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Introduction to D-branes

Larus Thorlaciusa∗

aDepartment of Physics, Princeton University,Princeton, NJ, 08544, USA

D-branes are string theory solitons defined by boundary conditions on open strings that end on them. They playan important role in the non-perturbative dynamics of string theory and have found a wide range of applications.These lectures give a basic introduction to D-branes and their dynamics.

1. Introduction

Dirichlet-branes, or D-branes for short, havemoved to center stage in string theory. In arelatively short time they have gone from beingesoteric, and largely ignored, extended objectsthat existed in certain string theories to play-ing a key role in the present understanding ofnon-perturbative dynamics and duality in stringtheory. One reason that the importance of D-branes was overlooked is that they are associatedwith open strings and, at the perturbative level,open string theory appears much less promisingthan closed string theories, such as the heteroticstring theory. With the development of string du-ality, attention shifted towards non-perturbativeissues including solitons, and it was realized thatD-branes had precisely the required attributes tobe partners with fundamental string states underduality transformations. This realization, alongwith the fact that D-branes are considerably sim-pler to describe than other string solitons andoften allow explicit calculations in places whereonly speculations went before, led to an explosionof activity involving D-branes that is still goingstrong.

These lecture notes are intended as a brief in-troduction to D-brane physics and no attemptwill be made to review all their applications. Wewill focus on D-branes in the context of the type IIoriented superstring theory. We begin with a dis-cussion of p-brane soliton solutions of type II su-

∗Lectures presented at the 33rd Karpacz Winter Schoolof Theoretical Physics, Karpacz, Poland, 13-22 February1997

pergravity theory, the low-energy effective fieldtheory of type II superstrings. We then introduceDirichlet-branes. They are string solitons definedin terms of open strings that end on them. Theworldsheet field theory of these open strings isa simple two-dimensional boundary field theoryand has exact conformal symmetry, as requiredby the classical equations of motion of string the-ory. Using worldsheet techniques, we show thatD-branes carry the appropriate conserved chargesin spacetime to be the string theory realization ofsome of the p-branes of the effective field theory.In the remainder of the lecture notes, we exploresome basic properties of D-branes and their dy-namics.

2. Solitons in Type II Supergravity

Our starting point for discussing string solitonsis the low-energy effective field theory that de-scribes strings on length scales that are large com-pared to the fundamental string length ls =

√α′.

In the case of type IIA and IIB string theoriesthe effective field theory is D = 10 supergrav-ity of type IIA and IIB, respectively. The la-bel II refers to the fact that these theories havetwo independent supersymmetries and the A andB refer to whether the two ten-dimensional su-percharges have the opposite (A) or the same(B) chirality. In the string theory this differ-ence corresponds to a freedom of choice betweentwo consistent GSO projections that remove thetachyon from the string spectrum. These theo-ries contain a number of massless bosonic fieldsand their fermionic superpartners. The bosonic

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fields are sufficient for our purposes here. Thetype IIA and IIB supergravities have in commona symmetric metric tensor gµν , an anti-symmetricrank two tensor potential bµν , and a scalar dila-ton φ. In string theory these fields are associ-ated with worldsheet fermions with anti-periodic(Neveu-Schwarz) boundary conditions in both theleft- and right-moving sectors of the closed string,and hence they are commonly referred to as NS-NS fields. The remaining bosonic fields in thesupergravity theory are associated with periodic(Ramond) worldsheet fermions. In the type IIA

theory the R-R fields are a vector potential a(1)µ ,

and an anti-symmetric rank three potential a(3)µνλ,

while in the IIB theory they are a scalar a(0), andanti-symmetric potentials of rank two and four,

a(2)µν and a

(4)µνλσ.

The bosonic terms in the effective action fortype IIA supergravity are

SIIA =1

2κ2

∫d10x

√−g[e−2φ

(R+ 4(∇φ)2

−1

2 · 3!H2)−(1

4h(2)2

+1

2·4!h′(4)2)]

−1

4κ2

∫h(4) ∧ h(4) ∧ b , (1)

where H = d∧b is the NS-NS antisymmetric ten-sor field strength, h(n) = d ∧ a(n−1) are R-R fieldstrengths, and h′(4) = h(4) +a(1)∧H. The NS-NSsector terms come with a prefactor of e−2φ ∼ gs−2

reflecting that they arise at lowest order in stringperturbation theory. The R-R sector terms, onthe other hand, do not have such a prefactor.This can be traced to the structure of the R-Rvertex operators (see for example [1]) and ulti-mately to the fact that fundamental string statesdo not carry R-R gauge charges.

The bosonic part of the type IIB supergravityaction is

SIIB =1

2κ2

∫d10x

√−g[e−2φ

(R + 4(∇φ)2

−1

2 · 3!H2)−

1

2(∇a(0))2

−1

2 · 3!(a(0)H + h(3))2

], (2)

plus some terms involving the self-dual h(5),which cannot be written in covariant form. The

equations of motion involving h(5) are nonethelesscovariant and can be deduced from the D = 10supersymmetry algebra [2].

These theories have classical solutions whichdescribe extended objects called p-branes. A p-brane extends in p spatial directions and tracesout a p+1 dimensional worldvolume in space-time. The p-branes are topological solitons carry-ing conserved charges that act as sources for thevarious anti-symmetric gauge fields.

The NS-NS sector is the same in the IIAand IIB theories, and includes a three-form fieldstrength H. Recall that a point charge in electro-dynamics is a source for a two-form field strength.This generalizes to higher-dimensional objects; ap-brane carries the charge of a (p+2)-form fieldstrength. The ‘electric’ charge that couples to His thus carried by a 1-brane. The corresponding‘magnetic’ field strength is a seven-form,

H = e−2φ ∗H , (3)

where ∗H is the Hodge dual of H. The magneticcharge that couples to H is carried by a 5-brane.

There is a two-parameter family of black stringsolutions labeled by the electric H charge and themass per unit length [3]. The term ‘black’ refersto the fact that for generic values of the massthe string is surrounded by an event horizon. Atthe extremal value, where the mass equals thecharge in appropriate units, the event horizonshrinks away and we are left with an extendedfundamental string [4]. The supersymmetry al-gebra dictates that the mass per unit length of ablack string must always be larger than or equalto the charge per unit length, and the fundamen-tal string saturates the inequality.

Similarly there is a two-parameter family ofblack 5-brane solutions [3], labeled by the massand magnetic charge per unit five-volume. Thesupersymmetry algebra again implies a lowerbound for the mass, M ≥ Q, which is satu-rated by the extremal NS-NS 5-brane describedin [5,6]. The mass bound is a generic feature ofp-brane solutions. The configurations that haveequal mass and charge per p-volume are referredto as BPS states (short for Bogomolnyi-Prasad-Sommerfield). They are annihilated by some frac-tion of the supercharges of the theory and as a

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result they sit in short multiplets of the super-symmetry algebra [7]. If the theory has enoughextended supersymmetry the mass and number ofshort multiplets in the spectrum is not changedby quantum corrections, even at strong coupling.BPS states have therefore played a key role in thestudy of strong/weak coupling duality in gaugetheory and string theory in recent years.

The p-branes of most interest to us in theselectures are associated with the R-R sector fields.The IIA theory has 0-branes that couple to thetwo-form field strength h(2) and 2-branes thatcouple to h(4). Then there are 4- and 6-branesthat couple to the magnetic fields ∗h(4) and∗h(2), respectively. In addition one can con-sider 8-branes. The corresponding ten-form fieldstrength is not dynamical, but a constant fieldcorresponds to a physical energy density [8,9].

In type IIB theory there are p-branes withp=-1,1,3,5,7. The p=-1 case describes instan-tons, which are events in Euclidean spacetime,and couple to the field strength of the R-R scalara(0). The R-R 1- and 5-branes couple to theelectric and magnetic field strengths h(3) andh(7) = ∗h(3), respectively. The 3-brane acts as asource for the self-dual field h(5) and the 7-branecouples to the magnetic field h(9) = ∗(da(0)).There are no 9-branes in type II supergravity.The corresponding field strength would have tobe of rank eleven and must therefore vanish inten dimensions. In type I string theory, however,9-branes play an important role in giving rise tothe SO(32) Chan-Paton gauge degrees of freedom[1].

The black p-branes of type IIA and IIB su-pergravity are approximate classical solutions ofstring theory valid for sufficiently large mass pervolume. Exact classical solutions of string theoryare described by two-dimensional superconformalfield theories. The NS-NS sector 1- and 5-branescan be systematically studied using two dimen-sional sigma models with the appropriate fieldcontent and couplings [4,10], but this approach isnot well suited for R-R sector backgrounds. For-tunately there is another way to proceed. It in-volves a somewhat indirect construction in termsof auxiliary open strings but, as we shall see be-low, the resulting two-dimensional conformal field

theory is very simple and the extended objectsso defined indeed carry R-R charge and have theright mass density required by string duality.

3. Dirichlet-Branes

In order to describe R-R solitons in type IIstring theory we introduce open strings into thetheory. We then have to prescribe boundary con-ditions for the worldsheet fields at the worldsheetboundary. The bosonic worldsheet fields Xµ, forµ = 0, . . . , 9, are interpreted as the embeddingcoordinates of the string in spacetime. Let themsatisfy

(N) na∂aXm = 0 m = 0, . . . , p,

(D) Xi = const i = p+ 1, . . . , 9, (4)

where na is normal to the boundary. The N and Drefer to Neumann and Dirichlet boundary condi-tions. Conventional open strings in type I stringtheory have Neumann boundary conditions for allthe Xµ fields. With the above prescription theopen strings are constrained to end on a p+1 di-mensional hypersurface in spacetime but the endsare free to move on this surface, see Figure 1.

X P

X 0

,..., X 9X P+1

FIGURE 1: Dirichlet p-brane with openstring attached.

This defines the p+1 dimensional worldvolumeof an extended object, called a Dirichlet-brane.At this stage the D-brane appears to be a rigidsheet infinitely extended in p spatial directions.Later on we shall see that a D-brane is in fact dy-namical object with certain massless open stringmodes describing its undulations. A D-brane canbe given a finite volume in a spacetime with com-pact dimensions by wrapping around non-trivialcycles. Finite volume D-branes that can contract

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to a point are presumably unstable and are notdescribed by simple classical solutions.

We also need to prescribe boundary conditionsfor the worldsheet fermions. The surface terms inthe fermion equations of motion cancel if we letthe left-and right-moving fermions, ψµ and ψµ,reflect into each other up to a sign, at the world-sheet boundary. Since the fermion fields are onlydefined up to an overall phase we can set ψµ = ψµ

at one boundary, for example at σ = 0. The signchoice at the other end must be correlated withthe N and D boundary conditions on the bosonicfields (4) in order to preserve worldsheet super-symmetry,

(N) ψm = ±ψm m = 0, . . . , p,

(D) ψi = ∓ψi i = p+ 1, . . . , 9. (5)

In flat spacetime the type II worldsheet fieldtheory contains only free bosons and fermions andis manifestly a conformal field theory. The lin-ear boundary conditions (4) and (5) preserve theconformal symmetry and thus define exact classi-cal solutions of string theory. The construction isindirect but it is very much simpler than the non-linear sigma model description of NS-NS solitons[11].

Dirichlet-branes are BPS states. This canbe seen as a simple consequence of the bound-ary conditions on the worldsheet fields. Thetwo supercharges of the type II theory are de-fined in terms of left- and right-moving world-sheet currents which get reflected into each otherat the boundary. Only the linear combinationQα + cpQα of the two supercharges is conserved,were cp is a phase factor that comes from a par-ity transformation in the transverse directions Xi

with i = p + 1, . . . , 9. In other words, a D-braneis invariant under half the supersymmetry of theoriginal type II theory, which is precisely the BPScondition.

4. D-Brane Tension and Charge

The tension in a p-brane is by definition theenergy per unit p-dimensional volume. In super-gravity theory the tension can be determined byan ADM procedure, considering the asymptoticfalloff of the gravitational field in the directions

transverse to the p-brane. In string theory onecould compute the one-point amplitude of a gravi-ton vertex operator inserted into a worldsheet ofdisk topology with the D-brane boundary condi-tions (4) and (5) applied. This amplitude directlyprobes the gravitational field of the D-brane andallows the tension to be read off [12]. The diskis an open string tree diagram and the amplitudetherefore depends on the dilaton as e−φ. It fol-lows that the D-brane tension is proportional togs−1. Similarly, a one-point disk amplitude in-

volving a vertex operator for the appropriate R-R sector gauge field yields the magnitude of anyR-R charge carried by the D-brane.

These calculations are straightforward but re-quire careful normalization, without which onecannot read off the correct mass and charge. Itturns out to be easier to instead work with aone-loop string amplitude which involves the ex-change of virtual closed strings between D-branes.Following Polchinski we consider a pair of parallelDirichlet p-branes, separated by a distance Y ina transverse direction, and compute the annulusgraph in Figure 2.

X = 09 X = Y9

FIGURE 2: Annulus amplitude for open stringsstretched between different branes.

Worldsheet duality allows us to either view thisdiagram as an open string vacuum loop with thestring endpoints living on the two separate D-branes or as a tree level closed string exchangeof closed strings between the D-branes, see Fig-ure 3. The diagram has a single real valued mod-ulus which, in the closed string channel, can betaken to be the worldsheet time τ between theends of the cylinder and runs from 0 to∞. In the

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limit of large τ the amplitude is dominated bythe exchange of massless closed sting states, thegraviton and dilaton from the NS-NS sector andR-R gauge particles. By identifying the separatecontributions from these states one can read offthe tension and R-R charge of the D-branes.

tτ

t τ =1/t

FIGURE 3: Open and closed string channelsfor annulus amplitude.

To evaluate the diagram we first view it as anopen string loop and write the vacuum amplitude

A = Vp+1

∫dp+1k

(2π)p+1

∫ ∞0

dt

2t

×∑{n}

e−2πt(k2+M2{n}). (6)

Vp+1 denotes the p-brane worldvolume, the inte-gral is over the zero mode momentum in the p+1directions along the worldvolume, and the sum isover the spectrum of physical modes of an opensuperstring with ends on separate D-branes. Themode sum and the momentum integral can becarried out explicitly, giving

A = Vp+1

∫dt

2t(8π2t)−

(p+1)2 e−Y

2t/2π

×F0 [F1 + F2 + F3]. (7)

The three terms in the brackets are

F0(e−πt) = e2πt/3∞∏n=1

(1− e−2nπt)−8,

F1(e−πt) = −16e−2πt/3∞∏n=1

(1 + e−2nπt)8,

F2(e−πt) = eπt/3∞∏n=1

(1 + e−(2n−1)πt)8,

F3(e−πt) = −eπt/3∞∏n=1

(1− e−(2n−1)πt)8, (8)

and they in fact cancel by Jacobi’s famous ab-struse identity. This is not surprising given thatwe are calculating a vacuum amplitude in a su-persymmetric theory. The vanishing of the am-plitude can also be given an interpretation interms of closed string exchange. The exchangeof a graviton or dilaton gives rise to an attractiveforce between the D-branes, while the exchangeof a R-R gauge boson gives rise to the repulsiveforce between two D-branes with the same sign R-R charge. When all the contributions are addedthe net force turns out to be zero. This force bal-ance, which is a familiar feature of BPS objects,only holds when both D-branes are at rest in somereference frame. Later on we will discuss the in-teraction between D-branes when one is movingwith respect to the other.

Even if the overall amplitude vanishes we canstill extract useful information from the separateterms. The amplitude (7) was expressed as anintegral over t, the modulus of the annulus in theopen string channel. The t → 0 end of the inte-gration range is an ultraviolet limit from the openstring point of view, but in the closed string chan-nel the modulus is τ = 1/t and the correspondinglimit is in the infrared. This is precisely where thecylinder amplitude is dominated by massless ex-change and we expect to find information aboutthe tension and R-R charge of the D-brane. In or-der to extract the t→ 0 behavior of the amplitude(7) we make use of the properties of the functionsFi defined in (8) under the modular transforma-tion t → 1/t which relates the open and closedstring channels,

F0(e−πt) = t−1/2F0(e−π/t),

F1(e−πt) = F3(e−π/t),

F2(e−πt) = F2(e−π/t). (9)

Inserting this into (7) and expanding out the in-finite products to leading orders one finds

A ≈ (1− 1)Vp+1

∫dt

t(8π2t)−

(p+1)2

×16t4e−Y2t/2π . (10)

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Carrying out the t integral one arrives at [1,9]

A ≈ (1− 1)Vp+12π(4π2)3−pGq−p(Y2), (11)

where Gd(Y2) is the radial Green function of a

massless scalar in d spatial dimensions.We now want to compare this result with a cor-

responding field theory calculation. Consider atheory of a rank p+ 2 field strength coupled to ap-brane source,

S =1

2(p+ 2)!

∫d10x(h(p+2))

2

+iµp

∫Vp+1

a(p+1). (12)

The leading order exchange amplitude is easilycomputed and has the same value as the R-R termin (11) provided we make the identification [9]

µp2 = 2π(4π2)3−p. (13)

The analogous calculation for graviton and dila-ton exchange in the effective field theory gives,upon comparing with (11), the following D-branetension [9]

τp =

√π(2π)3−p

gs. (14)

Dirichlet-branes are thus seen to be BPS ob-jects carrying R-R gauge charge and have ten-sion inversely proportional to the string cou-pling. Their definition in terms of boundary con-ditions on auxiliary open strings leads to a two-dimensional worldsheet theory with exact confor-mal symmetry as required for classical solutionsin string theory. This combination of propertiesmakes a very convincing case for the identifica-tion of D-branes as the R-R sector p-branes ofstring theory.

There is an important check on the above con-siderations. The p + 2 form field strength thatcouples to a p-brane is Hodge dual to a 8-p formfield strength that couples to a (6-p)-brane. Thisis analogous to the duality between electric andmagnetic point charges in four-dimensional space-time and accordingly the brane charge densitiesmust satisfy a quantization condition analogousto the Dirac condition [13],

µpµ6−p = 2πn, (15)

where n is some integer. The R-R charges ob-tained in (13) satisfy this relation with the mini-mum value of n = ±1.

We conclude this section with some remarkson the relation between D-branes and the cor-responding R-R charged solitons of supergravitytheory. They have the same tension and chargedensity but apart from that they appear quitedifferent. The D-branes are defined in terms ofweakly coupled open strings that propagate in flatspacetime, whereas a supergravity p-brane solu-tion describes a spacetime with non-trivial met-ric and curvature. It depends on the value ofthe string coupling which description is more ap-propriate. As gs → 0, the gravitational field ofthe p-brane vanishes and the flat space D-branepicture is valid. I on the other hand, the cou-pling is not small, we are not justified in ignoringthe gravitational response to the p-brane sourceand the supergravity solutions is a more accuratedescription of the geometry. Supersymmetry andthe BPS property of extremal p-branes are crucialingredients in any argument that takes a resultobtained at weak coupling, such as the numberof multi D-brane states with some macroscopiccharges, and extrapolates it to the strong cou-pling regime in order to draw conclusions abouta black hole of the same mass and charge.

5. Probing D-branes With Strings

The conformal field theory description of D-branes is rather simple and allows explicit calcula-tion of various processes involving D-branes. Forinstance one can compute amplitudes for closedstring scattering off a D-brane [14–16]. This pro-vides information about the structure of D-branesbeyond the classical picture of infinitely thin hy-persurfaces. The leading contribution to the p-brane form factor can be read off from the two-point function of closed-string vertex operators ona disk with the appropriate mixed Dirichlet andNeumann boundary conditions. This correspondsto a string colliding with a D-brane creating anexcitation that propagates along the worldvolumeuntil the D-brane returns to its ground state byemitting another string. For simplicity, we onlyconsider the scattering of NS-NS sector string

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states. The generalization to R-R sector statesis straightforward [15]. The amplitude takes theform

A = 〈V1(z1, z1)V2(z2, z2)〉Disk, (16)

where the vertex operator of a NS-NS sectormassless state (graviton or anti-symmetric ten-sor) with polarization εIJ is

V (z, z) = εIJ(∂XI + ikψψI)

×(∂XJ + ikψψJ )eikX(z,z). (17)

For evaluating the amplitude it is convenient tomap the disk to the upper half of the complexplane and perform Wick contractions using thefollowing propagators for the worldsheet fields,

〈Xµ(z, z)Xν(w, w)〉 = −2ηµν log | z − w |

−2aµν log | 1− zw |,

〈ψµ(z)ψν(w)〉 =ηµν

z − w,

〈ψµ(z)ψν(w)〉 =iaµν

1− zw, (18)

where aµν = diag{−1, 1, . . . , 1,−1, . . . ,−1} withthe -1 in the last 9-p entries reflecting the Dirich-let boundary conditions. The resulting integralscan be carried out explicitly with the result

A = f(εi, ki)Γ(1− 2s)Γ(−t/2)

Γ(1− 2s− t/2), (19)

where the functional form of the kinematic pref-actor f depends on the type of string states thatscatter off the D-brane [14]. The kinematic in-variants are

s = −k‖1

2= −k

‖2

2,

t = 2k‖1

2− 2k⊥1 · k

⊥2 , (20)

where k‖ and k⊥ are momenta parallel and per-pendicular to the D-brane worldvolume.

The amplitude (19) is similar to standard stringscattering amplitudes. This is hardly surprisingsince the D-brane by definition interacts with itsenvironment through the open strings that attachto it. The poles in the s-channel correspond toan open string propagating along the D-brane,while the t-channel poles correspond to a closed

string emitted from the D-brane interacting withthe external strings. The amplitude exhibits theRegge behavior characteristic of string scattering,

A ∼ st2−1 ≈

1

se

12 (log s)t, (21)

indicating that the transverse thickness of a D-brane, as seen by strings, is of order the stringscale at low energy and grows with increasing en-ergy of the probe. This conclusion holds for allDirichlet p-branes with p ≥ 0. For scattering offa D-instanton the kinematics is degenerate andthe form factor exhibits point like behavior.

It is not surprising to find that stringy probescannot see structure on scales smaller than thefundamental string length, but since D-branes in-teract by string exchange we can also draw someconclusions about high energy collisions of D-branes. If the intermediate string states carryhigh energy the Regge behavior will take over, soin order to probe short distances by scattering D-branes of each other the collisions should be softand involve relatively slow moving D-branes. Itturns out that sub-stringy distances can indeedbe probed by slow D-branes, but we need to in-troduce a few more tools before we address thatquestion.

One can also compute mixed closed and openstring scattering amplitudes in a D-brane back-ground using more or less standard techniques.The open string states correspond to excitationsof the D-brane itself and the closed strings repre-sent a coupling to the rest of the world. Such am-plitudes play an important role in the applicationof D-branes to black hole physics, for instance inthe calculation of Hawking emission from a near-extremal black hole modeled as a multi D-branecomposite object. Such calculations are discussedby Maldacena elsewhere in this volume.

6. D-brane Dynamics

So far we have only discussed single D-branesat rest. It is clearly important to develop a dy-namical theory for multiple D-brane configura-tions and D-brane motion. In full generality sucha dynamical description remains a technical chal-lenge. The formalism we have at our disposal de-scribes D-branes in terms of a fixed background

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conformal field theory and does not generalize inany easy way to time-dependent backgrounds. Inorder to proceed further we have to make approx-imations. Following Bachas [17], we consider theforward scattering of two parallel p-branes in theeikonal approximation, where one p-brane movesin a straight line past the other with impact pa-rameter b. The p-branes interact by string ex-change but no back-reaction on their trajectoriesor internal states is taken into account. The cal-culation involves an amplitude which is a straight-forward generalization of the one-loop open stringdiagram we used previously to identify the D-brane tension and R-R charge, with the only dif-ference being that the separation between the p-branes is no longer constant,

Y 2 = b2 + v2t2. (22)

The one-loop amplitude can still be evaluated ex-plicitly and gives the phase shift for the forwardscattering. The resulting expression is somewhatcomplicated and will not be repeated here. Thereare two interesting limits. One is large s, where sis the center-of-mass energy squared of the scat-tering process. The imaginary part of the phaseshift is then [17]

Im(δ) ∼ exp[−b2

log(s/M2)], (23)

where M is the mass of the D-brane. The phaseshift exhibits a characteristic length scale, b ∼[log(s/M2)]1/2, which grows with energy consis-tent with the Regge behavior of the string scat-tering amplitudes considered above.

The other limit is low velocity, v � 1, whereone finds [17]

Im(δ) ∼ exp[−b2

v]. (24)

In this case the characteristic length scale isb ∼ v1/2 in string units, which is very small ifv � 1. This is the first indication that slow-moving D-branes can probe distances below thestring length.

The two limits involve different values of thereal valued modulus of the annulus amplitude.The large s behavior is dominated by the region

in moduli space where the worldsheet is a longcylinder, i.e. the closed string channel, whereasthe low velocity amplitude is dominated by theopen string channel where the annulus is long andthin. In the v � 1 case, the light degrees offreedom are associated with short open stringsstretched between the D-branes. This suggestsan effective field theory description in terms ofmassless open string states, i.e. a gauge theory,which turns out to be very useful, as we shall seelater on.

The one-loop string amplitude that we havefound so useful can be generalized further [18,19].For instance, the two D-branes do not have tohave the same dimensionality. Consider the for-ward scattering of a q-brane past a p-brane withp ≥ q. We can only have p-q be an even numbersince we are either working in IIA theory where pand q are both even or IIB theory where they areboth odd. The new feature here is open stringsthat stretch between D-branes of different dimen-sionality, with some fields that have ND bound-ary conditions, i.e. Neumann at the p-brane butDirichlet at the q-brane. Once again the ampli-tude can be obtained explicitly but is somewhatmessy. It is useful to introduce an effective po-tential as follows

A(v, b2) =1√

2π2

∫ ∞−∞

dX0V (v, r), (25)

where r2 = b2 + (X0)2v2. This potential governsthe motion of a non-relativistic q-brane interact-ing with a p-brane with p≥q. The low velocitybehavior depends on the value of p-q.

For p=q we recover the case we discussed aboveand the potential takes the form

p = q : V (v, r) = −av4

r7−p

(1 +O

(v2

r4

)), (26)

where a is a positive constant. The velocity de-pendence means that identical D-branes at restfeel no force, consistent with the BPS property.At non-zero velocity the system no longer hasunbroken supersymmetry and there is a net at-tractive force between moving D-branes. Theexpansion parameter at low velocities is v2/r4

which again suggests a characteristic length scaler0 ∼ v1/2.

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For p = q+ 4 there is still no force between theD-branes when they are at rest, but the veloc-ity dependent potential is different from the p=qcase,

p = q + 4 : V (r, v) = −av2

r3−q

(1 +O

(v2

r4

)), (27)

The leading term goes as v2, so it is natural toabsorb it into the kinetic term

K ∼1

g

(1−

ag

r3−q

)v2. (28)

The low velocity scattering can then be describedin terms of geodesic motion on a moduli spacewith a non-trivial metric [19].

Another case of interest is q = p, the scatteringof a Dirichlet p-brane and anti-p-brane. Theseobjects do not satisfy a BBS condition even atrest and one expects a nontrivial potential alreadyat v=0. There is in fact an instability if the p-brane and antibrane get too close to each other[20]. To see this, consider the one-loop amplitude(7) for identical p-branes, with a small but criti-cal difference. In the present case the objects haveopposite sign R-R charge and the correspondingterm, F3 in the square brackets in (7) enters withthe opposite sign. Now there is no longer a cancel-lation but instead the terms add and the leadingbehavior of the integrand for large t is

A = Vp+1

∫ ∞0

dt

t(8π2t)−

p+12

×e(π−Y2

2π )t(1 +O(e−2πt)

). (29)

The integral diverges for Y 2 < 2π2. The systemis unstable against p− p annihilation at short dis-tance. The actual annihilation process is presum-ably complicated and results in the emission of alarge number of closed strings.

There is a similar instability in the amplitudewith p = q + 2. In this case the system containstwo D-branes of different dimensionality which donot annihilate each other but form a bound statewith non-vanishing binding energy.

7. Effective Action for D-Branes

We can generate an effective action to describelow-energy processes involving D-branes by the

usual string theory procedure. The first step is toidentify the appropriate two-dimensional world-sheet theory with general couplings. Since D-branes are associated with open strings it is thepossible boundary couplings that are of interesthere,

Sb =

∫ds[ p∑m=0

Am(X0, . . . , Xp)∂tXm

+

q∑i=p+1

νi(X0, . . . , Xp)∂nX

i], (30)

where ∂t and ∂n denote the tangential and normalderivatives at the worldsheet boundary. The pres-ence of the boundary action (30) leads to a gener-alization of the mixed Neumann/Dirichlet bound-ary conditions for the open strings attached to theD-brane. The above terms are the most generalboundary action of scaling dimension one (i.e.renormalizable) for scalar fields Xµ, µ = 0, . . . , 9.The function Am is interpreted as a U(1) gaugepotential that lives on the worldvolume and νirepresents a shift in the D-brane position in thetransverse directions. We have not written theaccompanying fermion terms required for world-sheet supersymmetry. The form of the boundaryaction (30) is constrained by the Dirichlet bound-ary condition in the free theory in that Am and νiare only functions of the worldvolume embeddingcoordinates.

The next step towards the effective action is tocouple these boundary terms to the general non-linear sigma model in the bulk of the string world-sheet and require the combined system to haveconformal symmetry in order to decouple nega-tive norm states from the string spectrum. Thisplaces a set of restrictions on the coupling func-tions of the theory which can be cast as dynam-ical field equations in the ten-dimensional targetspace. The final step is to identify a target spaceaction functional whose equations of motion areequivalent to the field equations that follow fromtwo-dimensional conformal invariance. This pro-cedure was carried out for D-branes in bosonicstring theory by Leigh [21] and the resulting D-brane effective action is a generalization of the

10

Born-Infeld action of non-linear electrodynamics,

Sp = −Tp

∫dp+1σe−φ

×√

det(gµν + bµν + 2πFµν), (31)

where the coefficient Tp is proportional to the D-brane tension, gµν , bµν and φ are spacetime fieldspulled back to the p-brane worldvolume, and thecomponents of the gauge field strength are

Fmn = ∂mAn − ∂nAm, m, n = 0, . . . , p

Fij = 0, i, j = p+ 1, . . . , 9,

Fmi = ∂mνi. (32)

The Fmn components are the worldvolume gaugefield strength while the Fmi components reflectthe D-brane velocity and tilt in the transversedirections. The full effective action also includesthe usual ten-dimensional action for gµν , bµν , andφ.

In type II superstring theory the NS-NS sectorcontributes a Born-Infeld term identical to (31)and in addition the R-R sector gives rise to Chern-Simons type terms [22,23] that can be neatly sum-marized as

S′p = iµp

∫Vp+1

a ∧ Str e(b+2πF ), (33)

where a =∑n a

(n) is the sum of all the R-Rgauge potentials in type II theory. The exponen-tial in (33) also involves a sum of terms but theintegral picks out forms of overall rank p+1 fromthe product. The couplings between the world-volume gauge field and the R-R sector potentialsplay an important role in the study of D-branebound states [1,22]. Consider for example a p-brane with a non-vanishing gauge field strengthFµν in its worldvolume. According to (33) sucha configuration carries (p-2)-brane charge in ad-dition to its p-brane charge. It turns out to havelower energy than the p- and (p-2)-branes sepa-rately and is in fact a realization of the (p,p-2)bound state mentioned at the end of the previoussection.

We have only kept track of the bosonic part ofthe D-brane action but there are of course alsofermionic terms as required by supersymmetry.

The supersymmetric generalization of the abelianBorn-Infeld action (31) has been written down[24]. It is rather involved and will not be neededhere.

The U(1) worldvolume gauge field is associatedwith massless open strings which end on the D-brane. An interesting new feature appears whenwe consider multiple D-brane configurations. Ifsome of the D-branes coincide then there are ad-ditional massless open strings that go betweendifferent D-branes. When the D-branes are wellseparated such strings have a large mass due totheir tension and are suppressed at low energies.When N D-branes are on top of each other thereare N2 massless open string modes that fill outthe adjoint representation of U(N) and the world-volume gauge symmetry is enhanced from U(1) toU(N) [25,26].

The D-brane position in the transverse direc-tions, νi, is also promoted to a U(N) matrix. Thishas played a crucial role in exploring D-brane dy-namics. It indicates that spacetime coordinatesare replaced by non-commuting variables whenwe get down to very short distances.

This formalism also leads to a nice geomet-ric picture for symmetry breaking when a gaugetheory can be realized as the worldvolume the-ory of a collection of D-branes. The unbrokenphase corresponds to all the D-branes coincidingand then the gauge symmetry can be partially orcompletely broken by moving some or all of theD-branes apart.

The non-abelian generalization of the D-braneeffective action remains a subject of investiga-tion at the present time. In particular the gen-eralization of the non-linear Born-Infeld actionis troublesome [27]. In flat spacetime when thegauge field strength is weak and the D-branes areall moving slowly the non-linear terms in the D-brane action will be small and the effective ac-tion is well approximated by the appropriate di-mensional reduction of D=10, supersymmetric,U(N) Yang-Mills theory [26]. Most studies ofD-brane dynamics to date have been based onthe Yang-Mills approximation rather than the fullnon-linear theory and we will describe an examplein the following section.

The dimensionally reduced Yang-Mills theory

11

will in particular include a potential term for thematrix valued D-brane positions which is propor-tional to Tr([νi, νj ][ν

i, νj ]) and comes from theFij components of the non-abelian field strength.The D-branes can only be separated along flatdirections of this potential but these occur pre-cisely when the νi all commute and the gaugegroup is broken from U(N) to U(1)N , as expectedfrom considering open strings attached to the D-branes.

8. D-Particle Scattering

As a simple application of the above ideaswe briefly consider the scattering of two non-relativistic 0-branes [19,28,29]. This involves thedimensional reduction of D=10 supersymmetricU(2) gauge theory to a 0+1 dimensional theoryon the D-particle worldline. The U(2) symme-try factorizes into U(1)×SU(2) in a natural way,where the U(1) factor represents the center-of-mass degrees of freedom and the interesting dy-namics is in the SU(2) sector.

The worldline action including the fermionicterms is

S =

∫dt[ 1

gsTr(FµνF

µν)− iT r(ψΓµDµψ)], (34)

where A0, νi, and ψ take values in the adjointrepresentation of U(2) and

F0i = ∂0νi + [A0, νi],

Fij = [νi, νj ],

D0ψ = ∂0ψ + [A0, ψ],

Diψ = [νi, ψ]. (35)

A simple scaling argument reveals the character-istic length scale of the dynamics [29]. In termsof the rescaled variables t = g1/3t, A0 = g−1/3A0,and νi = g−1/3νi, the action takes the form

S =

∫dt[Tr(Fµν F

µν)− iT r(ψΓµDµψ)], (36)

where the string coupling has been scaled out.The characteristic length scale is therefore l ∼g1/3ls, which is in fact the Planck length of theeleven-dimensional supergravity theory which isthe strong coupling dual of type IIA string theory.

At any rate, this length scale is short compared tothe fundamental string length ls when the stringcoupling gs is weak, and this is an indication thatslow moving D-branes can probe distances belowthe string length. We will not present a detailedstudy of the D-particle system here, but we notethat the dimensionally reduced Yang-Mills the-ory gives rise to a velocity dependent potentialV ≈ av4/r7 in agreement with our earlier consid-erations based on the one-loop open string ampli-tude. String duality requires there to be a boundstate at threshold in this system. The existenceof such a bound state has recently been proven[30]. This required some fairly heavy mathemat-ical machinery but relatively simple argumentscan be given [19,28] that in such a bound state

the D-particles would be separated by r ∼ g1/3s

and have velocities v ∼ g2/3s , which is consistent

with the relation r ∼ v1/2 obtained from the one-loop string amplitude.

9. T-Duality and D-Branes

Our final topic is T-duality in type II string the-ory and the transformation properties of D-braneconfigurations under this symmetry. D-braneswere in fact originally discovered when investi-gating how T-duality acts on open strings [12].

Consider the closed string mode expansionXµ(z, z) = xµ(z) + xµ(z), where z = eτ−iσ and

xµ(z) = xµ0 + i(−αµ0 logz +∑m6=0

1

mαµmz

−m)

xµ(z) = xµ0 + i(−αµ0 logz +∑m6=0

1

mαµmz

−m) (37)

The spacetime momentum is the sum of the left-and right-moving zero modes,

αµ0 + αµ0 = 2pµ, (38)

and their difference gives the change in Xµ(z, z)upon going around the closed string,

αµ0 − αµ0 =

1

2π

(Xµ(σ + 2π, τ)−Xµ(σ, τ)

). (39)

For non-compact spacetime directions Xµ mustbe single valued and it follows that αµ0 = αµ0 = pµ.If, on the other hand, one of the coordinates, say

12

X9, parametrizes a circle of radius R, then X9

only has to come back to itself up to an integermultiple of 2πR when σ → σ + 2π,

α90 − α

90 = mR. (40)

The momentum in the compact direction takesdiscrete values p9 = n/R so that

α90 + α9

0 =2n

R. (41)

The left- and right- moving zero modes in thecompact directions are then given by

α90 = (

n

R+mR

2),

α90 = (

n

R−mR

2). (42)

The mass spectrum and interactions of thestring theory is invariant under the T-dualitytransformation

T9 : R→2

R,m←→ n. (43)

In other words, the theories at radius R and2/R are exactly the same when the winding andmomentum quantum numbers are interchanged.The T-duality transformation can be written

T9 : X9(z, z)→ X ′9(z, z) = x9(z)− x9(z), (44)

i.e. as a parity transformation acting on rightmovers only. The corresponding transformationon the spin fields is

T9 : Sα ←→ Sα, Sα ←→ Sα. (45)

The chirality of the right-moving spin field ischanged under T9 which means that the dualitytransformation relates IIA and IIB theory. Thisremains true when T-duality acts on an odd num-ber of compact dimensions while acting with T-duality on an even number of dimensions does notchange the chiral character of the theory.

τσ

FIGURE 4: Worldsheet boundary inclosed-string channel.

In the open string sector T-duality acts on theboundary conditions in non-trivial way. To seethis it is convenient to view the worldsheet bound-ary in a closed string channel, as shown in Fig-ure 4. This can always be arranged by a con-formal transformation and choice of parametriza-tion. The Neumann boundary condition, ∂τX

µ =0, written in terms of complex coordinates z =eτ−iσ, is

0 = (z∂z + z∂z)Xµ(z, z)

= z∂zxµ(z) + z∂zx(z). (46)

Similarly, the Dirichlet condition, ∂σXµ = 0, be-

comes

0 = z∂zxµ(z)− z∂zx

µ(z). (47)

When T-duality acts on a compact coordinatethe right-moving part of that coordinate changessign and thus Neumann and Drichlet boundaryconditions are exchanged. Now act with a dual-ity transformation Tµ on a Dirichlet p-brane withworldvolume along (X0, . . . , Xp). If µ ≤ p the di-mension of the D-brane is reduced by one, whileit is increased by one if µ > p. This is consistentwith the observation that T duality on an oddnumber of coordinates changes IIA theory intoIIB and vice versa.

T-duality can often be used to relate problemsthat might at first sight seem quite different. Forexample, bound states of Dirichlet p- and (p-n)-branes for a given value of n with different valuesof p can all be mapped into each other by T-duality transformations. When T-duality is com-bined with the strong/weak coupling SL(2, Z)duality of IIB theory the class of related prob-lems is further enlarged.

10. Parting Words

D-branes are interesting and useful objects.They have enhanced our understanding of non-perturbative string theory and continue to findnew applications. There is already a vast lit-erature on D-brane physics and it seems likelythey will continue to play an important role inthe future development of string theory. In theselectures we described how D-branes are intro-duced into type II superstring theory and dis-

13

cussed some of their dynamical properties. Weonly scratched the surface of the subject andmany important topics were left out. For exam-ple, we did not cover unoriented string theory atall, where a lot of interesting work involving D-branes has been done. We nevertheless hope thatthe reader comes away with some impression ofthe power and beauty of D-brane technology.

The author thanks the organizers of the 33rdKarpacz Winter School for the opportunity to lec-ture and for their kind hospitality. He also thanksA. Peet for helpful comments on the manuscript.This work was supported in part by a US Depart-ment of Energy Outstanding Junior InvestigatorAward, DE-FG02-91ER40671.

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