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Preprint typeset in JHEP style - HYPER VERSION

DAMTP-2004-44 DFPD 04/TH/03

Intersecting D–branes, Chern–kernels and the in?ow mechanism

arXiv:hep-th/0406083v2 27 Jun 2004

Marco Cariglia

DAMTP, Centre for Mathematical Sciences, Cambridge University Wilberforce Road, Cambridge CB3 OWA, UK E-mail: m.cariglia@damtp.cam.ac.uk

Kurt Lechner

Dipartimento di Fisica, Universit` a degli Studi di Padova, and Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via F. Marzolo 8, 35131 Padova, Italia E-mail: kurt.lechner@pd.infn.it

Abstract: We analyse a system of arbitrarily intersecting D –branes in ten– dimensional supergravity. Chiral anomalies are supported on the intersection branes, called I –branes. For non–transversal intersections anomaly cancellation has been realized until now only cohomologically but not locally, due to short–distance singularities. In this paper we present a consistent local cancellation mechanism, writing the δ –like brane currents as di?erentials of the recently introduced Chern–kernels, J = dK . In particular, for the ?rst time we achieve anomaly cancellation for dual pairs of D –branes. The Chern–kernel approach allows to construct an e?ective action for the RR–?elds which is free from singularities and cancels the quantum anomalies on all D –branes and I –branes. Keywords: Supergravity, D –branes, anomalies, Chern–kernels; 11.25.Uv. PACS:04.65.+e,

Contents

1. Introduction and summary 2. Chern–kernels: de?nition and properties 2.1 Odd kernels 2.2 Even kernels 3. A new identity 4. Basic applications 4.1 Electromagnetically dual pairs of branes 4.2 Self–dual branes 5. Anomaly cancellation for intersecting IIB D–branes 5.1 Anomalies and the B ?eld 5.2 Bianchi identities and equations of motion 5.3 Action and anomaly cancellation 5.4 Born-Infeld actions and equations of motion on the branes 6. PST duality–invariant formulation 7. Conclusions and outlooks 8. Appendix: proof of the new identity 1 7 7 8 8 10 10 13 14 14 17 19 21 22 24 25

9. Appendix: Equations of motion for the U (1) theory on the D – branes 26 10. Appendix: the Wess–Zumino term 27

1. Introduction and summary

Anomaly cancellation represents a basic quantum–consistency check for extended objects in M –theory, and constrains eventually the physically allowed excitations. A particularly interesting case regards D –branes in ten dimensions. IIA–branes carry an odd–dimensional worldvolume and are trivially anomaly–free, while IIB –branes carry an even–dimensional worldvolume and are actually plagued by gravitational

–1–

anomalies; the problem of their cancellation has been addressed for the ?rst time in [1]. However, D –branes may also intersect with each other. The requirement of anomaly freedom for such con?gurations has indeed been used in [2] to deduce the anomalous (Wess–Zumino) couplings of the IIA and IIB RR–potentials to the ?elds on the branes: if the intersection manifold, called I –brane, is even–dimensional there are potential anomalies supported on it, which have to be cancelled by adding speci?c Wess–Zumino terms to the action and by modifying, correspondingly, the Bianchi– identities for RR–curvatures. Anomalies on I –branes represent the main topic of the present paper. One has to distinguish two kinds of I –branes. In a D –dimensional spacetime two D –branes with i–dimensional worldvolume Mi and, respectively, j –dimensional worldvolume Mj may indeed intersect in two di?erent ways, depending on the dimension of the I –brane manifold Mij = Mi ∩ Mj . In the ?rst (generic) case we have dim(Mij ) = i + j ? D, and the intersection is called transversal; for such intersections the normal bundles of the two branes do not intersect, Nij ≡ Ni ∩ Nj = ?. If i + j ? D < 0 it is understood that Mij = ?. An example of a transversal intersection are two planes in three dimensions that intersect along a line. In the second (exceptional) case the dimension of the I –brane satis?es dim(Mij ) > i + j ? D, and the intersection is called non–transversal. In this case Nij = ? and dim(Nij ) = dim(Mij ) + D ? (i + j ). If i + j ? D < 0 it is understood that Mij = ?. Examples in three dimensions are two coinciding planes, or two lines which intersect in a point. The anomaly cancellation mechanism presented in [2] applies to transversal I – branes, while the attempt of [1] was to include the case of non–transversal I –branes as well. Actually, the anomaly cancellation mechanism of [1] achieves only a “cohomological” cancellation, in a sense that we will specify more precisely in a moment. The main purpose of the present paper is to ?ll this gap, i.e. to present a cancellation mechanism for non–transversal intersections which works “locally”, point–wise, as explained below. The di?erence between these two kinds of I –branes can be translated in the language of di?erential forms as follows. Introduce the δ –function supported Poincar` e dual forms for Mi and Mj , i.e. their “currents” Ji and Jj , of degree D ? i and D ? j respectively, and the current Jij associated to Mij , of degree D ? dim(Mij ). Then for transversal intersections the product Ji Jj is well–de?ned and one has simply Ji Jj = Jij . (1.1)

–2–

For non–transversal intersections the degree of the product Ji Jj is greater then the degree of Jij but, moreover, the product itself is ill–de?ned. The reason is that since Nij is non empty, there exists at least one direction in Nij , parametrized by a coordinate say u, such that Ji as well as Jj contain a factor du δ (u). The product Ji Jj is therefore of the kind 0 (du ∧ du) times ∞ (δ (u)δ (0)). For what concerns anomalies on I –branes, the obstacle to their cancellation on non–transversal intersections arises as follows. Suppose ?rst that the intersection is transveral. Then the anomaly polynomial due to chiral fermions on the I –brane Mij is nonvanishing, and it amounts to a sum of factorized terms [1, 2], Pij = 2πPi Pj , Pi (Pj ) being supported on Mi (Mj ). The anomaly is given by the descent A = 2π (Pi Pj )(1) .

Mij 1

(1.2)

(1.3)

It is cancelled by a Wess–Zumino term in the action of the form SW Z = 2 π

Mi

Pi C,

(1.4)

where the RR–potential C entails an anomalous transformation supported on Mj , δ C = ?Pj Jj . The W Z –term varies according to δSW Z = ?2π Pj Jj Pi = ?2π

(1) (1)

(1.5)

Ji Jj (Pi Pj )(1) ,

(1.6)

Mi

which cancels A thanks to (1.1). For non–transversal intersections the addenda in the anomaly polynomial factorize only partially [1] Pij = 2πPi Pj χij , due to the presence of the Euler–form χij ≡ χ(Nij ) of the now no longer vanishing intersection of the normal bundles Nij – a form of degree dim(Mij ) + D ? (i + j ). The anomaly reads then A = 2π χij (Pi Pj )(1) .

Mij

(1.7)

The Wess–Zumino term is still given by (1.4) but its formal variation leads to the now ill–de?ned expression (1.6).

1

As usual for a generic polynomial we set P = dP (0) , δP (0) = dP (1) .

–3–

We can now make a more precise statement of the problem attacked in this paper and outline its solution. For non–transversal intersections the quantum anomaly (1.7) is still well–de?ned, what is ill–de?ned is the Wess–Zumino term (1.4) itself, since its variation leads to an ill–de?ned expression. The problem consists therefore in constructing a well–de?ned Wess–Zumino term and, afterwards, in checking whether its variation cancels (1.7) or not. The authors of [1] proposed a partial solution to this problem, maintaining the above Wess–Zumino term together with its variation (1.6), and trying to give a meaning to the product “Ji Jj ”. Clearly, to cancel the anomaly what one would need is the identi?cation Ji Jj ? Jij χij . (1.8) As it stands this identi?cation is rather contradictory since Ji Jj is simply a product of δ –functions, while the r.h.s. contains, apart from δ –functions, the gravitational curvatures present in χij . Moreover, the l.h.s. is ill–de?ned. The authors of [1] ?i . Then proposed ?rst to substitute say Ji by a smooth cohomological representative J ?i Jj is cohomologically equivalent to Jij χij , in the they showed that the product J sense of de Rham. Although this is clearly not enough to realize a local anomaly cancellation mechanism, the above identi?cation bears convincingly the correct idea. The main lines of our solution are, indeed, as follows. The expression (1.4) itself looks canonical and rigid: the unique feature one can try to change is the de?nition of the RR–potential C . Above it is indeed (implicitly) assumed that the RR–?eld strength is given in terms of C as 2 dR = Pj Jj ? R = dC + Pj Jj ,

(0)

(1.9)

which obliges C to the transformation law (1.5), carrying a δ –like singularity on Mj , meaning that C itself is singular on Mj , and therefore that (1.4) is ill-de?ned. Our strategy instead consists in keeping (1.4), while introducing a RR–potential that is regular on Mj , actually on all branes. A key step in this direction is to search for a convenient antiderivative Kj of the current, Jj = dKj . (1.10)

Then one can solve the Bianchi identity for R alternatively in terms of a di?erent potential (0) R = dC + Pj Kj , C = C ? Pj Kj , (1.11) where C is invariant and, for a convenient choice of Kj , regular on Mj . This time (1) δ C = ?dPj Kj and the variation of the Wess–Zumino term (1.4) amounts to δSW Z = ?2π

2

Mi

d (Pi Pj )(1) Kj = ?2π

d (Ji Kj ) (Pi Pj )(1) .

We focus here only on the above anomaly; the complete Bianchi identity for a RR–curvature is more complicated, especially for the presence of the B2 -?eld, and it is given in the text.

–4–

The di?erence w.r.t. (1.6) is that for a convenient choice of Kj the product Ji Kj may be well–de?ned together with its di?erential – contrary to what happens to Ji Jj . The apparent paradox is solved by the fact that, due to the singularities present, one is not allowed to use Leibnitz’s rule when computing d(Ji Kj ). The key observation of the present paper is that if one chooses for Kj a Chern– kernel [3] then the product is not only well–de?ned, but one also has the new fundamental identity d(Ji Kj ) = Jij χij , (1.12) realizing in some sense the “identi?cation” (1.8), which is precisely what is needed to cancel the anomaly. To be precise, this formula holds whenever Mi ? Mj . The extremal case Mi ? Mj needs a slight adaptation that is given in the text. For previous applications of the Chern–kernel–approach to anomaly cancellation see [4]– [7]. A special case of non–transversal I –branes is represented by a couple of electromagnetically dual branes, i + j = D ? 2, e.g. a D 1– and a D 5–brane, which have a non–empty intersection manifold Mij . Then the intersection of the normal bundles has dimension dim(Nij ) = dim(Mij )+2, and if it is even it has a nonvanishing Euler–form of the same degree. In this case the anomaly polynomial on Mij is given simply by the Euler–form itself [1], Pij = 2πχij , but its cancellation has not yet been achieved, not even “cohomologically”; for the solution of a particular example in D = 11 see however [7]. As stated in [1], the cancellation of these anomalies requires “a more powerful approach”: Chern–kernels provide actually such an approach. Indeed, in this case the relevant contribution in the Bianchi–identity, realizing the minimal coupling but ignored in [1], is dR = Jj ? R = dC + Kj , (1.13)

and the Wess–Zumino term is conveniently written as the integral over an eleven– dimensional manifold, with space–time R10 as boundary, of a closed eleven–form: SW Z = 2 π

M11

RJi ? Jij χij

(0)

.

(1.14)

The eleven–form is closed thanks to (1.12), and δSW Z = ?2π d Jij χij

(1)

M11

= ?2π

R10

Jij χij = ?2π

(1)

Mij

χij ,

(1)

which cancels the anomaly. Again, as we will see the de?nition (1.13) leads to a potential C that is regular on Mj .

–5–

A third case regards the anomalies on the (even-dimensional) D –branes of IIB – supergravity. Formally the anomalies supported on a D –brane can be interpreted as anomalies on the I –brane of two copies of the same D –brane (self–intersection). In light of this interpretation these anomalies are just a special case of anomalies on I –branes (their cancellation has been discussed in [1], again from a cohomological point of view). So the Chern–kernel approach furnishes automatically a consistent local cancellation mechanism also for IIB D–branes. For concreteness in this paper we consider a system of arbitrarily interacting and intersecting abelian IIB –branes (one for each woldvolume dimension), the case of abelian IIA–branes requiring only a straightforward adaptation. Actually, IIB – branes have a richer anomaly structure because, being even dimensional, they carry anomalies even in the absence of intersections. The generalization to non–abelian branes is exposed brie?y in the conclusions. Usually a magnetic equation of the kind dR = Jj requires the introduction of a Dirac–brane, as antiderivative of Jj , whose unobservability is guaranteed by charge quantization. The consistency of the employment of Chern–kernels as antiderivatives, instead of Dirac–branes, has been proven in [4]. In section two we recall the de?nition of odd (IIB ) and even (IIA) Chern– kernels, and review in a self–contained way their basic properties. In section three we present the basis for the fundamental identity (1.12). In section four we show how one arrives at formula (1.14) for a pair of dual branes, explaining the interplay between Dirac–branes and Chern–kernels. In section ?ve we recall the speci?c form of the anomalies produced by chiral fermions on D –branes and I –branes, we give the complete set of Bianchi–identities/equations of motion for the RR–?eld strengths of IIB –Sugra in presence of branes, and present their solutions in terms of Chern– kernels and regular potentials. This section is based on a systematic application and elaboration of our proposals for the introduction of regular potentials, made in (1.11) and (1.13). We take also a non–vanishing NS B2 –?eld into account, whose consistent inclusion is not completely trivial. In this section we write eventually the action, in particular the Wess–Zumino term, producing the correct equations of motion (for the “basic” potentials C0 , C2 and C4 ), verifying that it is well–de?ned and that it cancels all anomalies. Section six is more technical, in that there we write a manifestly duality–invariant (physically equivalent) action, in which the RR–potentials C0 , C2 , C4 and their duals C6 , C8 appear on the same footing. In this form the distinctive features of our action with respect to previous results emerge more clearly. Section seven is devoted to generalizations and conclusions. We remark brie?y on our conventions and framework. We will assume that there are no topological obstructions in spacetime, in particular closed forms in the bulk are then always exact. Since we are in presence of δ –like currents, for consistency di?erential forms are intended as distribution–valued, and the di?erential calculus is performed in the sense of distributions. This implies that our di?erential operator

–6–

d is always nihilpotent, d2 = 0. With our conventions it acts from the right rather than from the left.

2. Chern–kernels: de?nition and properties

In this section we review brie?y the de?nition of Chern–kernels and recall their main properties [4]. Since we will treat in detail only IIB –branes, that have an even–dimensional worldvolume, we concentrate mainly on odd Chern–kernels, but for completeness and comparison we report also shortly on even kernels. For more details we refer the reader to the above reference. 2.1 Odd kernels Let M be a closed (D ? n)–dimensional brane worldvolume in a D –dimensional spacetime, and introduce a set of normal coordinates y a , (a = 1, · · · , n) associated to M ; the brane stays at y a = 0. Then locally one can write the current associated to M as 1 a1 ...an a1 ε dy . . . dy an δ n (y ). (2.1) J= n! One can also introduce an SO (n)–connection Aab and its curvature F = dA + AA (both are target–space forms), which are only constrained to reduce, if restricted to M , respectively to the SO (n)–normal–bundle connection and curvature, de?ned intrinsically on M . For odd rank Chern–kernels (even currents, IIB –branes) n is even, n = 2m. Then one can de?ne the Euler n–form 3 associated to F and its Chern–Simons form, χ= 1 εa1 ...an F a1 a2 . . . F an?1 an = d χ(0) . m!(4π )m

Its anomaly descent is indicated as usal by δχ(0) = dχ(1) . Notice that the rank of the Euler–form equals the rank of the current. The Chern–kernel K associated to the even current J is written as the sum K = ? + χ(0) , dK = J, (2.2) (2.3)

where ? is an SO (n)–invariant (n ? 1)–form with inverse–power–like singularities on M , polynomial in F and D y ? = dy ? ? Ay ?, where y ?a = y a /|y |. For the expression of ? for a generic n see [4]; for example for n = 2 and n = 4 respectively, the formula reported there gives ?=? 1 a1 a2 a1 a2 ε y ? Dy ? , 2π 8 a3 a4 1 . εa1 ...a4 y ? Dy ?a1 D y ? ?a2 4F a3 a4 + D y ?=? 2 2(4π ) 3 (2.4) (2.5)

3

In the following odd dimensional Euler forms are taken to be zero by de?nition.

–7–

Chern–kernels are not unique due to the arbitrariness of normal coordinates and of A away from the brane, i.e. in the bulk, and due to the presence of the non– invariant Chern–Simons form χ(0) . But since for a di?erent kernel one has in any case dK ′ = J , one obtains K ′ = K + dQ, (2.6) for some target–space form Q. What matters eventually is the behaviour of Q on M . Since ? has a singular but invariant behaviour near the brane, it is only χ(0) that induces an anomalous but ?nite change on M , Q|M = χ(1) |M . (2.7)

The transformation (2.6) has been called Q—transformation in [4] and it is in some sense the analogous of a change of Dirac–brane. From (2.7) one sees that on M a Q– transformation reduces to a normal–bundle SO (n)–transformation, and it can give rise to anomalies supported on M . On the contrary, we demand a theory to be Q– invariant in the bulk. As we will see below, also the RR–potentials have to transform under Q–transformations, and so Q–invariance and anomalies are intimately related. 2.2 Even kernels For odd currents (IIA–branes) n is odd, and the kernel is even. In this case it is only constructed from an SO (n)–invariant (n ? 1)–form with inverse–power–like singularities on M , (K = ?) K = 2π n/2 (n Γ (n/2) εa1 ···an y ?a1 F a2 a3 · · · F an?1 an , ? 1)! (2.8) (2.9)

dK = J,

with F ab ≡ F ab + D y ?aD y ?b . The reason is that for an odd normal bundle the Euler form is vanishing. Also this kernel is de?ned modulo Q–transformations, K ′ = K + dQ, but since ? is invariant this time we have Q|M = 0. We can thus write in general K = ? + χ(0) , with the convention that for even kernels the Euler Chern–Simons form is set to zero.

3. A new identity

In this section we illustrate the new identity d (Ji Kj ) = Jij χij , Mi ? Mj , (3.1)

where it is understood that the Euler–form of an empty normal bundle is unity, χ(?) = 1. Its proof is worked out in the appendix. When i + j < D (3.1) is an

–8–

identity between forms whose degree exceeds D . In that case Mi and Mj have to be extended to worldvolumes in a larger space–time, keeping the degrees of the K ’s and the J ’s unchanged; see e.g. [4]. The “non–extremality” condition Mi ? Mj is needed to guarantee that the product Ji Kj is well–de?ned, implying that also its di?erential is so. In the appendix it is then shown that d(Ji Kj ) is 1) closed, 2) invariant, 3) supported on Mij and 4) constructed from the curvatures of Nij . The above identity follows then essentially for uniqueness reasons. To simplify some formulae of the following sections, and motivated by (3.1), we de?ne for arbitrary intersections (Ji Jj )reg ≡ Jij χij . Consider now an extremal intersection Mi ? Mj , where the product Ji Kj is ill– de?ned. Fortunately, as we will see, in the dynamics of intersecting D –branes such a product will never show up. However, for notational covenience it will be useful to de?ne ? ? ? Ji Kj if Mi ? Mj , (Ji Kj )reg ≡ Ji χ(0) (3.2) Kj odd j if Mi ? Mj , ? ?0 if Mi ? Mj , Kj even.

This de?nition is motivated as follows. If Mi ? Mj then for the normal bundles we have Nj = Nij , and therefore for the Euler–forms χj = χij . If Kj is of odd rank, then Jj is even and χj = 0; if Kj is even, then Jj is odd and χj = 0. This implies that with the above de?nitions we have in any case d (JiKj )reg = (Ji Jj )reg .

We conclude this section presenting an alternative, but equivalent, way of writing the information contained in (3.1). We may rewrite its l.h.s. in terms of the restriction of Kj to Mi , Ji Kj = Ji (Kj |Mi ). Since this restriction is well–de?ned, in this form we can apply Leibnitz’s rule to get Ji d (Kj |Mi ) = Jij χij . Denoting the δ –function supported Poincar` e–dual of Mij w.r.t. Mi with Jij – this is a form on Mi and not on target–space – we have Jij = Ji Jij . The target–space relation (3.1) is then equivalent to the relation on Mi , d (Kj |Mi ) = Jij χij . (3.3)

We can go one step further and observe that, if the intersection is e?ectively non– transversal i.e. χij = 1, then the above relation is equivalent to the existence of a form Lij on Mi such that (0) Kj |Mi ? Jij χij = dLij , (3.4)

–9–

transforming under Q–transformations of Kj and under normal bundle transformations of Nij respectively as δ Lij = Qj |Mi , δ Lij = ?Jij χij ,

(1)

apart from closed forms. If the intersection is extremal, Kj |Mi is not de?ned and according to above one (0) (0) would rather consider the expression χj |Mi ? Jij χij , which vanishes identically since Jij = 1. This suggests to de?ne, for Mi ? Mj , Lij = 0.

4. Basic applications

We present here two basic applications of the above identity, to dual pairs of branes and to self–dual branes. These cases enter as main building blocks in the construction of the action for arbitrary intersections, given in the next section. These two examples illustrate the role played by Q–invariance, which is fundamental also in the general case. For the sake of clarity we ignore here all other couplings but the minimal ones. We restore now the brane charges gi and Newton’s constant G, taken until now as gi = 1 and G = 1/2π . 4.1 Electromagnetically dual pairs of branes Suppose that a RR–?eld strength satis?es the Bianchi identity and equation of motion dR = gj Jj , d ? R = g i Ji , (4.1) (4.2)

where Jj (Ji ) is the current on the high–dimensional (low–dimensional) magnetic (electric) D –brane with wordvolume Mj , (Mi ) and charge gj (gi ). Mi and Mj form an electromagnetically dual pair whose dimensionalities satisfy i + j = 8. For the self–dual D 3–brane the eq. of motion has to be replaced by R = ?R, but we do not consider this case in the present section. To write an action for such a system one must ?rst solve (4.1), introducing a potential for R, and to do this one must search for an antiderivative of Jj or of Ji (or of both). There are two candidates for such antiderivatives. The standard one is a Dirac– brane Mj +1 , i.e. a brane whose boundary is Mj , with δ –function supported Poincar` e dual, say Wj . Then one has Jj = dWj . The same can be done for Ji . The (9 ? j )–form Wj carries by construction δ –like singularities on Mj +1 and hence also on Mj . Since the Dirac–brane is unphysical

– 10 –

one must eventually ensure that it is unobservable. In this case one would solve the Bianchi identity (4.1) through R = dC + gj Wj , and C would carry the known singularities along the Dirac–brane and on Mj The second candidate for an antiderivative is a Chern–kernel Kj , which carries invariant inverse–power like singularities on Mj , Jj = dKj , and due to this fact the potential C introduced according to R = dC + gj Kj is regular on Mj , because all singularities are contained in Kj . Since the Chern–kernel is de?ned modulo Q–transformations, one must eventually ensure that the theory is Q–invariant, i.e. independent of the particular kernel one has chosen. We recall now the recipe developed in [4] for writing an action for the system (4.1), (4.2) if Mi and Mj have a transversal intersection. Then we will present its adaptation to a non–transversal one. (Remember that for a dual pair of branes a transversal intersection amounts to no intersection at all, while a non–transversal one means simply Mij = ?). The recipe goes as follows. Introduce a Chern–kernel for the magnetic brane Jj = dKj , and a Dirac–brane for the electric brane Ji = dWi . Then solve the Bianchi–identity (4.1) according to R = dC + gj Kj . Under Q–transformations of Kj the potential must now also transform, C ′ = C ? gj Qj ,

′ Kj = Kj + dQj ,

(4.3)

(4.4)

to keep the ?eld–strength R invariant. For the restrictions on Mj (2.7) implies then, for an odd kernel, (1) δC |Mj = ?gj χj , (4.5) while for an even kernel C |Mj is invariant. From these transformations one sees that the potential C is a ?eld regular on Mj , because δC |Mj is ?nite. Equivalently, all singularities of R are contained in the Chern–kernel Kj , more precisely in the invariant form ?j . The action, generating the equation of motion (4.2), is given by S= 1 G 1 R ? R + gi R Wi . 2 (4.6)

Under a change of the (unphysical) electric Dirac–brane, Wi → Wi + dZi, this action changes by an integer multiple of 2π , if the quantization condition gi gj = 2πn G holds, see e.g. [10]. Moreover, the action S is trivially free from gravitational anomalies.

– 11 –

Two comments are in order. First, the eq. of motion (4.2) could also be obtained 1 1 i R ? R + gi dC Wi = 21 R?R+ g C . Howmore simply from the action G 2 G G Mi ever, this action is inconsistent in that it breaks Q–invariance in the bulk, because C is not Q–invariant. Second, introducing an eleven–dimensional manifold which bounds target–space, the above W Z –term can be rewritten equivalently as gi G RWi = gi G (RJi ? gj Jj Wi ) = gi G RJi

M11

mod 2π,

(4.7)

M11

i where in the last step we used that Jj Wi is integer [10]. The eleven–form g RJi G 4 is indeed closed modulo a well–de?ned integer form , because we are assuming that i Mi and Mj are not intersecting: d( g RJi ) = 2πnJj Ji . G Let us now adapt this recipe to a dual pair with a non empty intersection manifold i RJi is no Mij , starting from (4.7). In this case, due to the identity (3.1), the form g G longer closed, not even modulo integer forms, and there is the additional problem that for extremal intersections the product Kj Ji is ill–de?ned. But, in turn, thanks to this identity we know how to amend the W Z –term. Replace

gi G

M11

RJi → SW Z ≡

gi G

M11

(RJi )reg ? gj Jij χij

(0)

,

(4.8)

where the subscript reg refers to the product Kj Ji , de?ned in (3.2): the integrand is now again a well–de?ned closed eleven–form. This is the W Z –term anticipated in the introduction, see (1.14), holding now for extremal intersections as well. For an extremal intersection, Mi ? Mj , due to χj = χij and Ji = Jij , the above W Z simpli?es to gi gi SW Z = dCJi = C. G M11 G Mi Despite the formal cancellation of the “anomalous” term χij , the anomaly itself has not disappeared. Indeed, the potential C transforms under a Q–transformation, and thanks to (4.5), since Mij = Mi ? Mj , we have δC |Mi = ?gj χj = ?gj χij . The anomaly polynomial supported on Mij is then ?2πnχij also for extremal intersections, and the action remains Q–invariant in the bulk. As we observed, for non empty (non extremal) intersections it is Q–invariance that forces to put in (4.8) the Q–invariant combination RJi instead of the closed form dCJi. So it is eventually Q–invariance in the bulk that requires the presence

“Integer forms” are by de?nition forms that integrate over an arbitrary manifold (closed or open) to an integer. It can be shown that all such forms are necessarily δ –functions on some manifold M . In particular our currents J are integer forms, and a product of integer forms, whenever it is well–de?ned, is an integer form as well.

4

(0)

(1)

(1)

– 12 –

of the anomalous term Jij χij , needed to get back a closed eleven–form in the W Z . This interplay between Q–invariance and anomalies will be a guiding principle also in our construction of an anomaly–free e?ective action for an arbitrary system of intersecting D –branes in the next section. We may then summarize the properties of SW Z in (4.8) as follows. 1) It is written in terms of a potential C that carries an anomalous transformation law but that is regular on Mj . 2) It gives rise to the equation of motion (4.2). 3) It exhibits no singularities. 4) For transversal intersections (Jij = 0) it is well–de?ned modulo 2π , Q–invariant and anomaly free. 5) For non–transversal intersections it is well– de?ned, it carries the gravitational anomaly ?2πnχij supported on Mij , and it is Q–invariant in the bulk. 6) For empty intersections the second term drops, because Jij is vanishing, and one gets back the W Z –term for transversal intersections. We conclude and summarize this subsection, giving a ten–dimensional representation of (4.8). For transversal intersections one introduces an electric Dirac–brane for Mi , Ji = dWi , and uses that (4.8) reduces to (4.7), while for non–transversal ones one uses (3.4). The results are: SW Z gi = G C+

Mi gi gj G gi gj G

(0)

Kj Wi L Mi ij

for transversal intersections, for non–transversal intersections.

(4.9)

We recall that for extremal intersections Lij is zero by de?nition. The eleven– dimensional representation (4.8) for SW Z is, however, universal in that it holds for arbitrary intersections, transversal or not. 4.2 Self–dual branes As second example we consider a self–dual brane, i.e. a brane Mj with 2j = D ? 2, which is coupled to a ?eld–strength satisfying a self–duality condition rather than a Maxwell equation, dR = gj Jj , R = ?R. The brane we will be interested in is clearly the self–dual D 3–brane in IIB –Sugra. Strictly speaking, this case can be treated without using the identity (3.1), so we recall simply the results of [4]. One solves the Bianchi identity as above, R = dC + gj Kj , and using the covariant P ST –approach [8] to deal with the self–duality condition, one can write the action as S= 1 4G (R ? R + f4 f4 ) + gj 2G C.

Mj

The additional (overall) factor of 1/2 will be explained in the next section. A part from this the W Z appearing here coincides with the one of the extremal intersection of a dual pair discussed above; a self–dual brane amounts in some sense indeed to

– 13 –

Mi = Mj , a special case of Mi ? Mj , i + j = D ? 2. The potential C transforms under Q according to (4.5). This means that S entails the anomaly polynomial

2 gj 2πn χj = ? χj , ? 2G 2

supported on Mj . Due to the presence of the NS B –?eld and of the gauge–?elds on the branes, the technical details of the anomaly cancellation mechanism of a generic system of intersecting D –branes in IIB –Sugra appear slightly involved. For this reason we presented the new ingredients of the Chern–kernel approach, which are crucial for the entire construction, separately and in some detail in these ?rst four sections. We conclude this more general part by stressing again that the Chern–kernel approach, as we saw, allows to write well–de?ned actions which entail no ambiguities or singularities, despite the dangerous short–distance con?gurations represented by branes intersecting non–transversally. Furthermore, this approach does not require any regularization (or smoothing) of the currents, a procedure that would immediately run into troubles with the unobservability of the Dirac–brane, see [5, 10].

5. Anomaly cancellation for intersecting IIB D–branes

In this section we consider the full interacting system of IIB supergravity and abelian D –branes (one for each dimensionality) with arbitrary intersections. We recall ?rst the quantum anomalies of the system, including from the beginning the NS two– form B in the dynamics. Then we give the full set of Bianchi identities and equations of motion and solve them in terms of regular potentials, applying systematically the proposals of eqs. (1.11) and (1.13). Next we present the action, check it is well de?ned and show it cancels all the anomalies. Eventually we discuss Born-Infeld actions and equations of motion on the branes. 5.1 Anomalies and the B ?eld On each D –brane with worldvolume Mi the pullback of the NS 2–form B couples to the abelian gauge ?eld Ai living on Mi through the invariant ?eld strength hi = B ? 2πα′Fi , Fi = dAi . (5.1)

Under a gauge transformation δB = dΛ, the U (1) potential transforms as δAi = where pull backs are understood. 1 Λ, 2πα′ (5.2)

– 14 –

The anomaly polynomial that describes the anomalies of the system has been derived in [1] and is given by the 12–form P12 = ?π (?1) 2 eγ (hj ?hi )

i

i,j

?(Ti ) A ?(Ni ) A

?(Tj ) A J χ , ?(Nj ) ij ij A

(5.3)

? where γ ≡ 4π1 2 α′ , A is the roof genus and Ni and Ti are the normal and tangent bundles respectively. Since hj ?hi = 2πα′ (Fi ?Fj ) the polynomial P12 is independent of B and invariant under (5.2), as it should. The term in the sum is symmetric under exchange of i with j , so that on single branes (which can be considered as self–intersection) a factor of π appears since summation is on diagonal terms i = j , while for intersecting branes one recovers a factor of 2π . (5.3) reproduces indeed the anomaly polynomial for both single and intersecting D –branes. Notice that the anomaly on the (self intersecting) D (?1)– and D 1–brane (i = 0, 2) vanishes since the degree (10 ? i) of the Euler form χii exceeds i +2, while the anomalies on self intersecting D 3, D 5, D 7– branes are di?erent from zero. The case of the D 9 is special in that the worldvolume theory is the super Yang–Mills part of type I String Theory [9] and its anomaly is cancelled by the Green–Schwarz mechanism. We do not include this contribution in our discussion since we consider it well known. The anomalies that have not been dealt with both in [2] and [1] are those for pairs of dual branes, D (?1)–D 7, D 1–D 5, D 3–D 3, where the integrand in (5.3) for dimensional reasons reduces entirely to the Euler form. In this case it was not understood how to perform an in?ow of charge and, as mentioned in [1], “a more powerful approach is needed”. This can be achieved using Chern–kernels as we have seen in the previous sections. The charge of the D –brane with worldvolume Mi is given by [9] √ i?4 gi = 2πG γ 2 , gi g8?i = 2πG. (5.4) As noticed in [1, 2] a crucial step towards anomaly cancellation is to notice that the anomaly polynomial (5.3) is partially ’factorized’. On each D –brane one can introduce the closed and invariant polynomial Yi = e? 2π

Fi

?(Ti ) A (0) = d Y i + 1. ? A(Ni )

(5.5)

The forms Yi are those appearing in the Bianchi identities and equations of motion of [1, 2], where B was kept zero. For a non vanishing B –?eld one notices that (5.5) is not invariant under (5.2), so what should appear in the Bianchi identities and equations of motion is rather the invariant expression Y i = eγhi ?(Ti ) A = eγBi Yi . ? A(Ni ) (5.6)

– 15 –

These Y i are, however, no longer closed and satisfy dY i = γY i H, H = dB. (5.7)

In terms of these forms one can de?ne the following forms of degree n = 2, . . . , 10, ?n = g10?n

i

(?1) 2 Ji Y i ,

i

(5.8)

where it is understood that on the r.h.s. one has to extract the n–form part. (5.4) implies then the important relation d?n = ?n?2 H. (5.9)

These forms are crucial since they allow eventually to factorize the anomaly polynomial completely. Indeed, it is not di?cult to show that one can rewrite (5.3) as 1 (?1)n/2 (?n ?12?n )reg , (5.10) P12 = ? 2G n where the ’regularized’ product of two currents has been de?ned in the previous section. This formula holds in the case of non trasversal intersections for all couples of dual branes. In case these intersections are all transversal one has to add the term ?2π (J6 J2 ? J8 J0 ) which subtracts the same term from the above expression. Notice however that, since for transversal intersections the form J6 J2 ? J8 J0 is integer, it gives rise through the descent formalism to an anomaly which is a well de?ned integer multiple of 2π , and can be disregarded. Notice also that (5.10) is independent on B – despite its explicit appearance – as is obvious by construction. This can also be checked explicitly noticing that under a generic variation of B one has δ ?n = ?n?2 δB. (5.11)

δP12 vanishes under such a variation thanks to the alternating signs (?1)n/2 in eq. (5.10). This explains also the appearance of those signs. Evenutally, since P12 is invariant under (5.2) there exist a Chern–Simons form and a second descent which respect this symmetry, P12 = dP11 , δP11 = dP10 . The resulting anomaly A= P10 (5.12)

can therefore always be choosen to respect this symmetry, too.

– 16 –

5.2 Bianchi identities and equations of motion In this section we describe the theory of IIB supergravity in presence of arbitrary D –branes and with a non zero B ?eld turned on. We give new Bianchi identities and equations of motion that apply in the presence of D –branes . Using the techniques developed in the previous sections we solve Bianchi identities in terms of regular potentials, and then proceed to write down a classical action that gives the equations of motion. Q–invariance of this classical action generates an anomalous transformation law that exactly cancels the quantum anomaly. Start de?ning the formal sum of generalized currents as ?=

n

?n ,

(5.13)

where ?n is de?ned in (5.8). The full set of Bianchi identities and equations of motion is given by the compact formula dR = RH + ?, (5.14)

where R9 = ?R1 , R7 = ?R3 and R5 = ?R5 . Three comments are in order. First, in the limit where each brane charge gi is set to zero, (5.14) reduces to the Bianchi identities and equations of motion of free IIB supergravity. Secondly, (5.14) is well de?ned since the right hand side is a closed form, as can be seen using (5.9). Third point to mention is electro-magnetic duality. While an equation of the form dR = J is electro-magnetically symmetric, the full equation (5.14) is not, because of the RH and JY terms. These latter can be thought of as currents associated to smeared branes. The compact formula describing solutions of the Bianchi identities is R = dC ? CH + f, where fn , n = 1, 3, 5, is the n–form fn = g9?n

i

(5.15)

(?1) 2 Ki Y i ,

i

(5.16)

and it satis?es df = f H + ?. (5.17) Notice that under Q transformations δKi = dQi and the RR curvatures are invariant provided the potentials transform as δCn = ?g8?n

i

(?1)i/2 Qi Y i ,

(5.18)

and therefore the pullback of the potentials on the branes is regular. Such pullback amounts to an anomalous transformation of the potential, which plays an important role in cancellation of anomalies.

– 17 –

As a speci?c example, the corrected form of the Bianchi identities for R1 and R3 is dR1 = g8 J8 , dR3 = R1 H + g6 (?J6 + J8 Y 8,2 ), (5.19) (5.20)

(by Y 8,2 we mean the degree 2 part of the Y form de?ned on the D 7 brane). Now we introduce Chern-kernels K8 , K6 (and K4 for R5 ) associated to the D 7, D 5 (and D 3) branes that appear in the Bianchi identities. Further sources appearing in the equations of motion, J2 and J0 , have instead to be treated using Dirac branes W2 and W0 , as explained in [4]. The solution we propose for the Bianchi identities is R1 = dC0 + g8 K8 , R3 = dC2 ? C0 H + g6 (?K6 + K8 Y 8,2 ), (5.21) (5.22)

and similarly for R5 . It is straightforward to check that these de?nitions ensure that the Bianchi identities are satis?ed, using (5.7), (5.4). Notice the fact that this solution requires the Y forms to be extended to target space forms. One could object that a more standard way to solve Bianchi identites that involves only Y forms evaluated on branes would rather be ?0 + g8 K8 , R1 = dC ?2 ? C ?0 H + g6 (?K6 + J8 Y8(0) R3 = dC , 1 + K8 B ) , (5.23) (5.24)

and similarly for R5 . This is the approach used in [1], and we remark that there the solution of Bianchi identities is incomplete, in that it misses the terms were K appears without any Y form. However, this de?nition leads to singular potentials and therefore to inconsistencies. Consider for example an anomalous gauge trans(0) (0) (1) formation of Y8,1 in (5.24), δY8,1 = dY8,0 . Since the curvature R3 is invariant the ?2 transforms accordingly as potential C ?2 = ?g6 J8 Y8,0 . δC

(1)

(5.25)

?2 is always singular on the D 7–brane so that C ?2 itself is singular. The variation of C ?4 would be singular on the D 7 and D 5– Similarly, one shows that an analogous C branes. We now show that the RR curvatures are independent of the extension of the (0) Y forms. Consider target space forms Yi = dYi + 1. Under a change of extension they transform as (5.26) δYi = dXi , where Xi is an arbitrary target space form such that J i Xi = J i Xi | M i = 0 , (5.27)

– 18 –

since Yi is well de?ned on its D –brane. From here on we will refer to these as X – transformations. Here we explicitly consider the R3 curvature, but similar formulae apply to all the potentials C . Under (5.26) R3 is invariant if C2 shifts by δC2 = ?g6 K8 X8,1 . (5.28)

Such X –transformations of the potentials always exist due to consistency of the Bianchi identities. Even though K8 does not admit limit close to the D 7–brane, it remains ?nite and the product K8 X8,1 is well behaved and goes to zero since X8,1 goes to zero. This proves that the RR potentials as de?ned by (5.15) are completely regular close to the branes, and that the curvatures do not depend on the arbitrary extensions of the Y forms. In the next section we will present the action for the system and see that it does not depend on such extensions. 5.3 Action and anomaly cancellation In this section we present an action that gives rise to the equations of motion (5.14). There are two ways to discuss the Wess-Zumino part of the action. One possibility is write it as the integral of a closed 11–form. The advantage of this formulation is that it is the most clear one: it immediately displays Q and X –invariances of the theory and how the gravitational anomaly is cancelled. However, even if the procedure is rigorous and does not depend on the arbitrary extra dimension, nevertheless it is a natural expectation to ask for the existence of a well de?ned ten-dimensional action. Our approach to the problem will be that of presenting at ?rst the Wess-Zumino part of the action as the integral of an 11–form and discuss its properties. A well de?ned 10D action will be given in the end of the section, and its relationship with the former discussed in the appendix. We de?ne the total e?ective action of the theory as the sum of a classical and quantum part 1 Γ = (Skin + SW Z ) + Γquant , (5.29) G where Γquant generates the anomalies described in sect.5.1. Skin is given by Skin = +

1 2

√ 1 d10 x ?ge?2φ R + 2 e?2φ (8H1 ? H1 + H3 ? H3 ) 1 R1 ? R1 ? R3 ? R3 + 1 R ? R5 ? 2 f4 ? f4 , 2 5

(5.30)

where the ?eld f4 = ιv (R5 ? ?R5 ) appears as part of the PST approach [8], enforcing the self-duality equation, and v m is the unit vector ?m a vm ≡ √ , ?a?a vm v m = 1, (5.31)

where a is an arbitrary scalar ?eld. The PST formulation then guarantees that a is a non propagating, auxiliary ?eld.

– 19 –

To describe the Wess-Zumino term instead take an eleven dimensional manifold M11 whose boundary is the spacetime M10 of the theory, ?M11 = M10 . Assume no topological obstruction, and take the extra dimension to be parallel to the D –branes so that the degree of J forms, which counts the number of normal directions to a brane, is not changed. Then one can write the Wess-Zumino as SW Z =

M11

L11 ,

(5.32) (5.33)

dL11 = 0, and L11 is given by 1 L11 = ? R5 (R3 H + ?6 ) ? R3 ?8 ? R1 ?10 2 ? GP11 .

(5.34)

reg

P11 is the Chern-Simons form of the anomaly and is given by P11 = π

i,j (0)

(?1)i/2+1 Pij Jij χij + 2π (J2?6 χ2?6 ? J0?8 χ0?8 ) ? πJ4 χ4 ,

(0)

(0)

(0)

(0)

(5.35)

and Pij , in turn, is a Chern-Simons form of5 Pij = e ( F i ?F j )

2π

?(Ti ) A ?(Ni ) A

?(Tj ) A . ?(Nj ) A

(5.36)

Some comments are in order. A direct check shows that the classical part of the action gives the correct equations of motion. Moreover, Q and X –invariances in this picture are immediately displayed, since only RR invariant curvatures appear. Anomaly cancellation is guaranteed since the only term in the action which is not invariant under anomalous transformations is ? P11 , whose variation exactly cancels the quantum anomaly. Lastly, notice the factor of 1/2 in the minimal coupling R5 ?6 of eq.(5.34). On one side it is an artifact of the PST formalism (see the kinetic part of the action), and should not misunderstood as a novel feature. On the other (0) side, it combines exactly with πJ4 χ4 leaving only a C4 potential term, that cancels the anomaly on the self-dual D 3–brane according to what explained in sect.4.2. We now write down the Wess-Zumino term in a ten dimensional fashion. Here we simply display it as it is, leaving the proof of its equivalence with (5.34) to the appendix. We split the Wess-Zumino into a term depending on the potentials and a remainder: rem SW Z = (LC (5.37) 10 + L10 ). The part depending on the potentials is 1 1 LC 10 = ? C4 (R3 H + ?6 ) ? C2 Hf5 ? C2 ?8 ? C0 ?10 . 2 2

5

(5.38)

Notice that Pij = dPij + 1, which explains the form of (5.35).

(0)

– 20 –

The remainder can be written in two ways. In the case of transversal intersection for dual branes it is given by Lrem 10 = 1 f3 f7 ? f1 f9 + 2πG 2 (?1)i/2 Pij Ki Jj + 4πG(K6 W2 ? K8 W0 ) , (5.39)

(0)

i,j

where the forms f7 , f9 are formally de?ned as in (5.16), but using Dirac branes W2 and W0 instead of K2 and K0 . For non-transversal intersection the two last Ki Wj terms have to be modi?ed according to eq.(4.9). In this 10D picture, Q and X –invariances are hidden and have to be checked one (0) by one. Anomaly cancellation arises for non dual branes from the terms Pij Ji Kj . For dual branes with non extremal intersection it arises from the terms of the kind Lij Ji . If the intersection is extremal then Lij = 0 and the anomaly is cancelled by the anomalous transformation of the potentials. This is always true for the D 3–brane. rem All the other Q–variations instead are cancelled between terms in LC 10 and in L11 . 5.4 Born-Infeld actions and equations of motion on the branes In this section we describe the dynamics of U (1) ?elds on each D –brane and of the NS form B . The action (5.29) describes all the dynamics of RR ?elds but, as it stands, is not complete. One has to add to it Born-Infeld terms for the U (1) ?elds on each brane: 1 (5.40) Γ = (Skin + SW Z + SBI ) + Γquant , G with SBI =

i i IBI =? i gi IBI ,

(5.41)

i i ). ?det (gmn + Bmn ? 2πα′Fmn

dxi e?φ

Di

(5.42)

From such Born-Infeld term on can de?ne generalized ?eld strenghts ?i = h mn

i 2 δIBI , i δBmn ?detgmn

(5.43)

i ? i. justi?ed by the fact that under a variation of the ?eld B one has δIBI = Di δB ? h In terms of these one can obtain, after a straightforward but lenghty calculation, the equation of motion of B :

d ? H = R3 R5 ? R1 R7 +

i

? i, g i Ji ? h

(5.44)

and the equations of motion for the U (1) ?eld strenghts, that we report in appendix. These new equations of motion have three important properties. First of all, they are

– 21 –

invariant under Q and P –transformations. This is required by consistency since the action we wrote down is invariant in ?rst instance. Q–invariance happens because a direct check shows that such equations display no dependence on Chern-kernels Ki . P –invariance is evident since the equations are expressed in terms of Y forms pulled back on the appropriate branes and of RR curvatures. Second point is that the equations are explicitly invariant under gauge transformations of B δB = dΛ, 1 δAi = Λ|Di , 2πα′ (5.45) (5.46)

again as it should, by consistency. Third property is that, as expected, the U (1) theory on the branes is anomalous. If one writes the equations of motion in the form ?i = ? d?h ji , then an explicit check shows that d? ji = 0. (5.48) However the equation of motion for B (5.44), even though it contains the anomalous ? i , is non anomalous. ?eld strenghts h The action constructed so far only involves C0 , C2 , C4 potentials. Often in the literature the action is written in term of all the RR potentials, and hence in the next section we rewrite our results in a duality-invariant language. (5.47)

6. PST duality–invariant formulation

In order to make contact with the formulation of [1] in this section we construct the action for the same system but using all the possible RR potentials Ci , i = 0, 2, . . . 8, instead of the minimal ones C0 , C2 , C4 . A duality-invariant formulation may also be useful for the purpose of analysing the ?ux quantization of dual potentials, or for dimensional reductions involving dual branes and dual potentials. Let us then introduce the new RR potentials C6 , C8 and de?ne the forms f7 and f9 as in (5.16), but this time using proper Chern-kernels K2 and K0 instead of Dirac branes. Introduce RR curvature R7 and R9 using the same recipe of (5.15). In order to deal with C6 and C8 one has to exploit the PST formalism. Introduce an arbitrary scalar ?eld a and and construct the unit vector v m as in (5.31). In term of v m construct the forms r0 ≡ ιv (R1 ? ?R9 ), Then the PST duality-invariant action is Sdual = Skin + SW Z + SBI + 1 (r 2 ? r 2 ? r 0 ? r 0 ) . 2 (6.3) (6.1) (6.2)

r2 ≡ ιv (R3 ? ?R7 ).

– 22 –

Such action has all the PST symmetries necessary to prove that a is an auxiliary ?eld and that the conditions R9 = ?R1 , R7 = ?R3 are enforced (see [8], [4]). In order to make contact with the usual formulations one can use the following identities: R3 ? R3 ? r2 ? r2 = (R3 , R7 ) P (v ) R1 ? R1 ? r0 ? r0 = (R1 , R9 ) P (v ) where P (v ) is the operator valued matrix ?vιv ? vιv vιv ?vιv ? . (6.6) R3 R7 R1 R9 ? R3 R7 , + R1 R9 , (6.4) (6.5)

Substituting this into the action (6.3) gives Sdual = Skin,dual + SW Z,dual + SBI , where Skin,dual is a kinetic term for all the RR potentials given by 1 1 + R5 ? R5 ? f4 ? f4 , 2 2 (6.7) where there is symmetry under R1 ? R9 , R3 ? R7 , see [10]. SW Z,dual = Ldual is a modi?ed Wess-Zumino term whose 11D version reads Skin,dual = (R1 , R9 ) P (v ) R1 R9 ? (R3 , R7 ) P (v ) R3 R7 L11,dual = ? while the 10D one is

1 L10,dual = ? 2 n n

1 2

1 2

n

Rn+1 ?10?n ? GP11 ,

(6.8)

Cn ?10?nn + Rn+1 f9?n +

2π G 2 2π G 2

+ 2(K6 W2 ? K8 W0 ) . (6.9) Again the usual remark for non-trasversal intersections of dual branes applies, where eq.(4.9) should be used. Now we can try to make contact with eq.(2.11) of [1]. In our notation it says that on each brane the Wess-Zumino term goes like ? 1 2 ?i + R Y (0) . C i (6.10)

= ?1 2

i/2 (0) i,j (?1) Pij (Ki Jj )reg

i,j (?1)

i/2

Pij (Ki Jj )reg + 2(K6 W2 ? K8 W0 )

(0)

Mi

From eq.(5.16) one can decompose the forms fn , in the limit B ≡ 0, as fn = g9?n

i

(?1) 2 (Ji Yi

i

(0)

+ K i ) + d ( Ki Y i )

(0)

n

,

(6.11)

and plug them into the second line of (6.9). Consider the ?rst three terms in (6.11). fn on its own has only inverse power singularities near each brane, but the ?rst and

– 23 –

third term in the decomposition individually display δ –like singularities. Therefore, in (6.9) it is not allowed to multiply each single term times a RR curvature, but only the whole sum. Suppose however we want to formally forget about this di?culty. Then we can see that the ?rst term in (6.11) reproduces the second term of (6.10). The second term has two e?ects. Part of it is multiplied in eq.(6.9) times the potential (0) ?i part of part in R. Joint with some of the Pij (Ki Jj )reg terms, it reconstructs the C ?i is purely formal since the latter is (6.10). Remember that the passage from Ci to C ill-de?ned. The remaining part of the second term in (6.11), together with the third (0) term d(Ki Yi ), are dependent on the Chern-kernels Ki and cancel completely with (0) the rest of the Pij (Ki Jj )reg terms of (6.9). This is guaranteed since the (ill-de?ned) ?i of [1] are Q-invariant and so the K -dependent terms have to disappear. potentials C In conclusion, in the formal approximation when one can forget δ –like divergen?i potentials, one cies, in the limit B ≡ 0 and assuming it is possible to use the C exactly recovers the Wess-Zumino term of [1], plus the extra terms that cancel the anomaly for dual branes. The anomaly cancellation for the self-dual D 3–brane is given by transformations of C4 which are present only in the Chern-kernel formulation and cannot be reproduced in the context of [1].

7. Conclusions and outlooks

We conclude by summarizing our results and commenting on their extension. We have considered the system IIB supergravity interacting with all possible combinations of single D –branes, with arbitrary intersections as long as there are no topological obstructions. We have constructed a regular action that gives the equations of motion, which is written in terms of potentials that are everywhere well de?ned. We have provided a correct understanding of the mechanism of charge in?ow using Chern-kernel techniques. In particular we have shown that, for pairs of dual branes which had proven to be intractable before, charge in?ow is not produced by curvatures but either by potentials, in the case of the self-dual D 3–brane and extremal intersections of dual branes, or by the L forms for non-extremal, nontransversal intersections of dual branes. Another important part of the understanding of charge in?ow, that is used in order to implement a Chern-kernel analysis, is the new fundamental identity (3.1). The Wess-Zumino term we obtained di?ers from other expressions that appeared in the literature, like those of [2], [1], [11] and in particular it contains extra terms which are related to anomaly cancellation for pairs of dual branes. Moreover, it contains all the corrections due to the presence of the NS form B . We have obtained the full, corrected equations of motion for B interacting with RR ?elds, gravity and U (1) Yang-Mills ?elds, and for the U (1) ?elds themselves. The

– 24 –

classical U (1) theory on the branes is anomalous but quantum corrections restore the full symmetry. We insist on remarking that Chern-kernel techniques have wide application to all theories with extended objects, and not only D –branes of supergravity. They can be used for example to deal with orientifolds, like in problems considered in [12, 13], with O –planes [14], with non BPS branes [15]. Another possible system to which apply these techniques is IIA supergravity in presence of D –branes. There, branes have odd dimensional worldvolume but they still admit anomalies on their intersections. Another possible generalization is to couple our system of D –branes to an NS 5– brane, which is interesting since in that case the NS curvature would not be closed. Its treatment should go along the lines of [6] and we expect it to be straightforward to implement. Lastly, we discuss generalization to the U (N ) case. In this case, it is reasonable to argue that, in costructing physical U (N ) ?elds on each brane, the colourless NS form B will be coupled to some U (1) subgroup on U (N ). Let F then be the full U (N ) curvature, and decompose it into a U (1) part F , that couples to B as in (5.1), ? . Since the U (1) part commutes with the rest, and an SU (N ) part with curvature F it is easy to see that the non-abelian Chern character that enters in the anomaly has to be generalized to ? F F → e?γh ch . (7.1) ch 2π 2π This would be the ingredient necessary to form the new Y forms. Since B enters in the U (1) part the identity (5.7) continues to hold and from this one is able to impose again Bianchi identities, equations of motion, and to ?nd a Wess-Zumino term for the action from which they come from. The only limitation is that one does not have a Born-Infeld action that is uniquely ?xed so far, and therefore for the time being it is not possible to ?nd equations of motion for B and the Yang-Mills ?elds on the brane.

8. Appendix: proof of the new identity

The following proof holds for arbitrary Chern–kernels, even or odd. We begin by (0) considering the properties of Ji Kj . As we saw, Kj = ?j + χj is singular on Mj because ?j involves y ?a = y a /|y |, and Mj stays at y a = 0. But since Mi ? Mj the product J i K j = J i ( Kj | M i ) is well–de?ned and, therefore, in the sense of distributions also its di?erential is so. The subtle point is only that one can not apply Leibnitz’s rule to evaluate it, because the product has (inverse–power and δ –like) singularities on Mij . Away from Mij one

– 25 –

can apply Leibnitz and there the result is d(JiKj ) = 0. This means that d(Ji Kj ) is supported on Mij and hence proportional to Jij , d (Ji Kj ) = Jij Φ, for some form Φ de?ned on Mij . Furthermore, since the l.h.s. is closed also Φ must be a closed form. Moreover, Φ must be a completely invariant form as is the l.h.s., ′ because Ji is intrinsically de?ned and Kj transforms as Kj = Kj + dQj , where Qj is regular on Mj . This means that one can apply Leibnitz and d(Ji dQj ) = 0. Furthermore, Φ can depend only on the curvature components of the intersection of the normal bundles Nij . This can be seen as follows. Since Kj is made out only of gravitational curvatures belonging to Nj , also Φ is a polynomial made out only of (a subset of) those curvatures. If (in addition to Mi ? Mj ) we have also Mj ? Mi , we can apply Leibnitz to d(KiKj ) = Ki Jj ± Ji Kj , giving d(JiKj ) = ±d(Jj Ki ). This implies that Φ depends, moreover, only on the curvatures of Ni , and hence only on those of Nij . If on the contrary Mj ? Mi then, using e.g. the regularizations of [4], (0) one can show that d(Ki Kj ) = χi Jj ± Ji Kj . Applying the di?erential to this one gets directly (3.1), since in this case Jj = Jij and χi = χij . Eventually, Φ is a form of degree dim(Mij ) + D ? (i + j ) and it is odd under parity. Φ shares all these properties uniquely with the Euler–form of Nij and we conclude therefore that it is proportional to it. A cohomological argument can ?nally be used to ?x the proportionality coe?ε ε cient to one. Perform a regularization Kj → Kj , Jj → Jjε , Jjε = dKj , as for example ε the one given in the appendix of [4], where Jj is cohomologically equivalent to Jj ε and regular on Mj . Then d (Ji Kj ) = limε→0 d Ji Kj = limε→0 Ji Jjε , and as shown in [1], Ji Jjε is cohomologically equivalent to Jij χij for every ε. This proves that the l.h.s. of (3.1) is cohomologically equivalent to Jij χij , and it ?xes the proportionality coe?cient of our local i.e. point–wise derivation to unity.

9. Appendix: Equations of motion for the U (1) theory on the D–branes

The equations of motion obtained by the action (5.40) for the U (1) ?elds are: ? 2 = ?R1 , d?h

(9.1) g6 A4 ? A8 J8 , 2 2π 1 g4 A6 ? A8 F6 ? F8 R1 Y 6,4 + J8 , 2 γ 3 2π 2π 1 1 R3 Y 8,4 ? 3 R1 Y 8,6 2 γ γ (9.2) (9.3)

? 4 = +R3 ? 1 R1 Y 4,2 ? d?h γ ? 6 = ?R5 + 1 R3 Y 6,2 ? d?h γ ? 8 = +R7 ? 1 R5 Y 8,2 + d?h γ

– 26 –

?g2

1 A8 ? A4 1 A8 ? A6 F8 ? F6 . J4 ? J6 2 2π 3 2π 2π

(9.4)

On the right hand side the regularized products of currents and Chern-kernels are always understood.

10. Appendix: the Wess–Zumino term

Here we make contact between the Wess-Zumino written as the integral of an 11– form (5.34) and the one written in usual ten dimensional notation, (5.38) and (5.39). The procedure one realizes in practice is the following: ?rst of all construct (5.38), that is completely ?xed by equations of motion as showed in section (5.3). Then, the remaining part (5.39) is completely ?xed by asking invariance of the action under Q and P –transformations. The actual calculations are lenghty, though straightforward, and we do not include them here. Once the ten dimensional Wess-Zumino is ?xed, one can take its di?erential and get the much simpler form (5.34), which displays all invariances and anomaly cancellations at a ?rst sight. What we do here instead is to proceed in the opposite direction, that is to show how to transform (5.34) into the ten dimensional Wess-Zumino. As a ?rst step, consider (5.34) and extract all the terms dependent on the potentials. After some algebra and integration by part one shows that this amounts to 1 1 d ? C4 (R3 H + ?6 ) ? C2 Hf5 ? C2 ?8 ? C0 ?10 = dLC (10.1) 10 . 2 2 Now, take the remainder in (5.34). This is equal to + = = =

1 f (f H ? ?6 ) + f3 ?8 ? f1 ?10 ? GP11 2 5 3 1 1 d(f3 f7 ? f1 f9 ) + 2 (??2 f9 + ?4 f7 ? ?6 f5 + ?8 f3 ? ?10 f1 ) 2 n 1 1 d(f3 f7 ? f1 f9 ) + 2 Σn (?1) 2 +1 ?10?n fn+1 ? GP11 2 n 1 1 d(f3 f7 ? f1 f9 ) + 2 Σn (?1) 2 +1 ?10?n fn+1 B2 =0 ? GP11 , 2

? GP11

(10.2)

where in the last passage independence from B2 depends crucially on the alternating sign and can be checked using (5.11) and an analogous variation for f . Given this, one shows with some algebra that Σn (?1) 2 +1 ?10?n fn+1

n

B2 =0

= 2πG

i,j

(?1)i/2+1 Pij Ki Jj .

(10.3)

Putting together eqs.(10.2), (10.3) and the expression (5.35) for P11 one gets that the reminder is exactly given by (5.39).

– 27 –

Acknowledgements. The authors thank P.A. Marchetti for useful discussions. This work is supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00131 Quantum Spacetime. M. C. is funded by EPSRC, Fondazione Angelo Della Riccia, Firenze and Cambridge European Trust.

References

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