy in various dimensions [45–50]. There are excellentreviews in the literature [51].

In this section we try to present the simplest brane configurations that arise asclassical solutions of the field equations from the low-energy string effective actions.

We will limit our considerations to flat backgrounds and Type II string theories. Wemainly follow the discussion in [5]. More extensive and complete discussion can befound in, e.g., [42, 52,53].

3.1 Type IIA Supergravity

We discussed earlier the massless contents of Type II string theories. They naturallyinclude graviton and at low energies they give supergravities as effective theories. Atlow energies only the massless modes are relevant, massive modes decouple and donot appear in the low-energy effective theory14. Both IIA and IIB supergravities aredefined in ten dimensions and have 32 real supercharges corresponding to

N = 2

supersymmetry in D = 10. In Type IIA, the two supersymmetries have oppositechirality, while in Type IIB they have the same chirality. For simplicity, we will limitour considerations to purely bosonic sector of the Type II supergravity actions.

The massless bosonic content of Type IIA string theory consists of a graviton(Gµν ), an antisymmetric two-index tensor (B

(2)µν ), a dilaton (φ), a vector (A

(1)µ ) and

an antisymmetric tree-index form (A(3)µνρ). In the string frame, the dynamics of these

14A consistent truncation to massless modes is defined to be the one for which all solutions of the truncated theory are solutions of the original theory. It requires that the suppressed fields mustnever appear linearly in the action, otherwise setting them to zero would result in further constraintson the massless modes over and above the equations of motion of the massless theory.

19



fields is described by the action:

S IIA =1

2κ210

d10x e−2φ

√−GR(Gµν )

+

e−2φ(4dφ ∧ ∗dφ − 1

2H (3) ∧ ∗H (3)) − 1

2F (2) ∧ ∗F (2)

− 1

2F (4) ∧ ∗F (4) − 1

2B(2) ∧ F (4) ∧ F (4) , (3.1)

where R(Gµν ) denotes the Ricci scalar constructed from metric Gµν , which is com-monly called the sigma model metric or string metric. Further, in expression (3.1)κ10 is the gravitational constant in ten dimensions given by15

κ10 = 8π7/2α2gs , (3.2)

where the closed string coupling constant is the exponential of the vacuum expectationvalue of the dilaton, gs = eφ. Hence it is dynamically set, not a real free parameter.In this sense, the dilaton φ which appears explicitly in the action (3.1) is describingthe fluctuation around the vacuum expectation value. Furthermore

H (3) = dB(2) , F (2) = dA(1) , F (4) = dA(3) , F (4) = F (4) + A(1) ∧ H (3) .(3.3)

It is convenient to rewrite the action (3.1) in the Einstein frame to remove the dila-ton factor from the curvature term and to avoid mixed graviton-dilaton propagators.This is simply done by the redefinition of the metric tensor:

Gµν (string frame) = eφ/2gµν (Einstein frame) . (3.4)

After some algebra we find the effective action of the Type IIA string in the Einsteinframe,

S IIA =1

2κ210

d10x

√−gR(gµν )

− 1

2

dφ ∧ ∗dφ + e−φH (3) ∧ ∗H (3) + e

32φF (2) ∧ ∗F (2)

+ e12φF (4) ∧ ∗F (4) + B(2) ∧ F (4) ∧ F (4)

. (3.5)

3.2 Type IIB Supergravity

The massless bosonic content of the Type IIB string theory consists of a graviton(Gµν ), an antisymmetric NSNS two-index tensor (B

(2)µν ), a dilaton (φ), a zero-form

15Defined this way, Newton’s gravitational constant GD in D dimensions is related to κD by16πGD = 2κ2Dg

−2s , so G10 = 23π6α4g2s .

20



(A(0)), an antisymmetric RR two-index form (A(2)µν ) and a four-form (A

(4)µνρσ). In the

string frame we have the effective action:

S IIB =1

2κ210

d10x e−2φ

√−GR(Gµν )

+

e−2φ(4dφ ∧ ∗dφ − 1

2H (3) ∧ ∗H (3)) − 1

2F (1) ∧ ∗F (1)

−1

2˜

F (3)

∧ ∗˜

F (3)

−1

4˜

F (5)

∧ ∗˜

F (5)

−1

2A(4)

∧ H (3)

∧ F (3)

, (3.6)

where

H (3) = dB(2) , F (1) = dA(0) , F (3) = dA(2) , F (5) = dA(4) (3.7)

andF (3) = F (3) + A(0) ∧ H (3) , F (5) = F (5) + A(2) ∧ H (3) . (3.8)

We note that the self-duality constraint

∗F (5) = F (5) (3.9)

has to be imposed only at the level of the field equations and not inside the action.In other words, the field equations that follow from (3.6) are consistent with theself-duality of F (5), but they do not imply it.

In the Einstein frame (3.6) becomes

S IIB =1

2κ210

d10x

√−gR(gµν )

− 1

2

dφ ∧ ∗dφ + e−φH (3) ∧ ∗H (3) + e2φF (1) ∧ ∗F (1)

+ eφF (3) ∧ ∗F (3) + 12 F (5) ∧ ∗F (5) + A(4) ∧ H (3) ∧ F (3)

. (3.10)

3.3 Singly Charged Extremal and Black p-branes

We wish to look for solutions to the (bosonic) supergravity action (3.5) or (3.10)which contains only the metric, gµν , the dilaton, φ, and one of the antisymmetrictensors, say the ( p + 1)-form potential. It can be shown [42, 52] that the fields wekeep are not sources for the fields that are eliminated. We are thus left with thefollowing action

S n = 12κ2

10

d10x √−gR(gµν ) − 1

2

dφ ∧ ∗dφ + e−aφF (n) ∧ ∗F (n)

, (3.11)

where F (n) is the field strength of the chosen ( p + 1)-form potential, n = p + 2 and acan be read from the action (3.5) or (3.10). In particular,

21



• if we choose as the potential the NSNS two-form B(2), then p = 1, n = 3, anda = 1.

• if we choose the potential A( p+1) from the RR sector, then p = 0, 2, . . . in TypeIIA and p = −1, 1, . . . in Type IIB, n = p + 2 and a = ( p − 3)/2.

Varying the action (3.11), we obtain the classical field equations for the metric,dilaton and the antisymmetric potential,

Rµν −1

2gµν R = T µν (3.12)

∂ ν √−ge−aφF (n)νµ2...µn

= 0 (3.13)

1√−g∂ µ√−ggµν ∂ ν φ

= − a

2n!e−aφ(F (n))2 , (3.14)

where the energy-momentum tensor T µν is given by

T µν =1

2

∂ µφ∂ ν φ − 1

2gµν ∂ λφ∂ λφ

+

1

2n!e−aφ

nF

(n)µλ2...λn

F (n)λ2...λnν − 1

2gµν (F (n))2

.

(3.15)

We can also recast (3.12)-(3.14) in the forms

Rµν =

1

2∂ µφ∂ ν φ +

1

2n!e−aφ

nF (n)µλ2...λnF

(n)νλ2...λn

− n − 1

D − 2δµν (F (n))2

(3.16)

0 = ∂ ν √−ge−aφF (n)νµ2...µn

(3.17)

2φ = − a

2n!e−aφ(F (n))2 . (3.18)

Note that the statement that the n-form derives from a potential, can be replaced byimposing the Bianchi identities:

∂ [µ1F µ2...µn+1] = 0 , (3.19)

at the level of equations of motion (and not inside the action).The four sets of equations above (3.12)-(3.14), (3.19) let us present the explicit

form of the p-brane solution with p even in IIA and p odd in IIB. The name “ p-brane”actually covers a great deal of different objects. Most generally, it should be used toindicate, in the context of a theory containing gravity, a classical solution which isextended in p spatial directions, i.e., it has p spacelike translational Killing vectors.Black holes and, extrapolating to theories without gravity, magnetic monopoles areprototypes of p-branes for the case p = 0. As they are classical solutions, p-branes areoften called “solitons”. This is essentially in view of the role they play in quantum

theory which is behind the classical theory of which they are solutions.To describe a ( p + 1)-dimensional extended object we rearrange

D = d + (D − d) , (3.20)

22



where d ≡ ( p + 1) is the number of dimensions tangential to the p-brane, split theD = 10 spacetime coordinates into

xα=0,1,...,d−1 longitudinal coordinates

ya=d,...,D−1 transverse coordinates ,

and make henceforth the following ansatze:

•we require Poincare invariance in the d longitudinal directions; and

• we require rotational invariance in the remaining (D − d) transverse directions.

The most general metric that is compatible with these requirements can therefore bewritten as:

ds2 = eA(r)ηαβ dxαdxβ + eB(r)δabdyadyb , (3.21)

where A and B are some arbitrary functions depending only on the radial coordinater =

δabyayb in the transverse space. By requiring that the spacetime should be

asymptotically Minkowski, we have eA(r), eB(r) → 1 as r → ∞.Before presenting the ansatz for the field strength we calculate the Ricci tensor,

Rµν , and the Ricci scalar, R = gµν Rµν , for an arbitrary p-brane. First recall thedefinition:

Rµν = Rρµρν = ∂ ρΓρ

µν − ∂ ν Γρµρ + Γσ

µν Γρσρ − Γσ

µρΓρσν , (3.22)

where the connection Γρµν is defined as follows:

Γρµν =

1

2gρσ[∂ µgσν + ∂ ν gσµ − ∂ σgµν ] . (3.23)

By inserting (3.21) into (3.23) we find, after some algebra16,

Γγ αβ = Γb

aα = Γαab = 0

Γaαβ = −12eA−Bηαβ δab∂ bA

Γβ aα =

1

2δβ α∂ aA

Γcab =

1

2(δc

b∂ aB + δca∂ bB − δabδcd∂ dB) ,

and furthermore inserting these into (3.22), we find the Ricci tensor

Rαa = 0 (3.24)

16Notation: ∂ aA(r) = ∂r∂ya

∂A∂r = ya

r A.

23



Rab = −1

4δab

2B + (4(D − d) − 6)

B

r+ (D − d − 2)B2 + 2d

A

r+ dAB

− 1

4

yayb

r2

2d(A − A

r− AB) + dA2 − (D − d − 2)B2

+ 2(D − d − 2)(B − B

r)

(3.25)

Rαβ = −14

eA−Bηαβ

(D − d − 2)AB + 2A + 2(D − d − 1) A

r+ dA2

,(3.26)

and the scalar curvature

R = −1

4e−B

4d[A +

A

r(D − d − 1)]

+ 4(D − d − 1)[B +B

r(D − d − 1)]

+ d(d + 1)A2 + (D−

d−

2)(D−

d−

1)B2 + 2d(D−

d−

2)AB .(3.27)

If we require our solution to preserve half of the supersymmetry [54] we can assumethat the functions A(r) and B(r) can be written in terms of one function H (r) withdifferent exponents:

eA(r) = H (r)α , eB(r) = H (r)β . (3.28)

Using (3.28), (3.25)-(3.27) read

Rab = −1

4δab

1

H (2βH +

βH

r(4(D − d) − 6) +

2dαH

r)

+βH 2

H 2(−2 + β (D − d − 2) + dα)

− 1

4

yayb

r2

1

H (2dα + 2β (D − d − 2))(H − H

r) +

H 2

H 2

dα(α − 2β − 2)

− β (β + 2)(D − d − 2)

(3.29)

24



Rαβ = −1

4H α−β ηαβ

αβH 2

H 2(D − d − 2) + 2α(

H

H − H 2

H 2)

+2αH

rH (D − d − 1) +

dα2H 2

H 2

(3.30)

R = −1

4H −β

4

H (dα + (D − d − 1)β )(H +

H

r(D − d − 1))

+ H

2

H 2

dα((d + 1)α + 2(D − d − 2)β − 4)

+ (D − d − 1)β ((D − d − 2)β − 4)

. (3.31)

We proceed in solving the supergravity field equations by making an “electric”type ansatz for the antisymmetric potential:

A( p+1)01...p = eC (r) − 1 . (3.32)

For the dilaton we can simply take φ = D(r). The so far arbitrary functions H (r),

C (r) and D(r) are then uniquely determined by inserting all the ansatze into (3.12)-(3.14) and solving the resulting differential equations [42, 52, 53]. In this fashion weend up with the following results:

ds2 = H (r)−(7− p)/8ηαβ dxαdxβ + H (r)( p+1)/8δabdyadyb (3.33)

e2φ = H (r)(3− p)/2 (3.34)

A( p+1) = (H (r)−1 − 1)dx0 ∧ . . . ∧ dx p , (3.35)

where the prefactor17 H (r) is given by

H (r) = 1 +L

r7− p

(3.36)

with the characteristic length scale, L, defined by

L7− p =2κ10

(7 − p)Ω8− p(√

π(2π√

α)3− p) . (3.37)

In this expression we have denoted by Ωn the area of the unit n-dimensional sphereS n, i.e.

Ωn =2π(n+1)/2

Γ n+12 . (3.38)

We note that L is proportional to √α = ls, hence it has the correct dimension of alength.

17This is often also called the warp factor .

25



In the string frame the metric becomes rather simple-looking and it reads

ds2 = H (r)−1/2ηαβ dxαdxβ + H (r)1/2δabdyadyb , (3.39)

while all the other fields remain as before.One of the far-most reaching results of this subsection is to provide us a formula for

the tension of a p-brane or, more generally, of a configuration of p-branes giving riseto a metric like (3.33). Such a formula could be obtained using the ADM (Arnowitt-Deser-Misner) formalism (see, e.g., [55]) and it is derived for p-branes in [56]. For the

asymptotically flat metric of the form (3.33), it turns out that the ADM tension canbe simply read off the asymptotic behavior of the harmonic function H (r). Thus,from (3.36) and (3.37) we obtain [56]

T p = −Ω8− p

2κ210

limr→∞

r8− p∂ rH (r)

=

√π(2π

√α)3− p

κ10=

1

gs

√α

1

(2π√

α) p∼ 1

gs. (3.40)

To calculate the electric charge (2.33) we use Gauss’s law and find

Q p =

√π(2π

√α)3− p

κ10. (3.41)

Using this explicit expression for the p-brane charge, we have2κ2

10Q pQ6− p = 2π . (3.42)

This is a generalization of Dirac’s quantization condition [57,58] of the electric chargefrom which we can deduce that a p-brane and a (6 − p)-brane are electromagneticallydual to each other.

For the most general p-brane, for a given tension T p and “electric” charge Q p of anantisymmetric tensor field, there is a unique classical solution. If T p > Q p, then thesolution appears to have a horizon, as we approach it from the transverse directions.Therefore, this particular p-brane solution is called a black p-brane. When the tensionequals the charge, it is called an extremal black p-brane.

Comparing (3.41) with (3.40), we find that our solution is the extremal one (thisis related to the assumption (3.28) that the solution should preserve half of the su-persymmetry) and therefore saturates the BPS bound (2.38)

T p = Q p , (3.43)

which indicates that there is an exact cancelation between the attractive force of theNSNS fields due to the tension T p and the repulsive Coulomb-like force of the RRpotential due to the charge Q p. This precise cancelation of forces implies that we canpile up these extended objects on top of each other to form macroscopic configurations.Furthermore, it should be clear that there is a multicentre generalization of this p-

brane solution to represent N different branes located at arbitrary positions given bythe vectors ri, where we write for the harmonic function

H (r) = 1 +N i=1

L7− p

|r − ri|7− p . (3.44)

26



The function H (r) essentially represents the gravitational potential produced bythe branes. Therefore, for coincident branes

H (r) = 1 + N · L7− p

r7− p, (3.45)

the potential produced by stack of N branes is simply N times the potential producedby a single brane. Furthermore, the tension and the charge of the whole configurationis N times (3.40) and (3.41), respectively.

3.4 Region of Validity

As the distance r to the brane goes to zero, the metric apparently has a singularity.The effective curvature of the spacetime increases as it approaches this value, whichwe will consider to be the horizon of the black-brane. Thus the solution we have, saysnothing about the interior of the black-brane. We note that the dilaton backgrounddeviates from its value at asymptotic infinity except for the case of 3-branes. Thiswill be analyzed further below.

Several features of this solution are:

• For asymptotic region r L, H (r) → 1, the metric approaches that of Min-kowski.

• For intermediate region r < L, the deviation from flat metric is rapid.

• For throat region r → 0, the metric has an apparent singularity.

• For gsN < 1 we have L <√

α. This means that the characteristic length scaleof the solution is less than the string scale. Therefore we do not expect thelow-energy solution we have found to be a good approximation.

•For gsN > 1, the characteristic length scale is greater than the string scale. This

means that the geometry is expected to be smooth on the string scale, and sothe low-energy black p-brane solution we have found is a good approximation.

The calculation of the Ricci scalar, i.e. inserting d = p + 1, D = 10 with α =−(7− p)/8, β = ( p + 1)/8 with the function H (r) = 1 + N L7− p/r7− p into (3.31), gives

R = −( p + 1)(3 − p)(7 − p)2

32N 2L2(7− p)

1 + N

L7− p

r7− p

−( p+3)/8

r−2(8− p) . (3.46)

The implication of this equation is that the singularity at r = 0 in the metric is acoordinate not a spacetime singularity only for the case of 3- and 7-branes. This in

turn means that these branes are non-singular inside the horizon, since R = 0 for allr. The behavior for Ricci scalar (3.46) for different values of p is depicted in figures2 and 3, where the characteristic length scale is scaled to unity and the range of r ischosen around that value.

27



1,61 1,41,20

-0,5

-1

-2,5

-3

r

1,8

R

-1,5

-2

-3,5

2

p=0

p=1

p=2

Figure 2: Scalar curvature for p = 0, 1, 2.

In perturbative string theory with N coincident p-branes, gsN is the expansionparameter. Therefore, for small values of the coupling, gsN , perturbation theory can

be used, whereas for large values of gsN we cannot use the perturbative descriptionof string theory. Putting things together, we find that there is a strong-weak duality:in the small gsN regime, the characteristic length scales are very small, and we canconsider the perturbative string description with a collection of N weakly coupledRR D-branes. In the strong coupling regime, the characteristic length scale is largeand we do not require the string theory description (which is not available anyway);we can instead use the low-energy black p-brane description.

To be more concrete, we limit our considerations to p = 3. For N coincident3-branes, it is convenient to redefine the characteristic length scale by L ≡ 4

√NL, so

L4 = N 2κ10

√π

4Ω5= 4πg

sNα2 , (3.47)

28



r

0,1

0,3

0,2

108642

0,7

0,6

0,5

0,4

0

R

p=4

p=5

p=6

p=8

Figure 3: Scalar curvature for p = 4, 5, 6, 8.

thus the solution for N coincident 3-branes becomes

ds2 = H (r)−12 ηαβ dxαdxβ + H (r)

12 δabdyadyb (3.48)

eφ = 1 (3.49)

A(4) = (H (r)−1 − 1)dx0 ∧ . . . ∧ dx3 , (3.50)

with the function H (r) given by

H (r) = 1 +L4

r4. (3.51)

Note that now Einstein and string frames are equivalent due to constant value of thedilaton.

Now let us consider the detailed form of the metric (3.48) in both asymptoticregimes. For r L we can approximate H (r) by unity and the metric reduces tothat of flat ten-dimensional Minkowski spacetime

ds2 ηαβ dxαdxβ + δabdyadyb . (3.52)

Near the branes, i.e., for r L, we have a different scenario. In this case theharmonic function H (r) ≈ L4/r4, so that the metric (3.48) reduces to

ds2

r2

L2

ηαβ dxαdxβ +

L2

r2

δabdyadyb . (3.53)

29



Next we write (3.53) in terms of spherical coordinates in the transverse space

δabdyadyb = dr2 + r2dΩ25 , (3.54)

and define z = L2/r so that the metric is rewritten as

ds2

r2

L2

ηαβ dxαdxβ +

L2

r2

dr2

+ L2dΩ2

5 (3.55)

= L2

z2

(ηαβ dxαdxβ + dz2)

+ L2dΩ25 . (3.56)

This is one of the standard forms of the metric of five-dimensional anti-de Sitterspacetime times the five-dimensional sphere, AdS 5 × S 5, both with radius L.

We have found that 3-branes are classical non-perturbative solutions that inter-polate between

• the flat Minkowski spacetime in ten dimensions for r L; and

• the AdS 5 × S 5 spacetime for r L.

The geometry of AdS 5 × S 5

has been studied in the context of AdS/CFT cor-respondence, also known as Maldacena’s conjecture, which states that the Type IIBstring in an AdS 5 × S 5 background is dual to the N = 4 superconformal Yang-Millstheory in a flat four-dimensional Minkowski spacetime in the strong coupling limit.Detailed analysis on this subject is, however, beyond the scope of this thesis and thuswe refer to reviews [59–61].

3.5 Comparing Black-branes with D-branes

We have seen that in some sense the D-branes are the most interesting and intriguingconfigurations of string theory. D-branes can be understood from many points of

view. We have discussed branes in two very different fashions:

1. The ten-dimensional Type IIA and IIB supergravity theories each have a set of ( p + 1)-form fields A

( p+1)µ1···µ(p+1)

in the supergraviton multiplet, with p even/oddfor Type IIA/IIB supergravity. These are the Ramond-Ramond fields in themassless superstring spectrum. For each of these ( p + 1)-form fields, there is anon-trivial solution of the supergravity field equations that is invariant under( p + 1)-dimensional Poincare transformations, and which has the form of anextremal black-brane solution in the (9 − p) spatial directions that are notaffected by these Poincare transformations. These “black p-brane” solutionscarry charge under the RR fields A( p+1), and are BPS states in the supergravity

theory that preserve half the supersymmetry of the theory.

2. From a string-theory point of view branes are drastically different. Indeed, aD p-brane is a ( p +1)-dimensional extended object in the ten-dimensional space-time defined by the distinctive property that open strings can terminate on it.

30



In other words, a D p-brane is a hypersurface spanned by open strings with Di-richlet boundary conditions in the (9 − p) transverse directions and Neumannboundary conditions along directions parallel to the brane worldvolume (includ-ing time). When p is even/odd in Type IIA/IIB string theory18, the spectrumof the resulting quantum open string theory contains a massless vector Aα,α = 0, 1, . . . , p and scalars X a, a = p + 1, . . . , 9 from NS sector19. These fieldscan be associated with a gauge field living on the hypersurface and a set of de-grees of freedom describing the transverse fluctuations of the open string, whichdescribe a fluctuating ( p + 1)-dimensional hypersurface in spacetime. Thesebranes are called Dirichlet branes or simply D-branes.

The remarkable insight of Polchinski in 1995 [1] was the observation that thestable Dirichlet branes of superstring theory carry Ramond-Ramond charges, andtherefore should be described in the low-energy supergravity limit of string theoryby precisely the black p-branes discussed in 1. We would like to emphasize that thistwofold interpretation of the D-branes is a direct consequence of a duality betweenopen and closed strings which allows a double interpretation of the annulus/cylinderdiagram. The exchange of all closed-string modes, including the massless graviton,dilaton and ( p + 1)-form, is given in diagram of figure 4. Viewed as an annulus, this

same diagram also admits a dual and, from the field theory point of view, surprisinginterpretation: the two D-branes interact by modifying the vacuum fluctuations of (stretched) open strings, in the same way that two superconducting plates attract bymodifying the vacuum fluctuations of the photon field.

Figure 4: Two D-branes interacting through the exchange of a closed string. Thediagram has a dual interpretation as a Casimir force due to vacuum fluctuations of

open strings.

18We discuss the notion of “wrong-dimensional” branes in IIA/IIB in section 4.219The R sector contributes the supersymmetric partners.

31



This twofold nature of the D-branes is their most important feature; indeed be-cause of this they play a crucial role both from a gauge field theory point of view(i.e., in a theory of open strings) and from a gravitational point of view (i.e., ina theory of closed strings). This open/closed string duality is at the heart of thegauge/gravity correspondence which has recently been uncovered since Maldacena’swell-known conjecture [61,62].

3.5.1 T-duality Revisited

We found out that elementary Dirichlet p-brane solutions are characterized by a singlefunction H p that depends on the 9 − p transverse coordinates and is harmonic withrespect to these variables, i.e., δab∂ a∂ bH p(r) = 0. The metric (3.39), for all values of

p, 0 ≤ p ≤ 9, is written compactly as:

ds2 = H −1/2 p ds2 p+1 + H 1/2 p ds29− p . (3.57)

Recall that a D p-brane in ten dimensions is a state which satisfies Dirichlet bound-ary conditions for the 9 − p transverse directions and Neumann boundary conditionsfor the p + 1 worldvolume directions. Since under T-duality, Dirichlet and Neumannboundary conditions are interchanged, it follows that all D p-branes are T-dual ver-

sions of each other. It is natural that this T-duality is also realized on the p-branesolutions and indeed this has shown to be the case for all values of p [63].

Recall that T-duality relates solutions of Type IIA to solutions of Type IIB. Itis a symmetry of string theory, not of supergravity. But what is to follow, we justneed to work on the level of metric. Hence, as it can be shown, the only non-trivialT-duality rule involving the metric is given by

gxx = 1/gxx , (3.58)

where x labels the isometry direction over which the duality is performed. We as-sume here that the harmonic function H p is independent of the particular transversedirection which is dualized and therefore we can write

H p = H p+1 . (3.59)

Clearly, under this duality transformation the metric of a D p-brane becomes that of a ( p + 1)-brane if the duality is performed over one of the transverse directions of the

p-brane. In other words, one of the transverse directions of the p-brane has become aworldvolume direction of the ( p +1)-brane. But let us illustrate this effect concretely.Recall from subsection 2.2 that T-dualizing in a direction transverse to a D p-branemeans that we must place a set of branes on a circle of radius R and find an equivalentrepresentation for the system on a dual circle of radius α/R. Thus, let us take aninfinite array of identical branes on a line x p+1 separated with a distance 2πR. Then

we make an identification: x p+1 x p+1 + 2πR. Now the radius r can be written interms of a radius in the direction transverse to the D p-brane (in terms of x p+1) anda radius in the remaining directions:

r2 = (x p+1)2 + (x p+2)2 + . . . + (x9− p)2 = r2 + (x p+1)2 . (3.60)

32



Now we can write the harmonic function H p, which is an appropriate function repre-senting an infinite array of D p-branes (3.44) and including all of the images, as

H p = 1 +∞

n=−∞

L7− p p

|r2 + (x p+1 − 2πnR)2|(7− p)/2 . (3.61)

For small R, we can replace the sum with an integral, and after a substitutiontr = 2πnR − x p+1, rdt = 2πRdn, we get

H p 1 + L7− p p

2πRr6− p

∞

−∞

dt(1 + t2)(7− p)/2

= 1 + L7− p p

2πRr6− p√πΓ[12(6 − p)]

Γ[12(7 − p)]. (3.62)

Now plugging in the expression for L p from (3.37) we have established the relation(3.59),

H p 1 +

√α

R

L7−( p+1) p+1

r7−( p+1)= H p+1 , (3.63)

where the number of coincident branes N p+1 =√

α/R which can be realized by fixingR. That is, if we would like to have a single brane on the dual side, we set R =

√α.

Conversely, a ( p + 1)-brane becomes a p-brane if the duality is done over one

of the worldvolume directions of the ( p + 1)-brane. In order to establish a dualitybetween the two solutions, we assume that after T-duality the harmonic functionH p+1 becomes dependent on this particular worldvolume direction. In this sense, wewrite

H p+1 = H p . (3.64)

Strictly speaking the T-duality rules can only be applied as solution generatingtransformations to construct ( p + 1)-brane solutions out of p-brane solutions and notthe other way around [6]. Therefore, a 0-brane leads, via T-duality, to all other D-brane solutions. A priori it is not guaranteed that one can apply T-duality also as asolution-generating transformation to construct p-brane solutions out of ( p+1)-brane

solutions. In the case of D-branes, T-duality does generate new dual solutions, butthis is not true for other configurations [64, 65]. To see this, we start with the 9-branesolution written as

ds2 = H −1/29 ds210 , (3.65)

where H 9 is a constant, related to the ten-dimensional Minkowski spacetime volume.To obtain the 8-brane solutions, we dualize in one of the worldvolume directions, say,x9, and assume that after duality H 9 becomes dependent on x9, i.e., H 9 = H 8. Wethus obtain the 8-brane solution. Similarly, we can obtain all other Dirichlet branes.We conclude that not only does the 0-brane, via T-duality, lead to all other Dirichletbranes with 0 < p ≤ 9, but also the 9-brane lead, via T-duality, to all remaining

Dirichlet p-branes with 0 ≤ p < 9. However, it is not clear to us, how to correctlyget an increase of the power of r. We could integrate the expression for H p+1 over r,but the physical interpretation would then correspond to “smearing” the brane in thedirection which is T-dualized, i.e., integrating against a uniform density of branes.Such “brane distributions” are discussed in, e.g., [6].

33



4 Unstable D-brane Configurations

We now turn to the subject of tachyons. Certain D-brane configurations are unstableagainst decay to the vacuum, both in supersymmetric and non-supersymmetric stringtheories. This instability is manifested in the spectrum of open strings that endon the D-brane as a state with m2 < 0, a tachyon. For a standard quantum fieldtheory analogue consider, e.g., the Higgs field if we perturb around the wrong vacuumφ = 0. There the presence of a tachyon usually means that we are perturbingaround an unstable vacuum. In a physically sensible situation we expect the systemto roll down to some stable vacuum where the tachyon will disappear automatically.In the case of string theory this is a more involved question since it is difficult tostudy string theory off-shell. The proper framework to address this issue is stringfield theory. However, in this thesis we mainly consider first-quantized, on-shell,formalism, and often use bosonic theory as a toy model to illustrate the more realisticcase of superstrings.

In this section we list some elementary D-brane configurations where tachyonsarise and outline the strategy to study the case of rolling of tachyons to their minima,the tachyon condensation, see [66–97]20.

4.1 Sen’s Conjectures

Consider first the open bosonic string. By recalling the mass formula from (2.4)

m2 =1

α

∞

n=1

nN n − 1

, (4.1)

we find in the lowest level the tachyon: m2 = −1/α < 0.The existence of a tachyonic mode in the open bosonic string indicates that the

standard choice of perturbative vacuum for this theory is unstable. In the early daysof this subject, Bardakci and Halpern suggested that the tachyon could condense,

leading to a more stable vacuum [99, 101]. Despite of the fact that Kostelecky andSamuel argued [67] early on that this stable vacuum could be identified in string fieldtheory in a systematic way, the physical picture was not clear. In 1999, Ashoke Senreconsidered the problem of tachyons in string field theory. He suggested that theopen bosonic string should really be thought of as living on a space-filling D25-brane.He suggested further that the condensation of the tachyon should correspond to thedecay of the D25-brane. In particular, he made three conjectures:

1. The difference in the potential between the unstable vacuum and the perturba-tively stable vacuum should be the tension of the D25-brane.

2. Lower dimensional D-branes should be realized as soliton configurations of thetachyon and other string fields.

After condensation there is no brane left on which open strings could end, hence:

20For early work on tachyon condensation, see [98–101].

34



3. The perturbatively stable vacuum should correspond to the closed string vac-uum. In particular, there should be no physical open string excitations aroundthis vacuum.

There are analogous conjectures in superstring theory [71], e.g., for the D9-branein Type IIA. Open SFTs are appropriate to verify Sen’s conjectures. Indeed, we onlyneed the potential to verify the first conjecture. This kind of calculation fits very wellto the framework of effective field theory.

The other conjectures are harder to prove, since the dynamics of the tachyon on

a D-brane cannot be studied without taking into account its coupling to an infinitenumber of other fields living on the D-brane. This is associated with the infinitenumber of states of the open string. Thus the classical dynamics of the D-brane isgoverned by a coupled system of an infinite number of equations of motion for theinfinite number of fields, i.e., we arrive into string field theory. Therefore, the conjec-tures involving tachyon condensation should be interpreted as conjectures involvingproperties of the solutions of the infinite number of coupled equations. So, the goalwould then be to take a precise formulation of SFT, translate the conjectures into aset of precise questions about the classical SFT equations of motion, and then try toverify them.

The string field theory of Witten [102] is thought to be the correct framework fordiscussing the field theory of open bosonic strings. It contains an infinite number of spacetime fields, corresponding to every possible oscillation of the open bosonic string.Hence, Witten’s action has an infinite number of kinetic terms and an infinite numberof interactions between the fields and thus looks rather complicated. Using conformalfield theory, however, Witten’s open string field theory (OSFT) can be given a veryconcise formulation. However, Witten’s OSFT is not background independent. Thebackground has to be on-shell, i.e., the worldsheet theory of the open strings movingin this background has to be conformally invariant. A background independent openstring field theory has been proposed in [103]. In this theory, Sen’s first and secondconjecture have been proven exactly [104, 105]. For further background on Witten’s

OSFT see the reviews [106–109] and a set of lectures [110] on the subject.Zwiebach constructed a closed bosonic string field theory [111,112]. However, this

field theory is technically involved and difficult for concrete calculations. The storyin supersymmetric string field theories is a vast subject and would need a separatelong discussion. In what is to follow, we do not aim to cover this topic 21.

In this section we take a look on how to introduce lower dimensional D-branes assoliton configurations of the tachyon. We would like to emphasize that even thoughthe second Sen’s conjecture is verified, we lack information on the precise evolutionfrom the presence to the absence of the tachyon. That is, the detailed understanding of the time-dependent process of tachyon condensation into lower dimensional D-branesis still an open question. So we start by discussing unstable D-brane configurationsand their static properties.

21A lot of problems turned out in the naive attempts to construct superstring field theories [113,114], for conjecture part of tachyon condensation, see [115].

35



4.2 Stable and Unstable Non-BPS D-brane Configurations

In string theories with spacetime supersymmetry, i.e., Type I, Type II or heterotic,the tachyons are projected out by imposing the GSO projection22. However, evenin these cases open string tachyons can appear if we consider non-BPS D-branes.The open tachyon is associated with the instabilities of the non-BPS D-branes. Alsoa superposition of a brane with an anti-brane is unstable due to the presence of tachyons in the spectrum of open strings stretching between the branes [116]. We listsome simple examples of systems which contain a tachyon in their spectrum, these

including the following:

• Brane-Anti-brane: A configuration of a single D p- and a single D p -brane of Type II string theory with coincident worldvolumes. A prominent feature of thisconfiguration is that it is non-supersymmetric. Namely, there is no superchargeconserved by both the D p- and the D p -brane. Another way to see this is tonotice that the state as a whole is not BPS: denoting by T p the tension and byQ p the charge of a D p-brane, the state as a whole has tension 2T p but charge0. The tension of a BPS state in the sector of zero charge should be zero,hence the brane-anti-brane state is a non-BPS excited state. Notice that thereis a unique BPS state in the zero charge sector of the theory, namely the TypeII vacuum. Therefore we expect the non-BPS state given by the brane-anti-brane pair to be unstable against decay to the vacuum, since both states havethe same charges and the vacuum is energetically favored. There are two realtachyons, combined to form a complex tachyon, one from open string stretchingfrom brane to anti-brane and the other one from reversed direction.

• Wrong-dimensional: Type II string theory has stable BPS D p-branes with p = 0, 2, 4, 6, 8 in Type IIA, and p = −1, 1, 3, 5, 7, 9 in Type IIB. For the othervalues of p we find unstable non-BPS D p-branes p = −1, 1, 3, 5, 7, 9 in Type IIA,and p = 0, 2, 4, 6, 8 in Type IIB theory. The spectrum on an unstable D p-branein superstring theory is the spectrum of a single open string, but without theGSO projection. Hence there is a real tachyon.

• Bosonic D-branes: A D p-brane of any dimension in bosonic string theorycarries no conserved charge, and has a tachyon in the open string spectrum.Such a brane can decay into the vacuum without violating charge conservation.

The BPS branes are stable because of charge conservation, while the non-BPSbranes can decay, via tachyon condensation to be detailed in section 5, into thevacuum, or into lower dimensional (BPS or non-BPS) branes. A brane and anti-brane pair can decay similarly. In flat backgrounds, Type II branes are either BPSand stable or non-BPS and unstable.

It is interesting to look for backgrounds which admit non-BPS, but stable branes.In fact, quite often superstring theories have states in their spectra which are stable,

22In order to render superstring theories consistent it is necessary to impose a projection on stateswith definite fermion number. This projection is called the Gliozzi-Scherk-Olive or GSO projection.

36



but not BPS. These are in general the lightest states which carry some conservedquantum numbers of the theory. For these states there is no particular relation be-tween their mass and their charge; they receive quantum corrections and form longmultiplets of the supersymmetry algebra. However, they are the lightest states witha given set of conserved quantum numbers, so they are stable since they cannotdecay into anything else. This can be understood by realizing the following. Al-though there are unstable non-BPS D-branes in Type IIA/IIB string theory due tothe presence of the tachyonic mode, we may get stable non-BPS D-branes in certainorbifolds/orientifolds of IIA/IIB if the tachyonic mode is projected out under thisoperation . In particular, in Ref. [73] Sen considered a D-string/anti-D-string pair23

in Type IIB theory, and managed to prove that when the tachyon condenses to akink along the (compact) direction of the D-string, the pair becomes tightly boundand, as a whole, behaves like a D-particle. He also computed its mass and foundit to be a factor of

√2 larger than the one of the supersymmetric BPS D-particle

of the Type IIA theory. The presence of a D-particle in Type IIB spectrum seemsquite surprising at first sight, since we usually think that there are only D p-braneswith p odd in Type IIB. However, we should keep in mind that such a D-particle isa non-supersymmetric and non-BPS configuration. Furthermore, it is unstable dueto the presence of tachyons in the spectrum of open strings living on its worldline.

These tachyons turn out to be odd under worldsheet parity transformation, and hencedisappear if we perform the Ω parity projection to get the Type I string [117–120].Therefore, the D-particle found by Sen is a stable non-perturbative configuration of the Type I. It transforms as a spinor of SO(32). Since the action of Ω on Type IIBD-branes is as follows:

Ω : Dp → Dp ; p = 1, 5, 9

Ω : Dp → Dp ; p = −1, 3, 7 , (4.2)

one can see that Type I theory only has BPS D-branes for p = 1, 5, 9.

4.3 Static Properties of Unstable D-branes

The brane-anti-brane system and the non-BPS D-brane are unstable objects in TypeII string theory. The instability stems from the tachyon which exists on these D-brane systems. Thus, we can expect that they decay into something stable wheretachyons do not exist. It is known what types of objects those D-branes decay to.There are two kinds of decay channels. The main (and simpler) decay channel isdecay to the vacuum without any D-branes. Before going into this we discuss theother decay channel: a D p-D p system can decay to a non-BPS D( p − 1)-brane or toa BPS D( p − 2)-brane.

To understand the relations between different D-brane configurations in Type IIstring theories, we explain the notion of tachyonic kinks. We interpret the non-BPSbranes in Type IIA and IIB string theories as tachyonic kink solutions on a BPSD-brane-anti-D-brane pair of one higher dimension in the same theory. We also show

23A note on terminology: D-particle is a D0-brane and D-string is a D1-brane.

37



how a BPS D-brane (anti-D-brane) can be regarded as a tachyonic kink (anti-kink)solution on a non-BPS D-brane of one higher dimension. This gives us a set of descentrelations between BPS and non-BPS D-branes of Type II string theories 24. Beforeleaving this arena, we argue shortly how to extend the descent relations to bosonicD-branes.

4.3.1 Non-BPS D-brane as Tachyonic Kink on the Brane-Anti-brane Pair

Let us start with a coincident pair of D p-D p branes ( p

≥1 and odd) of Type IIB

string theory. Our discussion is easily extended to the case of Type IIA and similarrelations hold.

As mentioned in subsection 4.2, there is a complex tachyon field, denoted byT , living on the worldvolume of this system. This reflects the fact that T = 0 isthe maximum of the tachyon potential V (T ) obtained after integrating out all othermassive modes on the worldvolume. There is also a U (1) × U (1) gauge field livingon the worldvolume of the D p-D p system, and the complex scalar T picks up aphase under each of the U (1) transformations. The gauge invariance implies thatthe tachyon potential is a function of the modulus T , i.e., V (|T |). The diagonal U (1)subgroup decouples and will be irrelevant for the following discussion. The field theory

has soliton solutions, which correspond to topologically non-trivial worldvolume fieldconfigurations. Finite energy solitons must have a tachyon field asymptoting to aconstant value |T | = T 0, corresponding to the minimum of the potential at T = T 0eiθ

for some fixed T 0, but arbitrary θ, as shown in Fig. 5.At the minimum, the sum of the tension of the D p-D p pair and the (negative)

potential energy of the tachyon is exactly zero [71], i.e.,

2T p + V (T 0) = 0 , (4.3)

which indicates that the tachyonic ground state T = T 0 is indistinguishable from thevacuum, since it carries neither any charge nor any energy density.

Instead of the tachyonic ground state, we want to discuss physics on the top of the hill, T ∼ 0, by looking for a tachyonic kink solution. For the minimum energyconfiguration, let us consider the following profile, depicted in Fig. 6,

• (T ) = 0.

• (T ) is independent of time and p−1 of the p spatial worldvolume coordinates.

• (T ) depends on the remaining spatial coordinate, denoted by x, such that

T (x) → T 0 as x → ∞T (x)

→ −T 0 as x

→ −∞. (4.4)

As a concrete example, we could choose (T ) = 0 and (T ) = T 0 tanh(x/a). |T |differs from T 0 appreciably only in a region of thickness a in the vicinity of x = 0 and

38



T0

V(T)

T

Figure 5: The tachyon potential on D-brane-D-brane pair.

T0

T0

−

x

T(x)

Figure 6: Tachyonic kink solution on a brane-anti-brane pair.

the precise functional form of T (x) is not important for our discussion.

The kink solution admits the characteristic properties we required; as |x| → ∞ the

24As an aside, this forms the basis of identifying the D-brane charge with elements of K-theory,which is a mathematical theory extending the notion of cohomology [121–128], generalizing thepreviously mentioned D-string D-string pair example.

39



solution goes to vacuum configuration. Furthermore, it is topologically unstable; thekink can be continuously unwound into a trivial configuration. This happens becausethe manifold M describing the minimum of the tachyon potential V (T ) is a circle S 1,which is connected, thus we have homotopy π0(M) = 0, in contrast to topologicallystable situation, π0(M) = 025. Therefore, by turning on (T ) and increasing thethickness a, this configuration can be continuously deformed to pure vacuum.

It is clear that the energy density is concentrated around a subspace of dimension p − 1. Therefore we claim that, at least qualitatively, the p − 1 brane associatedwith the kink solution on a D p-D p pair describes a non-BPS D( p

−1)-brane of IIB

located at x = 0. This is indeed the case, however detailed analysis, see [76], is ratherlengthy. Intuitively we see that on the subspace x = 0 the tachyon field vanishes, andhence we expect the configuration to behave in a way that a D p-D p pair would havebehaved in the absence of the tachyon vev. More precisely, it can be shown that thetachyonic mode on the kink solution can be identified as the tachyonic mode on thenon-BPS D( p − 1)-brane of IIB.

4.3.2 The BPS D-brane as Tachyonic Kink on the Non-BPS D-brane

We have realized the non-BPS D( p − 1)-brane of IIB as a tachyonic kink on a D p-D p

pair. Now we want to apply a projection to construct a stable BPS brane out of it. As we already mentioned, the non-BPS brane has a real tachyon. We denote itby T , not to be confused with the tachyonic field studied earlier. The real tachyonresulted from the direction of instability for unwinding the kink in the vacuum modulispace. It can also be understood as the ground state of the open string connectingthe D-brane to itself. We can use the tachyon T to repeat the kink construction.The new vacuum moduli space in this case is two disconnected points T = ±T 0. T is odd under the action of either of the Z2 gauge groups, which is the symmetry onthe worldvolume of this non-BPS D-brane under which T changes sign, and thereforegauge invariance requires that V (T ) = V (−T ).

V now should have an isolated minimum at

|T

|= T 0, and by the same reasoning

as before,T p−1 + V (T 0) = 0 , (4.5)

where T p−1 is the tension of the non-BPS D-brane. The proposed potential has thesame form as in Fig. 5.

We now consider a kink solution on the non-BPS D( p − 1)-brane which has thesame form as in Fig. 6,

• T is independent of time as well as p−2 of the spatial worldvolume coordinates.

• T depends on the remaining spatial coordinate, denoted by x, such that

T (x)→

T 0

as x→ ∞T (x) → −T 0 as x → −∞ . (4.6)

25A necessary condition for the existence of topologically stable extended objects of space co-dimension n is the non-triviality of the homotopy group πn−1(M) for M the manifold of classicalvacua.

40



By the same argument as was outlined before, this describes a ( p − 2)-brane. Adetailed analysis, which can be found in the appendix in Ref. [76], shows that it canbe identified with a BPS D( p−2)-brane of Type IIB. Since the manifold M describingthe minimum of the tachyon potential V consists of a pair of points ±T 0, π0(M) = 0,and hence the kink is topologically stable as is expected for a BPS D-brane.

As in the previous subsection, it is intuitively clear why the kink should behaveas a D-brane near x = 0, but as vacuum for large |x|. It is also worth noting that thekink represents a D-brane rather than an anti-D-brane. This can be seen from thecoupling of a tachyon with the RR ( p

−2)-form

A( p−2) ∧ dT , (4.7)

and the fact that ∂ xT is non-zero at x = 0. Since ∂ xT has the opposite sign for theanti-kink, the anti-kink must represent the BPS D( p − 2)-brane.

4.3.3 Descent Relations

The previous results can be combined to give a set of “descent relations” betweenBPS and non-BPS D-branes. By combining these into results from subsection 4.3.1,

we can also represent a BPS D( p−2)-brane as a soliton solution on the D p-D p -branepair in the same theory.

Now let us consider another relation between BPS and non-BPS branes. Again westart from a D p-D p pair in Type IIB. The operator (−1)F L acts on the closed stringstates by simply changing the sign of every state that has a left-moving part in the Rsector. Thus, it is an exact symmetry of Type II theories. Now we want to see whathappens when we mod out the system by (−1)F L. In particular, (−1)F L transformsa D p-brane into an D p -brane and vice versa. Therefore a D p-D p pair is invariantunder this operator and so it is interesting to ask what happens when we mod it out.A detailed analysis of the open strings that live on the D p-D p pair shows that the

configuration we obtain after modding out (−1)F L

has the same features as that of a D p-brane of Type IIA. Since p is odd, this is a non-supersymmetric and non-BPSbrane. By modding out once more by (−1)F L we go back to the Type IIB theory andfind a stable and supersymmetric D p-brane.

By exploiting all the above relations, as shown in Fig. 7, we can conclude thatactually all D-branes of Type II theories descend from a bound state of D9- andanti-D9-branes.

4.3.4 Descent Relations Extended to Bosonic String Theory

In this subsection we extend the set of descent relations between different D-brane

configurations found in Type II string theories to the D-branes of bosonic stringtheory. As we already mentioned, this theory contains non-BPS D-branes of alldimensions, unlike Type IIA (IIB) string theory which contains non-BPS D-branes of odd (even) dimensions only.

41



FL(−1)

FL(−1)

FL(−1)

p−p−

T a c h y o n

c o n d e n s a t i o n

T a c h y o n


T a c h y o n


T−duality

T−duality

IIB

IIAIIB

IIB

IIB

IIA

p−1 p−1

p p

p−2

Figure 7: This figure resumes the relations between different D-branes in Type IIsuperstring theories. The squares represent the usual supersymmetric BPS D-branes,while the circles stay for the non-BPS configurations (unstable in Type II theories).Starting from a pair formed by a D p- and an anti-D p-brane, a non-BPS brane can beconstructed in two ways: one can mod out the system by (

−1)F L (horizontal arrows)

or condense the tachyon living on its worldvolume (vertical arrows). By repeatingthese operations twice, one finds a supersymmetric configuration. The diagonal linksrepresent the usual T-duality.

First of all it should be noted that there is a tachyonic mode even on a singleD-brane in the bosonic string theory, but when we bring two parallel D-branes ontop of each other, we get extra tachyonic modes from open strings stretched betweenthe two D-branes. We argue below, that the potential involving this tachyonic modeis even and it has a (local) minimum at a non-zero value of the tachyon field. Thus

there are two degenerate minima, and we can have a tachyonic kink solution whichinterpolates between the two minima. This kink solution describes a D-brane of onelower dimension. Thus, by starting with a pair of D p-branes, the tachyonic kinksolution describes a D( p − 1)-brane.

42



The steps involved in showing the equivalence of a tachyonic kink solution anda lower dimensional D-brane are very similar to the ones already discussed in thecase of Type II theories. That is, by taking a coincident pair of non-BPS D-branes,the D-brane effective field theory around T = 0 contains a U (2) gauge field, andthere are four tachyon states represented by a 2 × 2 Hermitian matrix valued scalarfield transforming in the adjoint representation of this gauge group. Again, the (ij)component represents the tachyon stretching from D-brane j to D-brane i. A familyof minima of the tachyon potential can be found by beginning with the configuration

T = T 0 1 0

0 −1

, which represents the tachyon on the first D-brane at its minimumT 0 and the tachyon on the second D-brane at its minimum −T 0, and then applyinga SU (2) rotation. This gives T = T 0n · σ, where n is a unit vector and σi are thePauli matrices. Let us denote by V (T ) the classical effective potential for the tachyonobtained after integrating out the other massive string modes. Since, in particular,SU (2) gauge transformation takes T to −T , it implies that the tachyon potential isan even function of T , i.e., V (T ) = V (−T )26. From now on we relate the discussionto the superstring case, but for definiteness, we shall argue shortly that V (T ) hasminima at some values ±T 0 such that

2T p + V (±T 0) = 0 , (4.8)

where T p denotes the tension of the D p-brane. Thus for T = ±T 0, the total energydensity on the brane pair vanishes and the system is indistinguishable from the vac-uum. The profile for the tachyon on the pair of D p-branes is exactly the same asdepicted in Fig. 6, but note that since bosonic D-branes do not carry any charge,there is no distinction between a kink and an anti-kink.

So far the construction has been very similar to the case of Type II string theories,but to describe a single non-BPS D-brane as a tachyonic soliton there is one importantdistinction. We argued that every single non-BPS D-brane in bosonic string theoryhas a tachyonic mode in the open string spectrum. Let us denote it by T . Following

Sen’s procedure, we denote the effective action by S eff (T , . . .), which is obtained afterintegrating out the field with positive mass squared and by V (T ) the effective potentialfor the tachyon T , a real scalar field with negative mass squared. Now setting all themassless fields to zero and considering spacetime-independent tachyons, gives

S eff (T ) = −

d p+1ξV (T ) . (4.9)

It is clear that the tachyon potential has a (local) maximum at T = 0. But does thepotential V (T ) have a (local) minimum? We should recall that the bosonic stringtheory is sick due to tachyons in the closed string spectrum. Furthermore, there is no

supersymmetry to guarantee that the potential is bounded from below. This causesinstability to the vacuum and is manifested as a (closed) tachyon. Open string theory

26Actually, by considering the tachyonic modes from all CP factors implies that the potential isinvariant under the whole SU (2) transformation.

43



at one-loop has properties that are similar to closed string theory at tree level27, thusit is natural to expect that the (open) tachyon potential is unbounded from below.But by recalling the Sen’s conjectures stated in subsection 4.1, this is not entirely thecase. Lower dimensional D-branes should be considered as soliton configurations of the tachyon, hence V (T ) has a (local) minimum at some value T = T 0. Furthermore,at the minimum, the tension T p of the initial D p-brane is exactly canceled by the(negative) value V (T 0) of the potential, i.e.,

T p + V (T 0) = 0 . (4.10)

The form of the potential is shown in Fig. 8. Note that unlike in the case of superstringtheory, in this case the tachyon potential does not have a global minimum.

T0 T

V(T)~

~~

~

Figure 8: The tachyon effective potential on a bosonic D-brane in bosonic stringtheory. In this case the potential has only one minimum and is unbounded frombelow on the other side.

Since the total energy density vanishes at T 0 = 0, it is natural to identify theconfiguration T = T 0 as the vacuum without any D-brane. This in turn implies

that there are no physical perturbative open string states around the minimum of thepotential. The equations of motion derived from the tachyon effective action (4.9) has

27There are many versions of open/closed string dualities, but they are not statements of equiva-lence between the open and closed string descriptions since the closed string theory could have stateswhich are not accessible to the open string theory.

44



non-trivial time-independent classical lump solutions of various co-dimensions. A co-dimension q lump on a D p-brane, for which T depends on q of the spatial coordinatesand approaches T 0 as any one of these q coordinates goes to infinity, represents aD( p − q)-brane of bosonic string theory. An example of a co-dimension 1 lump isshown in Fig. 9.

T0

T(x)~

x~

~

Figure 9: The co-dimension 1 lump solution on a D-brane in bosonic string theory.

4.4 Effective Field Theory Analysis

We now give a description of the phenomena associated with the Sen’s conjectures1 and 3, and the observation we made earlier, see the discussion leading to (4.3)or (4.5), using an effective field theory. Note that we have treated the process of tachyon condensation as a time-independent process, that is, we have just statedthe static properties of unstable branes. In this view, the tachyonic solutions werethought as solitonic solutions diluted on a higher dimensional brane. In order toproceed towards describing the decay of unstable D-branes, we need to take a step to

understand the time evolution in the framework of closed string theories with (open)tachyonic instability. An exact treatment of rolling tachyons, in context of stringtheory, will be covered in the next section. But now, let us consider essential featuresof the tachyon condensation.

The proposed effective action [129] for describing the dynamics of the tachyon T and gauge fields Aα on a D p-brane in the presence of a constant background metricGαβ , the antisymmetric tensor field Bαβ , and the dilaton φ is a generalization of theDirac-Born-Infeld action (2.21),

S = −T p d p+1ξe−φV (T )√− det A , (4.11)

where

Aαβ = Gαβ + Bαβ + 2παF αβ + ∂ αT ∂ β T + δab∂ αX a∂ β X b (4.12)

F αβ = ∂ αAβ − ∂ αAβ , (4.13)

45



and X a (a ≥ p + 1) denote coordinates transverse to the D p-brane. Note that inthis expression we have shifted the potential by a constant: V (T ) → −T p + V (T ),i.e., V (T = 0) = 1. Since the tachyon potential measures variable tension of theunstable D-brane, it should be a runaway potential connecting V (T = 0) = 1 andV (T = ∞) = 028. Most of the physics of tachyon condensation is irrelevant tothe detailed form of the potential once it satisfies the runaway property and theboundary values. For the case of superstring, Z2-symmetry around its maximum atthe origin is assumed, V (T ) = V (−T ). Various forms of the potential have beenproposed, e.g., V (T )

∼e−T 2 with different derivative terms from boundary string

field theory [104, 105, 115] or V (T ) ∼ e−|T | for large |T | [130]. For our discussion wetake the form29

V (T ) =1

cosh(αT 2 )

, (4.14)

which connects smoothly the small and large T regimes. In this expression α = 1 forbosonic string theory and

√2 for superstring theory.

To simplify our discussion, let us neglect the antisymmetric tensor and the dilaton,Bαβ = φ = 0, consider a flat brane, ∂ αX a∂ β X a = 0 in a flat background, Gαβ = ηαβ .Thus

S =

−T p d p+1ξV (T ) − det(ηαβ + 2παF αβ + ∂ αT ∂ β T ) . (4.15)

4.4.1 Homogeneous Rolling Tachyon

Let us first consider a spatially homogeneous time-dependent solutions, for which∂ iT = 0, in the absence of a gauge field Aα. The Lagrangian density L can be readfrom (4.15) and is given by

L = −T pV (T )

− det(ηαβ + ∂ αT ∂ β T ) = −T pV (T )

1 + ∂ αT ∂ αT

= −T pV (T ) 1 − T 2 , (4.16)

where T ≡ ddt

T (t). It is convenient to move to the Hamiltonian formalism, for thiswe calculate the conjugate momentum density Π,

Π ≡ ∂ L∂ T

=T pV (T )T

1 − T 2, (4.17)

28Note that this choice of the potential is in apparent contradiction with the potential discussedearlier, where the minima were finite distance away from the origin. It is believed that this paradoxdisappears via a complicated field redefinition which includes derivative terms. For instance, anaction of the form −

d p+1ξ(ηµν∂ µT∂ νT + V (T ) + · · · ) where · · · denote terms involving higher

powers of derivatives, can be transformed to an action of the form − d p+1ξ(ηµν∂ µT ∂ ν T + V (T ) +

· · · ) with a different form of the potential V (T ), by a field redefinition of the form T = f (T ) +g(T )∂ µT ∂ µT + · · · by appropriately choosing the functions f and g.

29This kind of potential has been derived in open string theory by taking into account the fluctua-tions around half-S-brane (introduced in the next section) configuration with the higher derivativesneglected, i.e., ∂ 2T = ∂ 3T = · · · = 0 [131, 132].

46



and the Hamiltonian density H follows:

H =

Π2 + (T pV )2 . (4.18)

The conservation of Hamiltonian density dH/dt = H, HP.B. = 0 implies thatthe energy density ρ is a constant of motion,

H = ρ =T pV

1 −T 2

. (4.19)

Note that this happens for any runaway potential and it forces T → ∞, T → 1 ast → ∞. The pressure P ≡ T ii /p, which is given by the Lagrangian density (4.16),vanishes as time elapses

P = L = −T pV (T )

1 − T 2 → 0 , t → ∞ . (4.20)

From equation of state P = wρ, we read

w = −(1 − T 2) = −T 2 p V (t)2

ρ2≤ 0 , (4.21)

where w = −T 2 p /ρ2 at T = 0 and w → 0 as T → ∞.Equation (4.19) leads to an integral

t + C =

dT

1 − (T pV )2

ρ2

, (4.22)

where C is a constant of integration. Now plugging in the preferred form of thepotential V (T ) = 1/ cosh(αT /2) into (4.22) one straightforwardly finds

sinhαT

2= C +eαt/2 + C −e−αt/2 , (4.23)

where C ± are written in terms of the initial conditions T (t = 0) ≡ T (0), T (t = 0) ≡T (0) as

C ± =1

2

T (0) cosh(αT (0)/2) ± sinh(αT (0)/2)

. (4.24)

By introducing a parameter E , as the total energy density of the original system, wecan write the solutions (4.23) in a more convenient form:

sinhαT

2=

λ cosh αt2

, ρ < T pλe

αt2 , ρ = T p

λ sinh αt2

, ρ > T p

, (4.25)

where

λ =

(T p/E )2 − 1 , ρ < T p1 , ρ = T p

1 − (T p/E )2 , ρ > T p

. (4.26)

47



4.4.2 Rolling Tachyons Coupled to U(1) Gauge Fields

Now let us generalize the above discussion a bit by considering a system, which isspatially homogeneous but with a time-dependent non-zero field strength F αβ ,

T = T (t) , F αβ = F αβ (t) . (4.27)

In [133] it is shown that in this case the Bianchi identity and the Dirac-Born-Infeldtype equations of motion force every component of the field strength tensor F αβ to

be constant. Hence we can rewrite the action (4.15) as

S = −T p

d p+1ξV (T )

− det(ηαβ + ∂ αT ∂ β T + 2παF αβ )

= −T p

d p+1ξV (T )

β p − α pT 2 , (4.28)

where α p is the 00-component of the cofactor of matrix Aαβ = ηαβ +∂ αT ∂ β T +2παF αβ ,

α p = C 00 ≥ 1 (4.29)

β p = − det(ηαβ + 2παF αβ ) . (4.30)

By requiring that the action (4.28) remains real, the positivity of α p forces the posi-tivity of β p. Now the rescaling of the time variable

t =t

α p/β p, (4.31)

renders the action (4.28) into form

S = −T p

dt

d pξV (T )

1 − ˜T 2 , (4.32)

where T p = T p√α p and ˜T = dT/dt. Now we can read the Lagrangian from theaction (4.32) and it subsequently has the same form as (4.16), hence the Hamiltonianformalism is analogous to the previous treatment. The energy density is [134], as in(4.19),

ρ ≡

β pα p

H =T pV (T )

1 − ˜T 2, (4.33)

and the three kinds of non-trivial rolling tachyon solutions are, as in (4.25),

sinh αT 2

=

√u2

−1cosh t

ζ , ρ < T p

etζ , ρ = T p√1 − u2 sinh t

ζ , ρ > T p

, (4.34)

where u = T p/ρ

β p and ζ = 2/α

α p/β p.

48



The energy-momentum tensor is given by the symmetric part of the cofactor C αβ S

of Aαβ , namely,

T αβ =T pV − det(Aαβ )

C αβ S . (4.35)

For general electromagnetic fields both momentum density, T 0i, and off-diagonalstress components, T ij, i = j, do not vanish. But as an easy exercise, we can considera pure electric field, E , with F ij = 0. Now the diagonal components of T αβ are theonly non-vanishing, since in this case β p = 1

− E 2, α p = 1, C 0iS = 0, and C ijS = 0

(i = j) [135–138]. We can further choose the electric field to be directed along x-axis,i.e., E = E x, so the pressure components of T αβ are

T 11 = −ρ(1 − ˜T 2) = −ρ(1 − T 2) (4.36)

T 22 = . . . = T pp = −ρ(1 − E 2 − T 2) . (4.37)

Since the equation of motion forces T → ±√1 − E 2 as t → ∞, the pressure again

vanishes asymptotically except for the component parallel to the x-axis, i.e., T 11 = 0.It is worth noting, that the time-scale, ζ = 2/α/

√1 − E 2, is enlarged as the electric

field approaches the critical value, | E | → 1. This might help us to slow down the

rolling of the tachyon, which is of interest for applications to cosmology.The properties of the energy-momentum tensor we have found open up a scenariofor cosmological speculations as of interpreting the decay product, the remnant, as acandidate for dark matter, i.e., a product with energy density but vanishing pressure.There is a wide interest in the literature in rolling tachyon fed cosmological scenarios[139], where various topics include inflation, dark matter, cosmological perturbationand reheating [140–181]. We, however, do not examine this area any deeper, but keepin mind the late time behavior when we study the evolution of the system from stringtheory point of view in section 5. Note that we cannot hope to have the effectiveaction description to be able to describe the full stringy results, but we might hopethat the late time behavior of the system may be describable by some effective action,

such as (4.11), (4.15) or (4.28).

49



5 Brane Decay

Unstable D-branes and their tachyons are a rich source of interesting problems instring theory. While the kinematics of tachyon condensation and the relation to D-brane charges is by now fairly well understood, the decay of unstable branes as a time-dependent process has attracted a considerable amount of attention only recently.Guided by intuition from ordinary stable D-branes, we are led to expect that thisprocess has both a microscopic (or “open string”) and a macroscopic (or “closedstring”) description, which might in some sense be “dual” to each other. In fact, itwas advocated in [182] that the process of unstable brane creation and decay shouldbe viewed as the direct spacelike analog of the familiar timelike branes.

A spacelike brane, or S-brane is almost the same as ordinary D-brane except thatone of its transverse dimensions includes time. S-branes can be also seen as time-dependent, soliton-like configurations in a variety of field theories. In string theory,the potential for the open string tachyon field on the worldvolume of unstable D-brane leads to S-branes in a time-dependent version of the construction of D-branesas solitons of the open string tachyon as we discussed in section 4. These S-branescan be thought as the creation and subsequent decay of an unstable brane. In theprevious section we took a look on a D-brane decay channel in the case of letting the

tachyon condense on the worldvolume of the relevant D-brane configuration giving alower dimensional D-brane. Quantitative results are still lacking, hence we now focuson letting the non-BPS D-brane, or equivalently the D p-D p pair, to decay all the wayto the vacuum. This is a simpler setting, but provides many interesting cosmologicalscenarios. From string theory point of view, the quantitative study of associatedrolling tachyons was initiated by A. Sen [130, 183–186]. It is of great interest tounderstand the “real time” behavior of this process [130,135,182–201].

As is well-known from the study of quantum field theory in curved spacetime,time-dependent background generally leads to the creation of particles during thetime evolution. Then we could expect that particles will be produced during unstableD-brane decay as well. In order to study this process we will follow [183,195,201].

5.1 Time-dependent Solution

We want to study solutions describing the rolling of the tachyon towards the mini-mum of the potential. We shall focus on bosonic string theory, since it provides asimpler setting for studying unstable D-brane systems. Although quantum mechani-cally superstring theory is much better behaved, the classical properties of unstableD-branes in IIA/IIB are very similar to those of bosonic string theory. We work inunits α = 1.

To describe a time-dependent solution we begin with a static solution that depends

on some spatial coordinate X . Next we Wick rotate X to iX

0

, where X

0

is the timecoordinate. The new configuration is then automatically a solution of the equationsof motion. There could be problems, e.g., the solution may not turn out to be real orthe solution may hit a singularity.

50



We begin with linearized equations of motion

(∂ 20 + m2)T (X 0) = O(T 2) , (5.1)

where m2 = −1 is the mass squared for the tachyon. The general solution to thisdifferential equation is

T = AeX0

+ Be−X0

, (5.2)

where A and B are constants determined from initial conditions. For

T (X 0 = 0) = λ, ∂ 0T (X 0 = 0) = 0 gives T (X 0) = λ cosh(X 0) (5.3)

T (X 0 = 0) = 0, ∂ 0T (X 0 = 0) = λ gives T (X 0) = λ sinh(X 0) (5.4)

T (X 0 = 0) = ∂ 0T (X 0 = 0) = λ gives T (X 0) = λeX0

, (5.5)

with all solutions valid for |λ| 1, X 0 1. Note that, by the redefinition of thetachyon field αT /2 → sinh(αT /2), with α = 1 for bosonic strings, we have one-to-one correspondence with the classical solutions of the effective field theory discussedin subsection 4.4.1. In studying properties of the solutions (5.3)-(5.5), it is moreconvenient to directly work with the deformed boundary CFT.

5.2 Conformal Field Theory Description

The boundary string field theory (BSFT) of Witten and Shatashvili [103–105] is aversion of open string field theory in which the classical configuration space is thespace of two-dimensional worldsheet theories on the disk which are conformal in theinterior of the disk but have arbitrary boundary interactions. Solutions of the classicalequations of motion correspond to conformal boundary theories. For early work onthe closely related sigma model approach to string theory see, e.g., [202–208].

When we expand the string field around a classical solution and insert it intothe original SFT action S , we obtain after suitable redefinition of the fluctuationmodes the SFT action S ’; defined on BCFT’ that is related to the original BCFTby inserting a marginal deformation on the boundary of the worldsheet. In otherwords, it can be shown that the two SFT actions S , S ’, written using two differentBCFT, BCFT’, which are related by a marginal deformation, are in fact two SFTactions expanded around different classical solutions. Keeping this in mind, we givea suitable spacetime action which is given by the partition function of the worldsheettheory, with the worldsheet couplings interpreted as spacetime fields. So, in the openstring sector we have

S (λi) ∝ Z disk(λi) =

[DX µ]e−S bulk−S bnd , (5.6)

where Z disk is the disk (of radius one) partition function and λi are exactly marginalcouplings for the boundary operators. The worldsheet action is

S bulk =1

2π

d2zηµν ∂X µ∂X ν , (5.7)

51



and

S bnd =

dτ T (X ) + · · · , (5.8)

where τ denotes the coordinate labeling the boundary of the disk. T is the tachyonand · · · indicate the other marginal boundary deformations. Above formulas arewritten for Euclidean spacetimes, as well as Euclidean worldsheets. As discussedearlier, the time-dependence is driven by replacing X by iX 0 into classical solutionand using the Minkowskian metric ηµν = diag(−1, 1, . . . , 1).

Consider the solution (5.3)

T (X 0) = λ cosh(X 0) . (5.9)

This corresponds to adding a boundary perturbation to the boundary conformal fieldtheory describing the original D-brane by

S bnd = λ

dτ cosh(X 0(τ )) , (5.10)

where λ = λ + O(λ2). This kind of a deformation gives rise to a new BCFT sincecosh(X 0) is an exactly marginal operator30, and hence generates a solution of the

equations of motion of open string theory. The tachyon profile (5.9) is interpreted asa configuration where the tachyon comes up from the minimum associated with theclosed string vacuum, gets closest to zero at the time chosen as X 0 = 0 and then rollsback. The sign of λ determines which side of the potential (Fig. 8) the motion startsfrom (in this case the whole trajectory takes place on either side) and since the tachyonpotential V (T ) is unbounded from below in the bosonic case, we restrict to λ > 0,corresponding to the stable side31. Note that these classical solutions are solitons intime as opposed to space as discussed earlier. We call (5.10) the full-S-brane.

The other profiles include (5.4),(5.5) giving the perturbations of the form

T (X 0) = λ sinh(X 0)

→λ dτ sinh(X 0(τ )) (5.11)

T (X 0) = λeX0 → λ

dτ eX0(τ ) . (5.12)

The case of the perturbation by (5.11) is related to the previous (5.10) case by X 0 →X 0 + iπ, λ → −iλ. The related spacetime interpretation would then be that thetachyon starts at the bottom on one side, reaching the maximum of the potential attime X 0 = 0, and then rolling to the bottom of the potential on the other side. Sincethe potential is unbounded from below on the λ < 0 side, we do not expect the profile(5.4) being physically sensible.

The simplest of the profiles is (5.5), which is described as simultaneously displacing

the tachyon and giving a velocity to it at X 0

= 0. Note that, in this case, λ is not a30Go to the Wick rotated theory, X 0 → iX , where the perturbation becomes λ

dτ cos(X (τ )).

This is known to be an exactly marginal deformation.31This qualitative difference between positive and negative values of λ is not visible in the Wick

rotated theory [209].

52



parameter, since we can set it to 1 by a time translation. Alternative, and better, thespacetime interpretation for this is that of a perturbation at X 0 = −∞: the tachyonstarts at the top of the potential, reaching the bottom of the potential at the latetimes. This can be seen by noting that the profile (5.5) can be obtained as limitingcase of the full-S-brane: starting from T (X 0 + C ) = λ1 cosh(X 0 + C ) and taking thelimit λ1 → 0, C → ∞ with fixed λ = 1

2λ1eC gives (5.5). Thus, the limit correspondsto tuning the displacement of the tachyon to zero, while translating the time at whichthe maximum is reached from X 0 to X 0 = −∞. Indeed, we realize this profile as aperturbation at X 0 =

−∞, intuitively thought as a spontaneously decaying brane,

and since this essentially corresponds to the future half of the full-S-brane, we call(5.12) the half-S-brane.

Note that since the Wick rotated version, X 0 → iX , of these deformed conformalboundary theories theories are exactly soluble [210–212], these theories are likely to besoluble as well. We shall now analyze the boundary state associated with the rollingtachyon solution to determine what kind of source it produces for closed strings.

5.3 Boundary State

Imagine an open string propagating with both ends attached to some D-brane. The

worldsheet of the string is topologically the disk (with appropriate operator insertionsat the boundary). This disk can equivalently be regarded as the half sphere glued tothe brane. But from this point of view it represents the worldsheet of a closed stringwith a certain source at the brane. Therefore the open string disk correlator on thebrane is physically the same as a closed string emission from the brane with a certainsource term corresponding to the open string boundary condition. The source termat the boundary of the half sphere can be represented by an operator insertion in thefull sphere. The state corresponding to this vertex insertion is the boundary state.

The boundary state for a D p-brane with the time-dependent boundary interaction(5.10), (5.12) has the structure

|B = N p|BX0 ⊗ p

α=1

|N α ⊗ 25

a= p+1

|Da ⊗ |Bghost ≡ |BX0 ⊗ |Bsp , (5.13)

where N p is the normalization constant [213]

N p = π11/2(2π)6− p , (5.14)

|N and |D denote the usual boundary states in free CFTs for directions with Neu-mann and Dirichlet boundary conditions, respectively, and |Bghost is in bc-ghostsCFT [214,215]. We define |BX0 below. We would like to emphasize that the bound-ary state associated with the decaying D-brane solution is determined by constructing

the boundary state in the Wick rotated theory, X 0

→ iX and then making an inverseWick rotation to get (5.13), since a clear picture of how to construct boundary statesdirectly in the Lorentzian signature is still lacking.

Since we are interested in dynamics of the rolling tachyon, discussion on explicitforms of |N , |D, and |Bghost need not be covered here. The λ-deformation of the

53



original CFT involves only the coordinate X 0, the part of the BCFT involving thespacelike coordinates X remains unchanged under this deformation. In particular, thecoordinates transverse to the original brane, carrying Dirichlet boundary conditions,will continue to carry Dirichlet boundary conditions. Thus we expect that as theconfiguration evolves in time, the energy density of the system is confined to theplane of the original plane with vanishing string coupling gs = 0. We will later showthat with non-vanishing string coupling gs = 0, the emitted closed string massivemodes slowly move apart from the initial unstable D-brane. Having said this, wefocus on

|B

X0 , which is in the BCFT with the time-dependent boundary interaction

(5.10) or (5.12). By noticing that the boundary interaction term becomes one of thegenerators of SU (2) current algebra we can explicitly construct the boundary state|BX0 : in compact space with self-dual radius, the boundary state is specified as theSU (2) rotated Neumann state, then the form in non-compact space is obtained byprojection of it. That is specified as a linear combination of the Ishibashi states withcoefficients of SU (2) rotation matrix elements [210,211]

|BX0 =

j=0, 12,1,...

jm=− j

D jm,−m(R)| j; m, m , (5.15)

where | j; m, m denotes the Ishibashi state for Virasoro algebra [216] and D jm,−m(R)

denotes matrix elements of a spin j representation with a rotation matrix [212]

R =

cos(πλ) i sin(πλ)

i sin(πλ) cos(πλ)

, (5.16)

for the full-S-brane, and

R =

1 2πiλ0 1

, (5.17)

for the half-S-brane.

It is more convenient to decompose the boundary state (5.15) in terms of oscilla-tors,|BX0 = ρ(x0)|0 + σ(x0)α0

−1α0−1|0 + · · · , (5.18)

where |0 denotes the SL(2,C) invariant vacuum and · · · represents states with higheroscillation modes. The wave functions ρ, σ turn out to take the following form byanalytic continuation [183,195], in the coordinate space,

ρ(x0) =1

1 + λex0+

1

1 + λe−x0− 1 , λ = sin(πλ) (5.19)

σ(x0) = cos(2πλ) + 1 − ρ(x0) , (5.20)

for the full-S-brane, and

ρ(x0) =1

1 + λex0, λ = 2πλ (5.21)

σ(x0) = 2 − ρ(x0) , (5.22)

54



for the half-S-brane.We summarize some essential aspects involving the boundary state (5.13):

• The full-S-brane is periodic in λ and it can be restricted to lie between 0 ≤λ ≤ 1

2. Recall that for half-S-brane, λ is not a free parameter. Furthermore, for

λ = 12 , the boundary state vanishes identically for all real values of time x0; the

system is sitting in the closed string vacuum. For λ = 0, the boundary staterepresents the usual stable D p-brane and letting λ > 0 drives the turning pointfurther away from the top of the tachyon potential.

• During the rolling process, the decay of the brane, the energy is conserved andthe energy-momentum tensor can be written as

T 00 =1

2T p(ρ(x0) + σ(x0)) (5.23)

T αβ = −T pρ(x0)δαβ , α, β = 1, . . . p (5.24)

T 0µ = T ab = 0 , µ = 1, . . . , 25 , a, b = p + 1, . . . , 25 , (5.25)

where T p is just the tension (3.40) of the initial brane, a normalization fixedto meet the stable D p-brane case, λ = 0. Note that T 00 is independent of

x0, which is just the statement of conservation of energy. Moreover, T αβ → 0as x0 → ∞, so the pressure vanishes in this limit, i.e., the decay product ispressureless (tachyon) matter [184]. The formula for ρ(x0) can also be obtainedfrom the effective field theory for the tachyon with the Lagrangian (4.16) withthe potential (4.14), see the Appendix of [201].

A D-brane is treated as a source for closed string modes. With a rolling tachyon,the D-brane becomes a time-dependent source. So, generically there will be closedstring creation as the tachyon rolls down from the top of the hill of the tachyonpotential. We now turn to analyzing quantitatively the energy emission from thedecaying brane.

5.4 Closed String Radiation

As argued above, at vanishing string coupling gs, the brane decays to “tachyon mat-ter” with energy but vanishing pressure. We now take a step further and let gs = 0.That is, we consider whether the energy stored in the initial unstable D p-brane couldbe diffused into the transverse directions by emitted closed strings during the decayprocess and compare the transmitted energy with the tension of the initial D p-brane.We evaluate the energy density carried by the emitted closed strings in the lowestperturbation level of string coupling gs in which interactions between emitted closed

strings may be neglected. We also evaluate the density of the emitted closed strings.The energy density has divergence in both IR for p ≥ 23 and UV region for p ≤ 2.The UV divergence is expected to be cut off by including higher order contributions.Furthermore, we argue that the average transverse velocity of emitted closed stringmassive modes is small, thus we expect that the remnant of the decay process, the

55



tachyon matter, may be identified with a collection of emitted closed string massivemodes of high density.

In lowest perturbation level, O(g0s), the interactions between the emitted closed

strings may be neglected and the problem reduces to that of a free particle productionby a time-dependent source. Thus the resulting closed string state is a coherent state,familiar from field theory. Up to an overall normalization factor it reads

|ψ ∼ eR

dka†k√ 2Es

A|0 (5.26)

where a†k represents a creation operator for a physical closed string mode and A isthe one-point amplitude of the closed string mode on the disk,

A = 0|V|B , (5.27)

where V is the corresponding on-shell vertex operator and E s in (5.26) is the spacetimeenergy of the emitted closed string mode with the same level N = N L = N R for leftand right movers, and 25 − p transverse momentum components k⊥,

E 2s = |k⊥|2 + 4(N − 1) . (5.28)

To evaluate the average total energy density E/V p,

E

V p=s

1

2|A|2 , (5.29)

and average total number density N/V p,

N

V p=s

1

2E s|A|2 , (5.30)

of emitted closed string modes, we need to calculate the one-point amplitude32 Ain the tachyonic background (5.10) or (5.12). The sum over s in (5.29) and (5.30)denotes the integrals over the transverse momenta k⊥ and the level sum with thedensity of states D(N ) giving the number of primary closed string oscillator statesthat are left-right identical at level N . V p is the spatial volume of the D p-brane, as(2π) pδ p(k = 0) =

d px = V p, where the delta functions making tangential momenta

zero come from the translational invariance of Neumann directions.Following the discussion in [201], we can choose a gauge in which the on-shell

closed string state with non-zero energy has no timelike oscillators [217,218], i.e., wecan put all the α0 oscillators to zero. Then the general closed string vertex operatorhas the form

V = eiEX0

V sp , (5.31)

where the spatial part V sp of the operator (5.31) is made out of 25 spatial fields and isconstrained to be a Virasoro primary state of conformal dimension ∆ = 1+ E 2/4. The

32Actually what we really need is the absolute value squared. As we will see later, this simplifiesour calculations drastically.

56



gauge (5.31) is convenient, since the computation of the one-point function factorizesinto a product of the one-point function for the time part AX0 and a one-point functionfor the spatial part Asp as

A = 0|eiEX 0V sp · |BX0 ⊗ |Bsp = 0|eiEX0|BX00|V sp|Bsp ≡ AX0Asp . (5.32)

One finds that the computation of Asp yields [219]

Asp = N peiθ , (5.33)

where eiθ is just a (possibly energy-dependent) phase and N p is given in (5.14). Thus,combining above results, we find that (5.27) can be identified, up to a phase, withthe one-point function for an operator of the form eiEX033

A N p

[DX 0]e−S bulk−S bndeiEX0

= N p0|eiEX 0|BX0 ≡ N pI (E ) . (5.34)

To evaluate (5.34), we separate out the zero mode from the worldsheet field asX 0(z, z) = x0 + X 0(z, z) with

d2zX (z, z) = 0 to give

I (E ) =

0

|eiEX0

|B

X0 =

∞

−∞

dx0eiEx0

0

|eiE X0

|B

X0 . (5.35)

Inserting the decomposition (5.18) into (5.35) reduces the expression (5.35) to34

I (E ) =

∞−∞

dx0ρ(x0)eiEx0 , (5.36)

where ρ(x0) is given in (5.19) or (5.21). The integral (5.36) is straightforward toperform35, and it gives

I (E ) =

e−iE ln λ − eiE ln λ −iπ

sinh πE (5.37)

for the full-S-brane, andI (E ) = e−iE ln λ −iπ

sinh πE (5.38)

for the half-S-brane.It is natural to interpret the two exponentials in (5.37) as from the incoming

and outgoing radiation processes during the rolling process as of a creation and asubsequent decay of the brane. Thus, we can combine above expressions (5.37),(5.38) into

I (E ) = e−iE ln λ −iπ

sinh πE , (5.39)

33Note that this is same as the closed string tachyon vertex operator, but not to be confused withthe open string tachyon discussed earlier.

34By a conformal transformation we can take the bulk vertex operator eiEX(z,z) to be at the originof the disk, z = 0. Hence the correlator for z in the bulk and w = eiτ in the boundary vanishes,X 0(z, z)X 0(τ ) = 0, as well as self-contractions in the bulk and in the boundary.

35Use the substitutions t→ λe±t to recognize the Beta functions.

57



which is the outgoing radiation part for both half- and full-S-brane36. Note that inthis expression the λ-dependence is precisely what we expect on the basis of timetranslational invariance and as we will see it drops out as we amount to take theabsolute value squared of (5.39).

Gathering everything together, we find the emitted average total energy densityand the emitted average total number density from the decaying brane to be

E

V p= N 2 p

s

π2

2sinh2(πE s)= N 2 p

N

d25− pk⊥(2π)25− p

π2

2sinh2(πE )(5.40)

N

V p= N 2 p

s

π2

2E s sinh2(πE s)= N 2 p

N

d25− pk⊥(2π)25− p

π2

2E sinh2(πE ), (5.41)

for both half- and full-S-brane with E = E (N, k⊥) =

k2⊥ + 4(N − 1) and where

the level sum N counts the closed string states which are left-right identical, and isin turn equivalent to summing over open strings. By expanding 1/ sinh2(πE )37 andmaking some manipulations, we can rewrite (5.41) as

N

V p

=

N 2 p

N

d25− pk⊥

(2π)

25− p

4π2

2E

∞

n=1

ne−2πEn

= N 2 pN

d25− pk⊥(2π)25− p

∞n=1

4π2n 1

2E e−2πEn

= N 2 pN

d25− pk⊥(2π)25− p

∞n=1

4π2n 1

2π

dk0

k20 + E 2

e2πik0n

= N 2 pN

dk0d25− pk⊥

(2π)26− p

∞n=1

4π2n

∞0

dt e−t(k20+k2⊥+4(N −1))e2πik0n

=

N 2 p

∞

n=1

4π2n ∞

0

dt N

e−4t(N −1) dk0d25− pk⊥

(2π)26− p

e−t(k20+k2⊥)e2πik0n

= (2π)−12− pπ24−p2

∞n=1

n

∞0

dt t−26−p2 e−

(πn)2

t

N

e−4t(N −1) . (5.42)

The sum over all open string states gives the Dedekind eta function [2]

η(q = e2πit) = eiπt/12

∞n=1

(1 − e2πint) , (5.43)

which has the modular transformation property

η(it) = t−1/2η(i/t) . (5.44)

36Note that λ are different depending on the profile.37−1 + 1

1−x =∞

n=1 xn → x

(1−x)2 =∞

n=1 nxn

58



In conclusion, we can write the average total number density of emitted closedstring modes in the forms

N

V p= 2−12− pπ12− 3p

2

∞n=1

n

∞0

dt t−26−p2 e−

(πn)2

t η2it

π

−24

(5.45)

= 2− pπ−p2

∞n=1

n

∞0

ds s−1−p2 e−n2sη

is

2π

−24

, (5.46)

where we have performed a modular transform s = π2

/t38

. A similar calculation forthe average total energy density leads to

E

V p= N 2 p

N

d25− pk⊥(2π)25− p

4π2

2E

∞n=1

n∂

∂ (−2πn)e−2πEn

...

= 2−12− pπ13− 3p2

∞n=1

n2

∞0

dt t−28−p2 e−

(πn)2

t η2it

π

−24

(5.47)

= 2− pπ−1−p2

∞

n=1

n2 ∞

0

ds s−p2 e−n2sη is

2π−24

. (5.48)

5.5 Identification of Emitted Closed String Modes

We shall analyze the behavior of E/V p in the alternative limits of modulus s to identifycontributions from the massless or massive modes of closed string, up to an overallnumerical factor. For further background see the discussion in section 7.4 in [2] vol.1.

5.5.1 Closed String UV limit: s → ∞ (t → 0)

For large s, we have that η(is/2π)−24

∼ exp(s)(1 + 24exp(−s)) plus exponentiallysuppressed corrections, thus from (5.48), in the UV,

E

V p∼ ∞

ds s−p2

1 + 24e−s + · · · . (5.49)

The leading term in (5.49) comes from the open string tachyon and the next is theopen string massless mode from viewpoint of s → ∞ or equivalently we can interpretthem as coming from massive modes of closed string from viewpoint of t → 0. Theemitted energy (5.49) is finite for p > 2 and infinite for p ≤ 2. If we cut off the energyat E ∼ 1/gs we see that we get an energy of the same order of magnitude as theinitial mass of the unstable D-brane.

We mentioned earlier that in the case of vanishing string coupling gs = 0, itis shown that the tachyon condensation leaves some pressureless objects (tachyon

38Here s is a length of annulus with width π and 2t is a length of cylinder with circumference 2π,hence they are related by s = π2/t.

59



matter) behind. They localize on a hyperplane along which the initial unstable D-brane was extended, and the energy density stored in it is the same as the tension of the initial unstable D-brane. With non-vanishing string coupling gs = 0, the tachyoncondensation causes the closed string emission. It is proposed that the remainder of D-brane decay, the tachyon matter, may be identified with a collection of emittedclosed string massive modes with small transverse velocity, hence of high density.

One can evaluate the transverse velocity of closed string massive modes for p < 25.The production probability of closed string modes with transverse momentum k⊥ andhigh level N is found from the equation (5.41) to be proportional to

ρ(k⊥, N ) ≡ 1 |k⊥|2 + 4(N − 1) sinh2(π |k⊥|2 + 4(N − 1))

∼ e−2π√

|k⊥|2+4N |k⊥|2 + 4N .

(5.50)Then the expectation value of transverse momentum squared behaves, up to theleading term for a fixed high level N , as

|k⊥|2 ≡

d25− pk⊥|k⊥|2ρ(k⊥, N )

d25− pk⊥ρ(k⊥, N )

∼√

N , N 1 . (5.51)

For non-relativistic particles, the corresponding velocity is determined by simply di-viding momentum by mass ∼ √N , thus the resulting expectation value of transversevelocity of emitted closed string massive modes behaves as N −1/4. Hence the emittedclosed string massive modes move slowly at speed N −1/4 away from the origin.

5.5.2 Closed String IR limit: s → 0 (t → ∞)

Again by using η(2it/π)−24 ∼ exp(4t)(1 + 24 exp(−4t)) and39

∞

n=1

n2e−π2n2

t =t3/2

4π5/2

∞

n=−∞e−n2t − t5/2

2π5/2

∞

n=−∞n2e−n2t ∼ t3/2 (5.52)

we can write (5.47), in the IR,

E

V p∼ ∞

dt t−25−p2

e4t + 24 + · · · . (5.53)

The leading parts come from the closed string tachyon and the closed string masslessmode from viewpoint of t → ∞. The closed string tachyon does not appear insuperstring theory, then we would not suffer from the divergence from it. The partof closed string massless mode is divergent in the case of p = 23, 24, 25 and finite inthe case of p

≤22. These divergences signal the breakdown of string perturbation

theory.39Expand

∞

n=−∞ δ(x − n) =∞

n=−∞ e2πinx and integrate ∞−∞

dxe−tx2

(. . .) to ob-

tain∞

n=−∞ e−n2t =

π/t

∞

n=−∞ e−π2n2/t. Further noticing that

∞

n=1 n2e−π

2n2/t =

∂/∂ (−π2/t)∞

n=1 e−π2n2/t yields (5.52) in the end of the day.

60



5.6 Concluding Remarks

The results on tachyon dynamics on an unstable D-brane were stated in section4 in terms of the effective action obtained by formally integrating out the heavyfields. In general it is difficult to do this in practice. Conformal field theory methodsdescribed in this section provide an indirect way of constructing solutions of theclassical equations of motion without knowing the effective action. But if we wanta more direct construction of the classical solutions, we need to explicitly take intoaccount the coupling of the tachyon to infinite number of other fields associated

with massive open string states. Hence a natural framework for a study of the fulltachyon condensation would be string field theory and although we found here thatthe emitted energy has divergences both in IR and UV depending on the numberof dimensions transverse to the decaying unstable brane, we obtained several keyresults. In particular, we managed to identify the most relevant closed string modesinto which the initial energy of the unstable brane decays. We end this thesis byexpecting the study of unstable branes to continue to be an active area of research.

61



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niko jokela- stable and unstable d-branes

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