CPHT/CL-615-0698

hep-th/9806199

Lectures on D-branes

Constantin P. Bachas^{1}^{1}1Address after Sept. 1:
Laboratoire de Physique Théorique, Ecole Normale Supérieure,
24 rue Lhomond, 75231 Paris, FRANCE, email :

Centre de Physique Théorique, Ecole Polytechnique

91128 Palaiseau, FRANCE

ABSTRACT

This is an introduction to the physics of D-branes. Topics covered include Polchinski’s original calculation, a critical assessment of some duality checks, D-brane scattering, and effective worldvolume actions. Based on lectures given in 1997 at the Isaac Newton Institute, Cambridge, at the Trieste Spring School on String Theory, and at the 31rst International Symposium Ahrenshoop in Buckow.

June 1998

Lectures on D-branes

Constantin Bachas

## 1 Foreword

Referring in his ‘Republic’ to stereography – the study of solid
forms – Plato was saying : … for even now, neglected and
curtailed as it is, not only by the many but even by professed
students, who can suggest no use for it, nevertheless in the face
of all these obstacles it makes progress on account of its
elegance, and it would not be astonishing if it were unravelled.
^{2}^{2}2Translated by Ivor Thomas in ‘Greek Mathematical Works’,
Loeb Classical Library, Harvard U. Press 1939.
Two and a half millenia later, much of this could have been said for
string theory. The subject has
progressed over the years by leaps and bounds, despite periods of
neglect and (understandable) criticism for lack of
direct experimental input. To be sure, the construction and
key ingredients of the theory –
gravity, gauge invariance, chirality – have a firm empirical
basis, yet what has often catalyzed progress is
the power and elegance of the
underlying ideas, which look (at least a posteriori) inevitable.
And whether the ultimate structure will be
unravelled or not, there
is already a name waiting for ‘it’: theory.

Few of the features of the theory, uncovered so far, exemplify this power and elegance better than D-branes. Their definition as allowed endpoints for open strings, generalizes the notion of quarks on which the QCD string can terminate. In contrast to the quarks of QCD, D-branes are however intrinsic excitations of the fundamental theory: their existence is required for consistency, and their properties – mass, charges, dynamics – are unambiguously determined in terms of the Regge slope and the asymptotic values of the dynamical moduli. They resemble in these respects conventional field-theory solitons, from which however they differ in important ways. D-particles, for instance, can probe distances much smaller than the size of the fundamental-string quanta, at weak coupling. In any case, D-branes, fundamental strings and smooth solitons fill together the multiplets of the various (conjectured) dualities, which connect all string theories to each other. D-branes have, in this sense, played a crucial role in delivering the important message of the ‘second string revolution’, that the way to reconcile quantum mechanics and Einstein gravity may be so constrained as to be ‘unique’.

Besides filling duality multiplets, D-branes have however also opened a window into the microscopic structure of quantum gravity. The D-brane model of black holes may prove as important for understanding black-hole thermodynamics, as has the Ising model proven in the past for understanding second-order phase transitions. Technically, the D-brane concept is so powerfull because of the surprising relations it has revealed between supersymmetric gauge theories and geometry. These relations follow from the fact that Riemann surfaces with boundaries admit dual interpretations as field-theory diagrams along various open- or closed-string channels. Thus, in particular, the counting of microscopic BPS states of a black hole, an ultraviolet problem of quantum gravity, can be mapped to the more familiar problem of studying the moduli space of supersymmetric gauge theories. Conversely, ‘brane engineering’ has been a useful tool for discussing Seiberg dualities and other infrared properties of supersymmetric gauge theories, while low-energy supergravity, corrected by classical string effects, may offer a new line of attack on the old problem of solving gauge theories in the planar (’t Hooft) limit.

Most of these exciting developments will not be discussed in the present lectures. The material included here covers only some of the earlier papers on D-branes, and is a modest expansion of a previous ‘half lecture’ by the author (Bachas 1997a). The main difference from other existing reviews of the same subject (Polchinski et al 1996, Polchinski 1996, Douglas 1996, Thorlacius 1998, Taylor 1998) is in the emphasis and presentation style. The aim is to provide the reader (i) with a basis, from which to move on to reviews of related and/or more advanced topics, and (ii) with an extensive (though far from complete) guide to the literature. I will be assuming a working knowledge of perturbative string theory at the level of Green, Schwarz and Witten (1987) (see also Ooguri and Yin 1996, Kiritsis 1997, Dijkgraaf 1997, and volume one of Polchinski 1998, for recent reviews), and some familiarity with the main ideas of string duality, for which there exist many nice and complimentary lectures (Townsend 1996b and 1997, Aspinwall 1996, Schwarz 1997a and 1997b, Vafa 1997, Dijkgraaf 1997, Förste and Louis 1997, de Wit and Louis 1998, Lüst 1998, Julia 1998, West in this volume, Sen in this volume).

A list of pedagogical reviews for further reading includes : Bigatti and Susskind (1997), Bilal (1997), Banks (1998), Dijkgraaf et al (1998), and de Wit (1998) for the Matrix-model conjecture, Giveon and Kutasov (1998) for brane engineering of gauge theories, Maldacena (1996) and Youm (1997) for the D-brane approach to black holes, Duff et al (1995), Duff (1997), Stelle (1997,1998), Youm (1997) and Gauntlett (1997) for reviews of branes from the complimentary, supergravity viewpoint. I am not aware of any extensive reviews of type-I compactifications, of D-branes in general curved backgrounds, and of semiclassical calculations using D-brane instantons. Some short lectures on these subjects, which the reader may consult for further references, include Sagnotti (1997), Bianchi (1997), Douglas (1997), Green (1997), Gutperle (1997), Bachas (1997b), Vanhove (1997) and Antoniadis et al (1998). Last but not least, dualities in rigid supersymmetric field theories – a subject intimately tied to D-branes – are reviewed by Intriligator and Seiberg (1995), Harvey (1996), Olive (1996), Bilal (1996), Alvarez-Gaume and Zamora (1997), Lerche (1997), Peskin (1997), Di Vecchia (1998) and West (1998).

## 2 Ramond-Ramond fields

With the exception of the heterotic string, all other consistent string theories contain in their spectrum antisymmetric tensor fields coming from the Ramond-Ramond sector. This is the case for the type-IIA and type-IIB superstrings, as well as for the type-I theory whose closed-string states are a subset of those of the type-IIB. One of the key properties of D-branes is that they are the elementary charges of Ramond-Ramond fields, so let us begin the discussion by recalling some basic facts about these fields.

### 2.1 Chiral bispinors

The states of a closed-string theory are given by the tensor product of a left- and a right-moving worldsheet sector. For type-II theory in the covariant (NSR) formulation, each sector contains at the massless level a ten-dimensional vector and a ten-dimensional Weyl-Majorana spinor. This is depicted figuratively as follows:

where and are, respectively, vector and spinor indices. Bosonic fields thus include a two-index tensor, which can be decomposed into a symmetric traceless, a trace, and an antisymmetric part: these are the usual fluctuations of the graviton (), dilaton () and Neveu-Schwarz antisymmetric tensor (). In addition, massless bosonic fields include a Ramond-Ramond bispinor , defined as the polarization in the corresponding vertex operator

(2.1) |

In this expression and are the covariant left- and right-moving fermion emission operators – a product of the corresponding spin-field and ghost operators (Friedan et al 1986), is the ten-dimensional momentum, and the ten-dimensional gamma matrix.

The bispinor field can be decomposed in a complete basis of all gamma-matrix antisymmetric products

(2.2) |

Here , where square brackets denote the alternating sum over all permutations of the enclosed indices, and the term stands by convention for the identity in spinor space. I use the following conventions : the ten-dimensional gamma matrices are purely imaginary and obey the algebra with metric signature . The chirality operator is , Majorana spinors are real, and the Levi-Civita tensor .

In view of the decomposition (2.2), the Ramond-Ramond massless fields are a collection of antisymmetric Lorentz tensors. These tensors are not independent because the bispinor field is subject to definite chirality projections,

(2.3) |

The choice of sign distinguishes between the type-IIA and type-IIB models. For the type-IIA theory and have opposite chirality, so one should choose the sign plus. In the type-IIB case, on the other hand, the two spinors have the same chirality and one should choose the sign minus. To express the chirality constraints in terms of the antisymmetric tensor fields we use the gamma-matrix identities

(2.4) |

It follows easily that only even- (odd-) terms are allowed in the type-IIA (type-IIB) case. Furthermore the antisymmetric tensors obey the duality relations

(2.5) |

As a check note that the type-IIA theory has independent tensors with and indices, while the type-IIB theory has and a self-dual tensor. The number of independent tensor components adds up in both cases to :

This is precisely the number of components of a bispinor.

Finally let us consider the type-I theory, which can be thought of as an orientifold projection of type-IIB (Sagnotti 1988, Hořava 1989a). This projection involves an interchange of left- and right-movers on the worldsheet. The surviving closed-string states must be symmetric in the Neveu-Schwarz sector and antisymmetric in the Ramond-Ramond sector, consistently with supersymmetry and with the fact that the graviton should survive. This implies the extra condition on the bispinor field

(2.6) |

Using we conclude, after some straightforward algebra, that the only Ramond-Ramond fields surviving the extra projection are and its dual, .

### 2.2 Supergraviton multiplets

The mass-shell or super-Virasoro conditions for the vertex operator imply that the bispinor field obeys two massless Dirac equations,

(2.7) |

To convert these to equations for the tensors we need the gamma identities

(2.8) |

and the decomposition (2.2) of a bispinor. After some straightforward algebra one finds

(2.9) |

These are the Bianchi identity and free massless equation for an antisymmetric tensor field strength in momentum space, which we may write in more economic form as

(2.10) |

The polarizations of covariant Ramond-Ramond emission vertices are therefore field-strength tensors rather than gauge potentials.

Solving the Bianchi identity locally allows us to express the -form field strength as the exterior derivative of a -form potential

(2.11) |

Thus the type-IIA theory has a vector () and a three-index tensor potential () , in addition to a constant non-propagating zero-form field strength (), while the type-IIB theory has a zero-form (), a two-form () and a four-form potential (), the latter with self-dual field strength. Only the two-form potential survives the type-I orientifold projection. These facts are summarized in table 1. A -form ‘electric’ potential can of course be traded for a -form ‘magnetic’ potential, obtained by solving the Bianchi identity of the dual field strength.

Neveu-Schwarz | Ramond-Ramond | |

type-IIA | ; | |

type-IIB | ||

type-I | ||

heterotic |

String origin of massless fields completing the N=1 or N=2 supergraviton multiplet of the various theories in ten dimensions.

From the point of view of low-energy supergravity all Ramond-Ramond fields belong to the ten-dimensional graviton multiplet. For N=2 supersymmetry this contains 128 bosonic helicity states, while for N=1 supersymmetry it only contains 64. For both the type-IIA and type-IIB theories, half of these states come from the Ramond-Ramond sector , as can be checked by counting the transverse physical components of the gauge potentials :

This counting is simpler in the light-cone Green-Schwarz formulation, where the Ramond-Ramond fields correspond to a chiral SO(8) bispinor.

### 2.3 Dualities and RR charges

A -form potential couples naturally to a -brane, i.e. an excitation extending over spatial dimensions. Let be the worldvolume of the brane (), and let

(2.12) |

be the pull-back of the -form on this worldvolume. The natural (‘electric’) coupling is given by the integral

(2.13) |

with the charge-density of the brane. Familiar examples are the coupling of a point-particle (‘0-brane’) to a vector potential, and of a string (‘1-brane’) to a two-index antisymmetric tensor. Since the dual of a -form potential in ten dimensions is a -form potential, there exists also a natural (‘magnetic’) coupling to a -brane. The sources for the field equation and Bianchi identity of a -form are thus -branes and -branes.

Now within type-II perturbation theory there are no such elementary RR sources. Indeed, if a closed-string state were a source for a RR -form, then the trilinear coupling

would not vansih. This is impossible because the coupling involves an odd number of left-moving (and of right-moving) fermion emission vertices, so that the corresponding correlator vanishes automatically on any closed Riemann surface. What this arguments shows, in particular, is that fundamental closed strings do not couple ‘electrically’ to the Ramond-Ramond two-form. It is significant, as we will see, that in the presence of worldsheet boundaries this simple argument will fail.

Most non-pertubative dualities require, on the other hand, the existence of such elementary RR charges. The web of string dualities in nine or higher dimensions, discussed in more detail in this volume by Sen (see also the other reviews listed in the introduction), has been drawn in figure 1. The web holds together the five ten-dimensional superstring theories, and the eleven-dimensional theory, whose low-energy limit is eleven-dimensional supergravity (Cremmer et al 1978), and which has a (fundamental ?) supermembrane (Bergshoeff et al 1987). The black one-way arrows denote compactifications of theory on the circle , and on the interval . In the small-radius limit these are respectively described by type-IIA string theory (Townsend 1995, Witten 1995), and by the heterotic model (Hořava and Witten 1996a, 1996b). The two-way black arrows identify the strong-coupling limit of one theory with the weak-coupling limit of another. The type-I and heterotic SO(32) theories are related in this manner (Witten 1995, Polchinski and Witten 1996), while the type-IIB theory is self-dual (Hull and Townsend 1995). Finally, the two-way white arrows stand for perturbative T-dualities, after compactification on an extra circle (for a review of T-duality see Giveon et al 1994).

Consider first the type-IIA theory, whose massless fields are given by dimensional reduction from eleven dimensions. The bosonic components of the eleven-dimensional multiplet are the graviton and a antisymmetric three-form, and they decompose in ten dimensions as follows :

(2.14) |

where . The eleven-dimensional supergravity has, however, also Kaluza-Klein excitations which couple to the off-diagonal metric components . Since this is a RR field in type-IIA theory, duality requires the existence of non-perturbative 0-brane charges. In what concerns type-IIB string theory, its conjectured self-duality exchanges the two-forms and . Since fundamental strings are sources for the Neveu-Schwarz , this duality requires the existence of non-perturbative 1-branes coupling to the Ramond-Ramond (Schwarz 1995).

Higher -branes fit similarly in the conjectured web of dualities. This can be seen more easily after compactification to lower dimensions, where dualities typically mix the various fields coming from the Ramond-Ramond and Neveu-Schwarz sectors. For example, type-IIA theory compactified to six dimensions on a K3 surface is conjectured to be dual to the heterotic string compactified on a four-torus (Duff and Minasian 1995, Hull and Townsend 1995, Duff 1995, Witten 1995). The latter has extended gauge symmetry at special points of the Narain moduli space. On the type-IIA side the maximal abelian gauge symmetry has gauge fields that descend from the Ramond-Ramond three-index tensor. These can be enhanced to a non-abelian group only if there exist charged 2-branes wrapping around shrinking 2-cycles of the K3 surface (Bershadsky et al 1996b). A similar phenomenon occurs for Calabi-Yau compactifications of type-IIB theory to four dimensions. The low-energy Lagrangian of Ramond-Ramond fields has a logarithmic singularity at special (conifold) points in the Calabi-Yau moduli space. This can be understood as due to nearly-massless 3-branes, wrapping around shrinking 3-cycles of the compact manifold, and which have been effectively integrated out (Strominger 1995). Strominger’s observation was important for two reasons : (i) it provided the first example of a brane that becomes massless and can eventually condense (Ferrara et al 1995, Kachru and Vafa 1995), and (ii) in this context the existence of RR-charged branes is not only a prediction of conjectured dualities – they have to exist because without them string theory would be singular and hence inconsistent.

## 3 D-brane tension and charge

The only fundamental quanta of string perturbation theory are elementary strings, so all other -branes must arise as (non-perturbative) solitons. The effective low-energy supergravities exhibit, indeed, corresponding classical solutions (for reviews see Duff, Khuri and Lu 1995, Stelle 1997 and 1998, Youm 1997), but these are often singular and require the introduction of a source. One way to handle the corrections at the string scale is to look for (super)conformally-invariant -models, a lesson sunk-in from the study of string compactifications. Callan et al (1991a, 1991b) found such solitonic five-branes in both the type-II and the heterotic theories. Their branes involved only Neveu-Schwarz backgrounds – being (‘magnetic’) sources, in particular, for the two-index tensor . Branes with Ramond-Ramond backgrounds looked, however, hopelessly intractable : the corresponding -model would have to involve the vertex (2.1), which is made out of ghosts and spin fields and cannot, furthermore, be written in terms of two-dimensional superfields. Amazingly enough, these Ramond-Ramond charged -branes turn out to admit a much simpler, exact and universal description as allowed endpoints for open strings, or D(irichlet)-branes (Polchinski 1995).

### 3.1 Open-string endpoints as defects

The bosonic part of the Polyakov action for a free fundamental
string in
flat space-time and in the conformal
gauge reads ^{3}^{3}3I use the label both for space-time
spinors and for the (Euclidean)
worldsheet coordinates of a fundamental string –
the context should, hopefully, help to avoid confusion.

(3.15) |

with some generic surface with boundary. For its variation

(3.16) |

to vanish, the must be harmonic functions on the worldsheet, and either of the following two conditions must hold on the boundary ,

(3.17) |

Neumann conditions respect Poincaré invariance and are hence momentum-conserving. Dirichlet conditions, on the other hand, describe space-time defects. They have been studied in the past in various guises, for instance as sources for partonic behaviour in string theory (Green 1991b and references therein), as heavy-quark endpoints (Lüscher et al 1980, Alvarez 1981), and as backgrounds for open-string compactification (Pradisi and Sagnotti 1989, Hořava 1989b, Dai et al 1989). Their status of intrinsic non-perturbative excitations was not, however, fully appreciated in these earlier studies.

A static defect extending over flat spatial dimensions is described by the boundary conditions

(3.18) |

which force open strings to move on a -dimensional (worldvolume) hyperplane spanning the dimensions . Since open strings do not propagate in the bulk in type-II theory, their presence is intimately tied to the existence of the defect, which we will refer to as a D-brane. Consider complex radial-time coordinates for the open string – these map a strip worldsheet onto the upper-half plane,

(3.19) |

The boundary conditions for the bosonic target-space coordinates then take the form

(3.20) |

Worldsheet supersymmetry imposes, on the other hand, the following boundary conditions on the worldsheet supercurrents (Green et al 1987) : , where in the Ramond sector, and in the Neveu-Schwarz sector. As a result the fermionic coordinates must obey

(3.21) |

To determine the boundary conditions on spin fields, notice that their operator-product expansions with the fermions read (Friedan et al 1986)

(3.22) |

with a similar expression for right movers. Consistency with (3.21) imposes therefore the conditions,

(3.23) |

where

(3.24) |

is a real operator which commutes with all and anticommutes with all . Since flips the spinor chirality for even, only even-dimensional D-branes are allowed in type-IIA theory. For the same reason type-IIB and type-I theories allow only for odd-dimensional D-branes. In the type-I case we furthermore demand that (3.23) be symmetric under the interchange . This implies , which is true only for and . All these facts are summarized in table 2.

type-IIA | = 0, 2, 4, 6, 8 |
---|---|

type-IIB | = –1, 1, 3, 5, 7, (9) |

type-I | = 1, 5, 9 |

The D-branes of the various string theories are (with the exception of the D9-brane) in one-to-one correspondence with the ‘electric’ Ramond-Ramond potentials of table 1, and their ‘magnetic’ duals. The two heterotic theories have no Ramond-Ramond fields and no D-branes.

The case is degenerate, since it implies that open strings can propagate in the bulk of space-time. This is only consistent in type-I theory, i.e. when there are 32 D9-branes and an orientifold projection. The other D-branes listed in the table are in one-to-one correspondence with the ‘electric’ Ramond-Ramond potentials of table 1, and their ‘magnetic’ duals. We will indeed verify that they couple to these potentials as elementary sources. The effective action of a D-brane, with tension and charge density under the corresponding Ramond-Ramond -form , reads

(3.25) |

where

(3.26) |

is the induced worldvolume metric. The cases are somewhat special. The D(-1)-brane sits at a particular space-time point and must be interpreted as a (Euclidean) instanton with action

(3.27) |

Its ‘magnetic’ dual, in a sense to be made explicit below, is the D-brane. Finally the D-brane is a domain wall coupling to the non-propagating nine-form, i.e. separating regions with different values of (Polchinski and Witten 1996, Bergshoeff et al 1996).

The values of and could be extracted in principle from one-point functions on the disk. Following Polchinski (1995) we will prefer to extract them from the interaction energy between two static identical D-branes. This approach will spare us the technicalities of normalizing vertex operators correctly, and will furthermore extend naturally to the study of dynamical D-brane interactions (Bachas 1996).

### 3.2 Static force: field-theory calculation

Viewed as solitons of ten-dimensional supergravity, two D-branes interact by exchanging gravitons, dilatons and antisymmetric tensors. This is a good approximation, provided their separation is large compared to the fundamental string scale. The supergravity Lagrangian for the exchanged bosonic fields reads (see Green et al 1987)

(3.28) |

where for type-IIA theory, for type-IIB, while for the self-dual field-strength there is no covariant action we may write down. Since this is a tree-Lagrangian of closed-string modes, it is multiplied by the usual factor corresponding to spherical worldsheet topology. The D-brane Lagrangian (3.25), on the other hand, is multiplied by a factor , corresponding to the topology of the disk. The disk is indeed the lowest-genus diagram with a worldsheet boundary which can feel the presence of the D-brane. These dilaton pre-factors have been absorbed in the terms involving Ramond-Ramond fields through a rescaling

(3.29) |

A carefull analysis shows indeed that it is the field strengths of the rescaled potentials which satisfy the usual Bianchi identity and Maxwell equation when the dilaton varies (Callan et al 1988, Li 1996b, Polyakov 1996).

To decouple the propagators of the graviton and dilaton, we pass to the Einstein metric

(3.30) |

in terms of which the effective actions take the form

(3.31) | |||||

and

(3.32) |

To leading order in the gravitational coupling the interaction energy comes from the exchange of a single graviton, dilaton or Ramond-Ramond field, and reads

(3.33) |

Here , and are the sources for the dilaton, Ramond-Ramond form and graviton obtained by linearizing the worldvolume action for one of the branes, while the tilde quantities refer to the other brane. and are the scalar and the graviton propagators in ten dimensions, evaluated at the argument (), and the total interaction time. To simplify notation, and since only one component of couples to a static planar D-brane, we have dropped the obvious tensor structure of the antisymmetric field.

The sources for a static planar defect take the form

(3.34) |

where the -function localizes the defect in transverse space. The tilde sources are taken identical, except that they are localized at distance away in the transverse plane. The graviton propagator in the De Donder gauge and in dimensions reads (Veltman 1975)

(3.35) |

where

(3.36) |

Putting all this together and doing some straightforward algebra we obtain

(3.37) |

where is the (regularized) p-brane volume and is the (Euclidean) scalar propagator in transverse dimensions. The net force is as should be expected the difference between Ramond-Ramond repulsion and gravitational plus dilaton attraction.

### 3.3 Static force: string calculation

The exchange of all closed-string modes, including the massless graviton, dilaton and -form, is given by the cylinder diagram of figure 2. Viewed as an annulus, this same diagram also admits a dual and, from the field-theory point of view, surprising interpretation: the two D-branes interact by modifying the vacuum fluctuations of (stretched) open strings, in the same way that two superconducting plates attract by modifying the vacuum fluctuations of the photon field. It is this simple-minded duality which may, as we will see below, revolutionize our thinking about space-time.

The one-loop vacuum energy of oriented open strings reads

where

(3.39) |

is the usual spin structure sum obtained by supertracing over open-string oscillator states (see Green et al 1987), and we have set . Strings stretching between the two D-branes have at the th oscillator level a mass , so that their vacuum fluctuations are modified when we separate the D-branes. The vacuum energy of open strings with both endpoints on the same defect is, on the other hand, -independent and has been omitted. Notice also the (important) factor of in front of the second line: it accounts for the two possible orientations of the stretched string,

The first remark concerning the above expression, is that it vanishes by the well-known -function identity. Comparing with eq. (3.37) we conclude that

(3.40) |

so that Ramond-Ramond repulsion cancels exactly the gravitational and dilaton attraction. As will be discussed in detail later on, this cancellation of the static force is a consequence of space-time supersymmetry. It is similar to the cancellation of Coulomb repulsion and Higgs-scalar attraction between ’t Hooft-Polyakov monopoles in N=4 supersymmetric Yang-Mills (see for example Harvey 1996) .

To extract the actual value of we must separate in the diagram the exchange of RR and NS-NS closed-string states. These are characterized by worldsheet fermions which are periodic, respectively antiperiodic around the cylinder, so that they correspond to the , respectively open-string spin structures. In the large-separation limit () we may furthermore expand the integrand near :

(3.41) |

where we have here used the standard -function asymptotics. Using also the integral representation

(3.42) |

and restoring correct mass units we obtain

(3.43) |

Comparing with the field-theory calculation we can finally extract the tension and charge-density of type-II D-branes ,

(3.44) |

These are determined unambiguously, as should be expected for intrinsic excitations of a fundamental theory. Notice that in the type-I theory the above interaction energy should be multiplied by one half, because the stretched open strings are unoriented. The tensions and charge densities of type-I D-branes are, therefore, smaller than those of their type-IIB counterparts by a factor of .

## 4 Consistency and duality checks

String dualities and non-perturbative consistency impose a number of relations among the tensions and charge densities of D-branes, which we will now discuss. We will verify, in particular, that the values (3.44) are consistent with T-duality, with Dirac charge quantization, as well as with the existence of an eleventh dimension. From the string-theoretic point of view, the T-duality relations are the least surprising, since the symmetry is automatically built into the genus expansion. Verifying these relations is simply a check of the annulus calculation of the previous section. That the results obey also the Dirac conditions is more rewarding, since these test the non-perturbative consistency of the theory. What is, however, most astonishing is the fact that the annulus calculation ‘knows’ about the existence of the eleventh dimension.

### 4.1 Charge quantization

Dirac’s quantization condition for electric and magnetic charge
(Dirac 1931)
has an analog for extended objects in higher
dimensions (Nepomechie 1985, Teitelboim 1986a,b).
^{4}^{4}4 Schwinger (1968) and Zwanziger (1968)
extended Dirac’s argument to dyons. The
generalization of their argument to higher dimensions
involves a subtle sign discussed recently by Deser et al
(1997, 1998).
Consider a D-brane sitting at the origin, and
integrate the field equation of the Ramond-Ramond form
over the transverse space. Using Stokes’ theorem one finds

(4.45) |

where is a (hyper)sphere, surrounding the defect, in transverse space. This equation is the analog of Gauss’ law. Now Poincaré duality tells us that

(4.46) |

where the potential is not globally defined because the D-brane is a source in the Bianchi identity for . Following Dirac we may define a smooth potential everywhere, except along a singular (hyper)string which drills a hole in . The hole is topologically equivalent to the interior of a hypersphere . These facts are easier to visualize in three-dimensional space, where a point defect creates a string singularity which drills a disk out of a two-sphere, while a string defect creates a sheet singularity which drills a segment out of a circle, as in figure 3.

The Dirac singularity is dangerous because a Bohm-Aharonov experiment involving -branes might detect it. Indeed, the wave-function of a -brane transported around the singularity picks a phase

(4.47) |

For the (hyper)string to be unphysical, this phase must be an integer multiple of . Putting together equations (4.1-4.3) we thus find the condition

(4.48) |

The charge densities (3.44) satisfy this condition with . D-branes are therefore the minimal Ramond-Ramond charges allowed in the theory, so one may conjecture that there are no others.

Dirac’s argument is strictly-speaking valid only
for ^{5}^{5}5The D3-brane is actually also special,
since it couples to a self-dual four-form.. In order to extend it
to the pair , note that a D7-brane creates a
(hyper)-sheet singularity across which the Ramond-Ramond scalar,
, jumps discontinuously by an amount
. Dirac quantization ensures that the
exponential of the (Euclidean) instanton action
(3.27) has no discontinuity across the sheet, whose
presence cannot therefore be detected by non-perturbative physics.
It is the four-dimensional analog of this special case that is, as a
matter of fact,
illustrated in figure 3.

A final comment concerns the type-I theory, where the extra factor of in the charge densities seems to violate the quantization condition. The puzzle is resolved by the observation (Witten 1996b) that the dynamical five-brane excitation consists of a pair of coincident D5-branes, so that

(4.49) |

This is consistent with heterotic/type-I duality, as well as with the fact that the orientifold projection removes the collective coordinates of a single, isolated D5-brane (Gimon and Polchinski 1996).

### 4.2 T-duality

T-duality is a discrete gauge symmetry of string theory, that transforms both the background fields and the perturbative (string) excitations around them (see Giveon et al 1994). The simplest context in which it occurs is compactification of type-II theory on a circle. The general expression for the compact (ninth) coordinate of a closed string is

(4.50) |

Here and are the quantum numbers corresponding to momentum and winding, and with . A T-duality transformation inverts the radius of the circle, interchanges winding with momentum numbers, and flips the sign of right-moving oscillators :

(4.51) |

It also shifts the expectation value of the dilaton, so as to leave the nine-dimensional Planck scale unchanged,

(4.52) |

The transformation (4.51) can be thought of as a (hybrid) parity operation restricted to the antiholomorphic worldsheet sector :

(4.53) |

Since the parity operator in spinor space is , bispinor fields will transform accordingly as follows:

(4.54) |

Using the gamma-matrix identities of section 2, we may rewrite this relation in component form,

(4.55) |

for any . T-duality exchanges therefore even- with odd- antisymmetric field strengths, and hence also type-IIA with type-IIB backgrounds. Consistency requires that it also transform even- to odd- D-branes and vice versa.

To see how this comes about let us consider a D-brane wrapping around the ninth dimension. We concentrate on the ninth coordinate of an open string living on this D-brane. It can be expressed as the sum of the holomorphic and anti-holomorphic pieces (4.50), with an extra factor two multiplying the zero modes becuase the open string is parametrized by . The Neumann boundary condition at real , forces furthermore the identifications

(4.56) |

This is consistent with the fact that open strings can move freely along the ninth dimension on the D-brane, but cannot wind.

Now a T-duality transformation flips the sign of the antiholomorphic
piece, changing the Neumann
to a Dirichlet condition, ^{6}^{6}6For general curved backgrounds with
abelian isometries this has been discussed by Alvarez et al
1996, and by Dorn and Otto 1996.

(4.57) |

The wrapped D-brane is thus transformed, in the dual theory, to a D-brane localized in the ninth dimension (Hořava 1989b, Dai et al 1989, Green 1991a). Open strings cannot move along this dimension anymore, but since their endpoints are fixed on the defect they can now wind. The inverse transformation is also true: a D-brane, originally transverse to the ninth dimension, transforms to a wrapped D-brane in the dual theory. All this is compatible with the transformation (4.55) of Ramond-Ramond fields, to which the various D-branes couple. Furthermore, since a gauge transformation should not change the (nine-dimensional) tension of the defect, we must have

(4.58) |

Using the formulae (4.51-4.52) one can check that the D-brane tensions indeed verify this T-duality constraint. Conversely, T-duality plus the minimal Dirac quantization condition fix unambiguously the expression (3.44) for the D-brane tensions.

### 4.3 Evidence for d=11

The third and most striking set of relations are those derived from the conjectured duality between type-IIA string theory and theory compactified on a circle (Witten 1995, Townsend 1995). The eleven-dimensional supergravity couples consistently to a supermembrane (Bergshoeff et al 1987), and has furthermore a (‘magnetic’) five-brane (Güven 1992) with a non-singular geometry (Gibbons et al 1995). After compactification on the circle there exist also Kaluza-Klein modes, as well as a Kaluza-Klein monopole given by the Taub-NUT space (Sorkin 1983, Gross and Perry 1983). The correspondence between these excitations and the various branes on the type-IIA side is shown in table 3. The missing entry in this table is the eleven-dimensional counterpart of the D8-brane, which has not yet been identified (for a recent attempt see Bergshoeff et al 1997). The problem is that massive type-IIA supergravity (Romans 1986), which prevails on one side of the wall (Polchinski and Witten 1996, Bergshoeff et al 1996), seems to have no ancestor in eleven dimensions (Bautier et al 1997, Howe et al 1998).

tension | type-IIA | on | tension |
---|---|---|---|

D0-brane | K-K excitation | ||

string | wrapped membrane | ||

D2-brane | membrane | ||

D4-brane | wrapped five-brane | ||

NS-five-brane | five-brane | ||

D6-brane | K-K monopole | ||

D8-brane | ? | ? |

Correspondence of BPS excitations of type-IIA string theory, and of theory compactified on a circle. Equating tensions and the ten-dimensional Planck scale on both sides gives seven relations for two unknown parameters. Supersymmetry and consistency imply three Dirac quantization conditions, leaving us with two independent checks of the conjectured duality.

Setting aside the D8-brane, let us consider the tensions of the remaining excitations listed in table 3. The tensions are expressed in terms of and the Regge slope on the type-IIA side, and in terms of and the compactification radius on the -theory side. To compare sides we must identify the ten-dimensional Planck scales,

(4.59) |

Equating the fundamental string tension () with the tension of a wrapped membrane fixes also in terms of eleven-dimensional parameters. This leaves us with five consistency checks of the conjectured duality, which are indeed explicitly verified.

How much of this truly tests the eleven-dimensional origin of string theory? To answer the question we must first understand how the entries on the -theory side of table 3 are obtained. Because of the scale invariance of the supergravity equations, the tensions of the classical membrane and fivebrane solutions are a priori arbitrary. Assuming minimal Dirac quantization, and the BPS equality of mass and charge, fixes the product

(4.60) |

An argument fixing each of the tensions separately was
first given by Duff, Liu and Minasian (1995) and further developped by
de Alwis (1996,1997) and Witten (1997a). ^{7}^{7}7 See also
Lu (1997), Brax and Mourad (1997, 1998) and Conrad (1997).
It uses the Chern-Simons term of the eleven-dimensional Lagrangian,

(4.61) |

where et al 1978),
but in the presence of electric and magnetic sources it is
also subject to an independent quantization condition.
^{8}^{8}8The quantization of
the abelian Chern-Simons term in the presence of a magnetic
source was first discussed in 2+1 dimensions (Henneaux and Teitelboim
1986, Polychronakos 1987).
is the three-index antisymmetric form encountered already
in section 2.
In a nutshell, the coefficient of this
Cherm-Simons term is fixed by supersymmetry (Cremmer

Let me describe the argument in the simpler context of five-dimensional Maxwell theory with a (abelian) Chern-Simons term,

(4.62) |

Assume that the theory has both elementary electric charges (coupling through ), and dual minimally-charged magnetic strings. If we compactify the fourth spatial dimension on a circle of radius , the effective four-dimensional action reads

(4.63) |

where and . The scalar field must be periodically identified, since under a large gauge transformation

(4.64) |

Such a shift changes, however, the -term of the four-dimensional Lagrangian, and is potentially observable through the Witten effect, namely as a shift in the electric charge of a magnetic monopole (Witten 1979). This latter is a magnetic string wrapping around the compact fourth dimension. To avoid an immediate contradiction we must require that the induced charge be an integer multiple of , so that it can be screened by elementary charges bound to the monopole.

In order to quantify this requirement, consider the -term resulting from the shift (4.64). In the background of a monopole field it will give rise to an interaction (Coleman 1981)

(4.65) |

where we have here integrated by parts and used the monopole equation . The interaction (4.65) describes precisely the Witten effect, i.e. the fact that the magnetic monopole has acquired a non-vanishing electric charge. Demanding that the induced charge be an integer multiple of leads, finally, to the quantization condition

(4.66) |

This is the sought-for relation between the coefficient of the (abelian) Chern-Simons term and the elementary electric charge of the theory.

Let us apply now the same reasonning to -theory. Compactifying to eight dimensions on a three-torus gives an effective eight-dimensional theory with both electric and magnetic membranes. The latter are the wrapped five-branes of -theory, which may acquire an electric charge through a generalized Witten effect. Demanding that a large gauge transformation induce a charge that can be screened by elementary membranes leads to the quantization condition

(4.67) |

This relates the electric charge density or membrane tension, , to the coefficient, , of the Chern-Simons term. The membrane tension predicted by duality corresponds to the maximal allowed case .

We can finally return to our original question : How much evidence for the existence of an eleventh dimension in string theory does the ‘gedanken data’ of table 3 contain? Note first that Dirac quantization relates the six tensions pairwise. Furthermore, since the maximal non-chiral 10d supergravity is unique, it must contain a term obtained from the 11d Chern-Simons term by dimensional reduction. An argument similar to the one described above can then be used to fix the product of D2-brane and fundamental-string tensions. Thus, supersymmetry and consistency determine (modulo integer ambiguities) all but two of the tensions of table 3, without any reference either to the ultraviolet definition of the theory or to the existence of an eleventh dimension. We are therefore left with a single truly independent check of the conjectured duality, which we can take to be the relation

(4.68) |

This is a trivial geometric identity in -theory, which had no a priori reason to be satisfied from the ten-dimensional viewpoint.

The sceptic reader may find that a single test constitutes little
evidence for the duality conjecture.
^{9}^{9}9
To be sure, the existence of threshold bound states of D-particles –
the Kaluza-Klein modes of the supergraviton – constitutes further, a priori
independent, evidence for the duality conjecture (Yi 1997, Sethi and Stern
1998, Porrati and Rozenberg 1998).
The above discussion, however,
underscores what might be the main lesson of the ‘second string
revolution’ : the ultimate theory may be unique precisely
because reconciling quantum mechanics and gravity
is such a constraining enterprise.

## 5 D-brane interactions

D-branes in supersymmetric configurations exert no net static force on each other, because (unbroken) supersymmetry ensures that the Casimir energy of open strings is zero. Setting the branes in relative motion (or rotating them) breaks generically all the supersymmetries, and leads to velocity- or orientation-dependent forces. We will now extend Pochinski’s calculation to study such D-brane interactions. Some suprising new insights come from the close relationship between brane dynamics and supersymmetric gauge theory – a theme that will be recurrent in this and in the subsequent sections. Two results of particular importance, because they lie at the heart of the M(atrix)-model conjecture of Banks et al (1997), are the dynamical appearance of the eleven-dimensional Planck length, and the simple scaling with distance of the leading low-velocity interaction of D-particles. Since space-time supersymmetry plays a key role in our discussion, we will first describe in some more detail the general BPS configurations of D-branes.

### 5.1 BPS configurations

A planar static D-brane is a BPS defect that leaves half of the space-time supersymmetries unbroken. This follows from the equality , and the (rigid) supersymmetry algebra, appropriately extended to take into account -brane charges (de Azcarraga et al 1989, see Townsend 1997 for a detailed discussion). Alternatively, we can draw this conclusion from a worldsheet point of view. On a closed-string worldsheet the thirty-two space-time supercharges are given by contour integrals of the fermion-emission operators,

(5.69) |

Holomorphicity allows us to deform the integration contours, picking (eventually) extra contributions only from points where vertex operators have been inserted. This leads to supersymmetric Ward identities for the perturbative closed-string S-matrix in flat ten-dimensional space-time.

Now in the background of a D-brane we must also define the action of the (unbroken) supercharges on the open strings. The corresponding integrals, at fixed radial time , run over a semi-circle as in figure 4. Moving the integration to a later time, is allowed only if the contributions of the worldsheet boundary vanish. This is the case for the sixteen linear combinations

(5.70) |

for which the holomorphic and antiholomorphic pieces add up to zero on the real axis by virtue of the boundary conditions (3.23). The remaining sixteen supersymmetries are broken spontaneously by the D-brane, and cannot thus be realized linearly within the perturbative string expansion.

Consider next a background with two planar static D-branes, to which are associated two operators, and . These operators depend on the orientation, but not on the position, of the branes. More explicitly, we can put equation (3.24) in covariant form

(5.71) |

where

(5.72) |

is the (oriented) volume form of the D-brane, and we have done some simple -matrix rearrangements. There is of course a similar expression for the tilde brane. In the background of these two D-branes, the linearly-realized supercharges are a subset of (5.70), namely

(5.73) |

with an appropriate projection operator. Demanding that the corresponding contour integrals cancel out on a worldsheet boundary that is stuck on the tilde brane leads to the condition

(5.74) |

which admits a non-vanishing solution if and only if

(5.75) |

The number of unbroken supersymmetries is the number of zero eigenvalues of the above matrix. Every extra D-brane and/or orientifold imposes of course one extra condition, which has to be satisfied simultaneously.

A trivial solution to these BPS equations is given by two (or more) identical, parallel D-branes at arbitrary separation . This background preserves sixteen supersymmetries and has, of course, a -independent vacuum energy, consistently with the cancellation of forces found by Polchinski. Flipping the orientation of one brane sends , thus breaking all space-time supersymmetries. The resulting configuration describes a brane and an anti-brane, attracting both gravitationnally, and through Ramond-Ramond exchange. In the force calculation of section 3.3, this amounts to reversing the sign of the spin structure, i.e. of the GSO projection for the stretched open string. The surviving Neveu-Schwarz ground state becomes, in this case, tachyonic at a critical separation , beyond which the attractive force between the brane and the anti-brane diverges (Banks and Susskind 1995, Arvis 1983).

Other solutions to the BPS conditions can be found with two orthogonal D-branes. For such a configuration

(5.76) |

where denotes the set of dimensions spanned by one or other of the branes but not both, and the overall sign depends on the choice of orientations. The eigenvalues of the above operator depend only on the even number () of dimensions in . For the eigenvalues are all purely imaginary, and supersymmetry is completely broken. For or , on the other hand, half of the eigenvalues are , so eight of the supersymmetries are linearly-realized in the background. Examples of