D-Brane Wess-Zumino Terms and U-Duality

We construct gauge-invariant and U-duality covariant expressions for Wess-Zumino terms corresponding to general Dp-branes (for any p<D) in arbitrary 2<D<11 dimensions. A distinguishing feature of these Wess-Zumino terms is that they contain twice as many scalars as the 10-D compactified dimensions, in line with doubled geometry. We find that for D<10 the charges of the higher-dimensional branes can all be expressed as products of the 0-brane charges, which include the D0-brane and the NS-NS 0-brane charges. We give the general expressions for these charges and show how they determine the non-trivial conjugacy class to which some of the higher-dimensional D-branes belong.


Introduction
Since their invention in 1995 D-branes have played a crucial role in understanding the (non-perturbative) nature of string theory [1]. D-branes occur as brane solutions of the low-energy supergravity limit of string theory [2]. Their dynamics and interactions can be described by an appropriate worldvolume action. Unlike the usual brane actions the Dbrane actions do not only contain worldvolume embedding scalars describing the position of the brane but also a worldvolume vector field describing the fact that Fundamental strings may end on the brane. It is well-known that the kinetic terms for the embedding scalars and worldvolume vector are given by a so-called Dirac-Born-Infeld (DBI) action. In view of this the worldvolume vector is often called the Born-Infeld (BI) vector. The coupling of the D-brane to the Ramond-Ramond (RR) gauge potentials is described by a so-called Wess-Zumino (WZ) term. 1 Using a short-hand notation the WZ terms of all Dp-branes, with 0 ≤ p ≤ 9, in Type II string theory are given by [3] 2 L WZ (D=10) = e F 2 C . (1.1) Here C is defined as the formal sum 3 where C n (1 ≤ n ≤ 10) are the pull-backs of the RR n-form potentials, i.e.
C i 1 ...in = ∂ i 1 X µ 1 · · · ∂ in X µn C µ 1 ···µn . (1. 3) The X µ (µ = 0, 1, · · · , 10) are the embedding worldvolume scalars and ∂ i (i = 0, 1, · · · , p) is the derivative with respect to the worldvolume coordinates of the Dp-brane. Furthermore, F 2 is the worldvolume 2-form curvature tensor of the BI vector V 1 given by where B 2 is the Neveu-Schwarz (NS-NS) 2-form potential. Using the above notation and the Bianchi identity one can show that the WZ term is invariant under where λ is the formal sum of the RR gauge parameters λ n (0 ≤ n ≤ 9): λ = n λ n . (1.7) Type IIA string theory has Dp-branes for all even p (p = 0, 2, 4, 6,8). The D0-brane is a particle, the D2-brane a membrane etc. The D8-brane is a domain-wall (one transverse direction) which couples to the RR 9-form potential [6] that is dual to the Romans parameter m in massive IIA supergravity [7]. On the other hand, Type IIB string theory has Dp-branes for all odd p (p = 1, 3,5,7,9). The D1-brane is the D-string etc. The D9-brane is a so-called space-filling brane that couples to the RR 10-form C 10 of IIB supergravity. Note that the 10-form potential does not describe any physical degree of freedom and is therefore often not given in combination with the standard IIB supergravity multiplet. Nevertheless, it is needed to describe the coupling of the D9-brane to IIB supergravity and it is perfectly consistent with the IIB superalgebra to add this potential to the standard multiplet [8].
A distinguishing feature of Type IIB string theory is that it has a manifest SL(2, R) S-duality under which the D-branes transform non-trivially. For instance, the D1-brane transforms, together with the Fundamental string F1, as a doublet of SL (2, R). This can be seen from the fact that IIB supergravity has a doublet of 2-form potentials A α 2 (α = 1, 2). The two components of the doublet describe the NS-NS and RR 2-forms. Similarly, the other even-form potentials of IIB supergravity transform as a singlet, doublet, triplet and quadruplet+doublet, respectively: A α 2 , A 4 , A α 6 , A αβ 8 , A αβγ 10 , A α 10 . (1.8) The 10-forms are special in the sense that they occur as a reducible representation of the duality group: a quadruplet and a doublet. It turns out that it is the quadruplet that contains the RR 10-form that couples to the D9-brane [8].
In order to write down the WZ terms for the D-branes of Type IIB string theory one needs to relate the S-duality covariant A-fields given in (1.8) to the RR-fields C occurring in the expression (1.1) of the WZ term. In order to do this in a duality-covariant way one first introduces charge vectorsq α , q α and defines [9] withq α q β ǫ αβ = 0, 4 such that C 2 is the RR 2-form coupling to the D1-string and B 2 is the NS-NS 2-form coupling to the Fundamental string. Assuming thatq α and q α transform as doublets under SL(2, R) this fixes in a duality-covariant way which component of the 2-form doublet we define as the RR 2-form and which component as the NS-NS 2-form. The leading term in the WZ-term of the D3-brane, i.e. (the pull-back of) the RR 4-form can be written as [9] C 4 =q α q β −iǫ αβ A 4 − 1 16 A α 2 A β 2 . (1.10) The difference between A 4 and C 4 is that A 4 transforms under both the RR gauge transformations of C 2 , C 4 and the NS-NS gauge transformations of B 2 whereas C 4 only transforms under the RR gauge transformations. The requirement that C 4 transforms as in (1.6), which is required for the gauge invariance of the WZ term (1.1), fixes the relation (1.10) between the RR-field C 4 and the duality-covariant A-fields. Extending this to the higherrank n-forms the general WZ-term can be written in the universal form (1.1). All C-fields are related in a duality-covariant way to the A-fields, using the basic chargesq α , q α . The world-volume 2-form curvature F 2 is given by (1.11) Here we have introduced, together with the doublet of 2-form potentials A α 2 , a doublet of worldvolume vectors V α 1 . In this way the charges of all Type IIB Dp-branes with p ≥ 3 can be expressed as products of the basic 1-brane chargesq α , q α , corresponding to D1-branes and F1-strings, respectively [9]. The same analysis shows that • in the case of the 10-forms, which belong to a reducible representation of SL(2, R), it is the quadruplet, that is the highest-dimensional representation, that contains the RR field C 10 . One cannot write down a gauge-invariant WZ term for the doublet; • for both the triplet of 8-forms and the quadruplet of 10-forms, not all n-forms inside the representation can be reached by a duality rotation of the RR-field. More specifically, there is one 8-form out of the triplet of 8-forms and there are two 10-forms out of the quadruplet of 10-forms that cannot be viewed as a duality transformation of the RR n-form. For these potentials one cannot write down a gauge-invariant WZ term.
It is well-known how the SL(2, R) symmetry of IIB supergravity gets generalised to other symmetry groups for the D < 10 maximal supergravities [10,11]. These symmetries have been identified as the U-dualities, containing S-and T-dualities, of superstring theory [12]. They have also been discussed in the context of worldvolume actions of extended objects in supergravity backgrounds [13]. The different duality groups for 3 ≤ D ≤ 11 are given in Table 1. The generic global symmetry group in D dimensions (D ≤ 9) can be denoted (generalising what one gets in 3,4 and 5 dimensions) by E 11−D , where 11 − D is the rank of the group.
In each dimension the corresponding maximal supergravity theory contains a number of n-forms that transform in given representations of the U-duality group. These representations naturally follow by making a level decomposition of the very extended Kac-Moody Table 1: The U-duality groups for all maximal supergravities in dimensions 3 ≤ D ≤ 11. The group is always over the real numbers and of split real form. In D = 10 we distinguish between IIA and IIB supergravity.
algebra E 11 [14,15,16,17,18]. Following the notation of [19], for each n-form we denote these representations with a lower M n index. The forms are thus denoted by All fields decompose into representations of the T-duality group SO(10 − D, 10 − D), which is a subgroup of the U-duality group E 11−D . It is convenient to make this decomposition since all RR fields transform as irreducible representations of the T-duality group. A convenient way to see whether a given n-form potential is a RR field is by calculating the corresponding brane tension. For a RR field this tension should scale as 1/g s in the string frame, with g s the string coupling constant. It turns out that in each dimensions D the RR fields of odd rank transform in the spinor representation of SO(10 − D, 10 − D) and we denote them with C 2n−1,a , with a an SO(10 − D, 10 − D) spinor index, while the RR fields of even rank transform in the conjugate representation and we denote them with C 2n,ȧ [20]. Besides the RR-fields we need to consider the Fundamental 2-forms and 1-forms that couple to the Fundamental string and Fundamental 0-branes, i.e. wrapped Fundamental strings, respectively. These Fundamental fields have corresponding brane tensions that are independent of the string coupling constant, again in the string frame. In each dimension the Fundamental 1-forms transform in the vector representation of SO(10 − D, 10 − D) and we denote them with B 1,A , while the Fundamental 2-form is a T-duality singlet, B 2 . As we will discuss in detail in section 4, the equivalent of the chargesq α , q α introduced in Type IIB string theory will be chargesq M 1 a , q M 1 A , such that define the RR and Fundamental 1-forms in a U-duality covariant way. We will see in section 4 how all the charges of the higher-dimensional Dp-branes, with p ≥ 1, can be expressed as products of these basic 0-brane charges. Unlike in Type IIB string theory, all charges can be expressed in terms of 0-brane charges only, no 1-brane charges are involved. This has to do with the fact that, for D < 10, the basic gauge symmetries generating the whole gauge algebra are always the ones corresponding to the 1-form potentials only. The aim of this paper is to construct gauge-invariant and duality-covariant expressions for general WZ terms using the ingredients introduced above. These WZ terms will describe the coupling of general Dp-branes in dimensions 3 ≤ D ≤ 10 to the target space supergravity fields in a duality-covariant way. For D = 10 there have been attempts to construct such WZ terms. For instance, an SL(2, R)-invariant formulation of 1-branes has been given [21,22]. This formulation made use of the fact that in two spacetime dimensions the Born-Infeld vector is equivalent to an integration constant describing the tension of a string. Similarly, the case of 3-branes has been discussed [23]. In this case one makes use of the fact that in 4 spacetime dimensions the electric-magnetic dual of a Born-Infeld vector is again a vector. Such special properties do not occur for the branes with p > 3 and indeed constructing an SL(2, R)-invariant formulation of 5-branes turns out to be problematic [24]. For D = 10 this gap was filled and gauge-invariant and duality-covariant expressions for all the D-branes of IIB string theory were given [9]. In this paper we extend this work to D < 10 dimensions.
A basic difference with D = 10 dimensions is that, to write down a WZ-term in D < 10 dimensions, we need to introduce, together with the standard worldvolume scalars describing the position of the brane in D dimensions, not only a worldvolume vector V 1 but also additional worldvolume scalars. This is due to the fact that the Fundamental string can wrap around each of the 10 − D compactified dimensions. Therefore, in D < 10 not only strings but also a number of particles can couple to the D-brane. The coupling of these particles are described by the extra worldvolume scalars. We find 2(10 − D) of such scalars transforming as a vector V 0,A under the T-duality group SO(10 − D, 10 − D), with corresponding curvatures F 1,A . In this paper we will show that, using these curvatures, the general WZ term can be written in the following elegant form: where Γ A are the SO(10 − D, 10 − D) gamma matrices and C is a formal sum of all RRpotentials. Note that F 1,A = 0 in D = 10 since there is no T-duality in that dimension. Therefore, the expression (1.14) reduces to the usual expression (1.1) for D = 10.
To show that (1.14) is the correct gauge-invariant and duality-covariant WZ term for any Dp-brane in D ≤ 10 dimensions, our strategy will be as follows. First, for the convenience of the reader, we will shortly review in Section 2 the WZ terms for the D-branes of D = 10 Type IIA and Type IIB string theory. Next, we will present in Section 3 the general gauge algebra of maximal supergravity in any dimension 3 ≤ D ≤ 10. The structure of this gauge algebra follows from the underlying E 11 algebra. We will first give a general analysis and, next, work out the formulae for each specific dimension. In Section 4 we will, starting from the duality-covariant A-basis presented in Section 3, derive the expressions for the RR potentials C that couple to the Dp-branes using the basic charge vectors (1.13). In particular, we will give expressions for the charge vectors of the higher-dimensional branes in terms of products of the basic charges (1.13). Like in Section 3, we will first give the general analysis and then give explicit expressions for different dimensions. Next, in Section 5 we will derive the main result of this paper, i.e. the gauge-invariant and duality-covariant WZ term (1.14). Finally, in Section 6 we will present our conclusions and indicate a few natural extensions of this work.

D-branes in Ten Dimensions
For the convenience of the reader we shortly summarise in this Section what is known about the WZ terms corresponding to the D-branes of Type IIA and Type IIB superstring theory. We first discuss the IIB case [9].

Type IIB
Our starting point is the set of SL(2, R)-covariant IIB n-form potentials given in (1.8). Note that the 4-form field A 4 in (1.8) is self-dual and that A 2 and A 6 are each other's dual. The triplet of 8-forms is dual to the 2 scalars (axion and dilaton) of IIB supergravity. The counting works (2 scalars are dual to 3 8-forms) since the 8-forms satisfy a single constraint [29,26,8]. We next introduce the basic charges q α ,q α as in (1.9), where q α andq α are such that ǫ αβq α q β = 0. This fixes our choice of the RR potential C 2 and NS-NS potential B 2 . We are using the normalisations of [8], and convert these results in form language in the usual way: Our first task is to select among all n-form fields of IIB supergravity the RR-potentials who have the defining property that they do not transform under the NS-NS gauge transformations or, equivalently, that they couple to D-branes whose tensions scale as 1/g s . To see how this works, consider the gauge transformation of the 4-form potential A 4 which transforms both under RR and NS-NS gauge transformations, with parametersq α Λ α and q α Λ α , respectively. One may verify that there is a unique combination of A 4 and A 2 A 2 terms that is invariant under the NS-NS gauge transformations. This combination defines the RR 4-form potential C 4 : Applying the same procedure to the higher-form potentials we obtain the following expres-sions for the RR-potentials in terms of the SL(2, R)-covariant A-fields: These RR n-form potentials occur as the representations 1 n/2−2 in the decomposition of the SL(2, R)-covariant A-fields under where the sub-index n/2−2 indicates the R + -charge w of the representation. The complete decomposition is given in Table 2. Each field can be associated with a p-brane (p = n − 1) if it exists, whose brane tension in string frame scales as 5 We distinguish between the following objects: Solitonic Object , α < −2 : Rest , which in the Table are given in the columns RR, F, S and Rest, respectively. We find the following expression of the charge of a Dp-brane in terms of the D1-brane and F1-brane charges: Indicating with n(q) and n(q) the number of q andq charges that occur in the expression (2.7) we have the following relations: Note that there are Dp-branes for each odd p but that there is only one Fundamental string F1. The 1 1 6-form is dual to the 1 −1 2-form and represents the duality between the D5-brane and D1-brane. The 1 −1 6-form is dual to the 1 1 2-form and represents the duality between the NS 5-brane NS5B and the Fundamental F1 string. field U repr RR F S Rest The other two 8-forms in the same triplet have charges: (2.10) We observe that the 1 2 D7-brane and its S-dual, the 1 −2 S7-brane, both have charges that are proportional to the product of two uncontracted SL(2, R) vectors (either q α orq α ) that are the same whereas the charge corresponding to the 1 0 8-form contains the product of two different charges. This means that under a U-duality the D7-brane and S7-brane can be transformed into each other but that one can never rotate one of these branes into a brane corresponding to the 1 0 8-form. In other words, the D7-brane and S7-brane belong to the same conjugacy class that forms a (non-linear) doublet embedded into the triplet. The fact that they belong to the same conjugacy class can also be deduced from the fact that, viewed as a 2 × 2 matrix, both charges have zero determinant, i.e. det [q α q β ] = det [q αqβ ] = 0. It is not clear whether the 1 0 8-form couples to a brane since under supersymmetry it transforms to the gravitino with an axion-dependent coefficient, thereby violating one of the assumptions that go into the general formula (2.6). What is clear is that for such an object one can not write down a gauge-invariant WZ term as in [9] because its charge is proportional to the product of two different uncontracted 1-brane charge vectors. We finally consider the 10-forms. We first consider the quadruplet of 10-forms. The 1 3 10-form potential couples to the D9-brane and has charge The other three objects in the quadruplet have charges Applying the same reasoning as in the case of the 7-branes, we conclude that the D9-brane is in the same conjugacy class as its S-dual, the 1 −3 brane since these are the only two objects whose charge is proportional to the product of three SL(2, R) vectors q α that are the same. Together, they form a non-linear doublet embedded into the quadruplet. It is not clear whether the other, 1 1 and 1 −1 , 10-forms couple to a 9-brane since they violate the assumptions underlying (2.6). Anyway, for these two quantities it is impossible to write a gauge-invariant WZ term as in [9]. We next consider the charges associated to the doublet of ten-forms. These are It turns out that using these expressions it is impossible to write down a corresponding gauge-invariant WZ term along the lines of [9].

Type IIA
Although IIA superstring theory has only an R + -duality symmetry the situation is similar to the IIB case. In the IIA case the chargesq (n) of the higher-dimensional Dp-branes can be written as products of the D0-brane chargeq and the Fundamental F1-string charge q as follows:q (m) =q q m/2−1/2 , m = p + 1 . (2.14) Note that, unlike in the IIB case, the basic charges correspond not only to 1-branes but also to 0-branes. The difference with the IIB case can be traced back to the fact that the IIA and IIB gauge algebras have different so-called fundamental symmetries. The fundamental symmetries are the basic gauge symmetries out of which all other symmetries can be generated by taking multiple commutators. It turns out that the fundamental symmetries of the IIB gauge algebra are the ones corresponding to the doublet of 2-form potentials, i.e. the RR and NS-NS 2-form potentials, whereas those of the IIA gauge algebra are given by the gauge symmetries corresponding to the RR 1-form and NS-NS 2-form potentials.
We have collected the different n-form potentials of IIA supergravity in Table 3. The corresponding brane tensions scale as , 6 where w is the weight under R + , n is the rank of the form and n(q) is the number ofq charges that occurs in the expression of the p-brane charge, with p = n − 1. In the second column we have indicated all RR n-form potentials with n odd. The 3d column contains the Fundamental 2-form that couples to the Fundamental string F1. The

Gauge Algebra in Any Dimension
In [19] the gauge algebra of all maximal ungauged and gauged supergravities in any dimension was derived from E 11 . Under E 11 the fields transform as where we denote with A the form fields and with a the corresponding constant parameters. In the ungauged case, which is the one on which we will focus from now on, these transformations are promoted to gauge transformations via the identification 7 a → dΛ , (3.2) and the resulting gauge algebra can be schematically written as Although the E 11 algebra is non abelian, one can make, in any dimension, suitable field redefinitions, as well as field-dependent redefinitions of the gauge parameters, such that the resulting gauge algebra is abelian, that means that the gauge transformations are gauge invariant. 8 This corresponds to writing the gauge transformations as where F 's denote the gauge-invariant field strengths. In this section we will derive the gauge algebra in this basis.
In D dimensions the global symmetry is E 11−D , and each n-form carries a representation of E 11−D that we denote with a lower M n index [19], see eq. (1.12). Using this notation the field strength of the 1-form is and it is invariant under the gauge transformation The field strength of the 2-form is where f M 1 N 1 M 2 is an invariant tensor of E 11−D and an upstairs index M n denotes the conjugate representation, which means that there is an invariant tensor δ Mn One also gets the Bianchi identities The part of the invariant tensor f M 1 N 1 M 2 which is antisymmetric in M 1 and N 1 can always be eliminated by means of a field redefinition We can therefore assume that f M 1 N 1 M 2 is symmetric in M 1 and N 1 . This condition naturally follows from the Jacobi identities of the E 11 algebra.
The field strength of the 3-form is One can always redefine suitably the fields in such a way that the E 11−D invariant tensors satisfy the constraints which we therefore assume. These can also be seen as coming from E 11 Jacobi identities. To summarise, the field strength is 12) and the gauge transformation of the 3-form is One also has the Bianchi identity For the 4-form one gets and with the constraints and the constraint that f M 2 N 2 M 4 is antisymmetric in M 2 and N 2 . This analysis can easily be extended to forms of higher rank. For the convenience of the reader we write the explicit form of the various invariant tensors in dimension D from 9 to 4, for each dimension separately. This reproduces the results of [19] in the abelian basis. The notations we use here are taken from that paper. The results are summarised in Table 4.

D=9
In nine dimensions the global symmetry is GL(2, R) = SL(2, R) × R + and the fields are where α = 1, 2 is an SL(2, R) doublet index and the sub-index indicates the R + -charge.
The only non-trivial invariant tensor is the epsilon symbol of SL(2, R), and from Table 4 one can see that the conditions (3.11) and (3.17) are satisfied.

D=7
In seven dimensions the symmetry is SL(5, R), while the representations of the fields up to the 5-form and the corresponding notations are (M = 1, . . . , 5) Given the invariant tensors of Table 4, one may verify that the conditions (3.11) and (3.17) are satisfied.

D=5
In five dimensions the global symmetry is E 6 . The representations of the fields up to the 5-form, and their corresponding notations, are (M = 1, . . . , 27; α = 1, . . . , 78) It should be noted that we are considering the split, or real form of E 6 , therefore the representations are real. With 27 we simply mean the representation that is dual to the 27. This will be the case in all dimensions. The conditions (3.11) and (3.17) are satisfied by the invariant tensors in Table 4.
The index A with which we denote the 912 representation should not be confused with the A vector index of the T-duality group in any dimension. The reducible representation to which the 4-forms belong is the antisymmetric product of two adjoint (i.e. 133) indices. Therefore the αβ indices of the 4-form are meant to be antisymmetrised. Note that in all dimensions the representation of the 4-form is the antisymmetrised product of two 2-form representations.
Given the invariant tensors listed in Table 4, eq. (3.11) becomes These two equations are both satisfied, see [19].

From the A-fields to the C-fields
In this section we wish to generalise the IIB construction of [9], thus determining in any dimension the RR C-fields starting from the duality-covariant A-basis of the previous section. Schematically, we expect the C fields to transform under the RR gauge transformations as where λ are the gauge parameters of the RR fields and H are the field strengths of the Fundamental fields, that is the fields corresponding to the branes whose tension does not depend on the dilaton in the string frame. In ten dimensions, H is a 3-form. By dimensional reduction, we expect H in D < 10 dimensions to be either a two-form or a three-form.
That is why we have not yet indicated the rank of the forms in (4.1). The precise form of this equation will be given below, see eq (4.14).
The T-duality subgroup of the U-duality group E 11−D in D dimensions is SO(10 − D, 10 − D). It is known that the RR fields transform in the spinorial representations of the T-duality group [20]. This is consistent with the fact that, upon reduction over a single dimension, each Dp-brane gives rise to two branes: a Dp-brane (dimensional reduction in the transverse direction) and a D(p − 1)-brane (dimensional reduction in the worldvolume direction). Therefore, the total number of D-branes doubles when going one down in the dimension D. This is precisely what happens with the dimension d spinor of a spinor representation of the T-duality group SO(10 − D, 10 − D) which is given by d spinor = 2 9−D . It turns out that, more precisely, the forms of odd rank transform as spinors of a given chirality, while the forms of even rank transform as spinors of the opposite chirality. As we will see in this section, this is completely general and applies to RR (D − 1)-and Dforms as well.
We now decompose in any dimension each duality-covariant form, belonging to a given E 11−D representation denoted by M n , in terms of representations of SO(10 − D, 10 − D). More precisely, we have in all cases D < 10. In D = 4 and D = 3 dimensions there are extra symmetry enhancements, such that in D = 4 the decomposition is and in D = 3 one has E 8 ⊃ SO (8,8) . From the point of view of this paper these symmetry enhancements are not practical since they combine the RR-fields together with n-forms that couple to other kind of branes in one multiplet. We will therfore consider the further decompositions for D = 4 and for D = 3. In the second part of this section we will discuss the above decompositions for each dimension separately. Here, we first anticipate the main outcome of this analysis. Using the fact that eq. (4.1) relates a RR n-form to the gauge parameter of a RR (n − 2)-form, it follows that considering the decomposition (4.2), the R + -charges satisfy the relations where we denote with w RR (n) the charge of the RR n-form and with w F (1) and w F (2) the charge of the Fundamental 1-forms and 2-forms. In Tables 6-11 the decomposition of all the forms with respect to T-duality is performed in any dimension 4 ≤ D ≤ 9. From an analysis of the R + -charges in the tables, it is straightforward to see that the condition (4.7) on the charges has only one solution, and, moreover, it is the same solution in any dimension. That solution is: • All RR-forms, including the (D − 1)-forms and the D-forms, belong to the spinor representations of the T-duality group. The RR-forms of odd rank transform as spinors of a given chirality, and the RR-forms of even rank transform as spinors of the opposite chirality.
• In the cases in which the representation of the U-duality group is reducible, as is always the case for D-forms, and also for (D − 1)-forms in dimension higher than six and for (D − 2)-forms in dimension higher than seven, the RR-forms are always inside the highest-dimensional representation.
• The Fundamental 1-forms always belong to the vector representation of the T-duality group. We denote these fields by B 1,A , with A a vector index of SO(10 − D, 10 − D). Half of these represent the wrapped Fundamental strings, while the other half correspond to reduced pp-waves which, in D = 10, are T-dual to the Fundamental string.
• The Fundamental 2-form always transforms as a singlet under T-duality. This is not surprising because this is the form associated to the Fundamental string.
In the four-dimensional case, in which the decomposition of eq. (4.3) occurs, the situation is more subtle due to the further decomposition (4.5) but the final result is exactly the same, as we will show in the last subsection of this section. The general result is summarised in Table 5. A straightforward way of reaching this conclusion for the lower rank forms is to consider explicitly the dimensional reduction of IIA and IIB supergravity. The same dimensional reduction also gives the right answer for the higher rank RR-forms that do not describe physical degrees of freedom. It turns out that all RR-forms in D < 10 dimensions arise from dimensional reduction of the IIA field RR F or IIB RR-forms. This is not the case for the forms that we have collected in the 'Rest' columns of the different Tables which have dilaton couplings different from the RR and Fundamental fields. Their higher-dimensional origin resides in the mixed representations predicted by E 11 . These 'Rest' fields are a minority in D = 10 dimensions but become a majority in lower dimensions. Besides Solitonic objects they might describe other exotic objects in string theory, with unconventional dilaton couplings, like 1/g 3 s , 1/g 4 s , etc. The precise meaning of these objects, if they exist et all, is not understood.
Given these general results, we now proceed with analysing how the C-basis is defined in terms of the duality-covariant A-basis. As we have already stressed and as Table 5 shows, the RR-fields transform in the fermionic representations of SO(10 − D, 10 − D) and we denote them by C 2n−1,a , C 2n,ȧ (4.8) denoting with a andȧ the 2 9−D -dimensional spinor representations of SO(10 − D, 10 − D). It is useful to list the conventions for the SO(10 − D, 10 − D) Gamma matrices that we are using. In particular, we are using a Weyl basis, so that the Gamma matrices have the form where a,ȧ = 1, ..., 2 9−D . They satisfy the Clifford algebra where η AB is the Minkowski metric. Because of R + -charge conservation, we expect the NS-NS field strengths of B 1,A and B 2 to be whose gauge invariance fixes the gauge transformations of the fields to be We write down the field strengths and gauge transformations of all the RR fields in the compact form and (4.14) Here we denote with C the sum of all the RR forms, in the spinor representation of SO(10 − D, 10 − D), where each odd form in the sum is projected on one chirality and each even form on the opposite chirality, and the opposite projection occurs for the gauge parameters λ. In components the two equations read and We now introduce the chargesq M 1 a and q M 1 A that project the 1-forms on the RR and Fundamental 1-forms respectively. That is Up to field redefinitions, the most general expression for the RR and Fundamental 2-forms is where the parameter a will now be determined by consistency. Indeed, varying both expressions according to eqs. (3.8), (4.12) and (4.14), and using the fact that f M 1 N 1 M 2 is symmetric in M 1 and N 1 , one obtains a = 1 2 . Furthermore, consistency implies the constraintsq and q These relations can be inverted using the invariant tensorf M 1 N 1 M 2 , where in general f MmNn P m+n is such that and thus they determine the charges of the 2-forms entirely in terms of the charges of the 1-forms:q To summarise, the expressions for the gauge fields and the gauge parameters are and We now move to the 3-forms. Before determining the actual expression for the RR 3-form in terms of the A-fields, we first analyse the corresponding charge. In principle, the charge q M 3 a can be written as either .

(4.26)
Substituting the relations (4.22) and using the conditioñ which is the conjugate of the second of the constraints of eq. (3.11), one can see that the two expressions for q M 3 a are actually proportional, and therefore one can writẽ (4.28) We are now ready to determine the relation between the RR 3-forms and the A-fields, which also determines the coefficient in (4.28). The procedure is completely general: we know the variation of the C field, that is eq. (4.14), and we known the transformation of the 3-form A field, that is eq. (3.13). We then compare the two expressions solving for C in terms of A. The final result is Proceeding this way one can determine the relation for all the RR C-fields in terms of the duality-covariant A-fields. These results are general and apply to any dimension. We give the expression for the charge of any D-brane in terms of the chargesq M 1 a and q M 1 A . For instance, for the 4-form one gets This expression is unique, in the sense that there is no other independent way of contracting oneq M 1 a with three q M 1 A 's to get an object with the right indices. The reason is, like in the discussion of the 3-forms above, the constraints that thef generalised structure constants satisfy for consistency of the gauge algebra. In this case the relevant constraint is the conjugate of eq. (3.17). One can show that this is true in all cases, and the general expression for the charge is for odd forms and (4.32) for even forms.
Clearly one can in general construct charges with n(q) = 1, which do not correspond to D-branes. It turns that that there is a simple formula for the dilaton scaling of the tension of the brane to which the n-forms couple. It is given in terms of the number n(q) of basic q charges that one uses in the above expressions. All tensions scale as g α s , α = −n(q) . This formula assumes that the corresponding n-form transforms under supersymmetry to the gravitino with a non-zero coefficient that only depends on the dilaton scalar. The structure for the charges we found above simplifies if we consider only those charges that are generated by 2-form basic charges. This corresponds to considering the gauge algebra of the even-form fields whose gauge transformations are generated by the 2-form gauge transformation, that is considering only the structure constants f MmNn P m+n where m and n are even. The 2-brane charges are in fact given in terms of the 1-form charges in (4.22), but now we consider them as basic charges and we indicate them schematically with Q in order to distinguish them from the original 1-form charges q. We will make use of this observation when we discuss the issue of conjugacy classes for the D = 8 case below. The simplification is that when one projects on this sector, in the original expression for the higher-form charges in terms of the 1-form basic charges q all 1-form basic charges pair up to 2-form basic charges Q ∼ qq. More specifically, assuming that the 2-form sector only contains RR and Fundamental 2-forms, which is true for D > 6, 9 the basic charges are given byQ These basic charges define the RR and Fundamental 2-forms as We next consider the gauge algebra restricted to this sector and, for simplicity, we consider only the gauge transformations of the 2-forms and 4-forms: The remarkable thing about this subsector is that in all cases, that is in any dimensions, the invariant tensor f M 2 N 2 M 4 does not put any constraint, that is the representation M 4 is in all cases precisely the antisymmetric product of two M 2 representations [19]. The consistency of the gauge transformations requires the C 4,ȧ RR-field to be expressed in terms of the U-duality-covariant A-fields as The difference with respect to the full case is that no invariant tensor of SO(10−D, 10−D) takes part in the expression (4.38). This is because the Fundamental 2-form charge is a singlet. The only reason why this is consistent is that neither doesf M 2 N 2 M 4 pose any constraint on the representations. We therefore arrive at the conclusion that there is a collaboration between the universal unconstrained structure of the gauge algebra when restricted to the forms generated by the even-form basic charges only, and the fact that there are fundamental strings in the theory that are singlets under T-duality. In other words, the gauge algebra knows about strings! Below we show how things work out for each dimension 4 ≤ D ≤ 9 separately, starting with the highest dimension.

D=9
The U-duality symmetry of maximal supergravity in nine dimensions is SL(2, R)×R + . One can consider the nine-dimensional theory (as well as any lower dimensional one) as coming from dimensional reduction of either the IIA or the IIB theory. It is instructive to review how the branes in the IIA theory can be seen from 11 dimensions, which is summarised as (4.39) Here we denote with ♯ the compact 11th coordinate, and the 6-form A 6 is the magnetic dual of the 3-form A 3 in eleven dimensions.
Reducing to nine dimensions results in the fields collecting in SL(2, R) multiplets, as it is obvious from the 11-dimensional or IIB origin of the theory. We analyse the fields from the IIA/11-dimensional viewpoint. 10 Denoting with 9 the compact 10th coordinate, the 1-forms, and the corresponding 0-branes, are

-form (and 3-brane) is
The fact that we have called F0 the brane associated to the field g µ 9 , which is a reduced pp-wave, is straightforward from considering the T-dual IIB picture. For all the other cases the assignments are straightforward from the reduction of eq. (4.39).
There are a doublet and a singlet of 1-forms, and what eq. (4.40) shows is that the RR-form always belongs to the doublet. Besides, no matter how we choose the RR-1form within the doublet, eq. (4.41) reveals that the RR-2-form must correspond to the same component. 11 A similar analysis can be performed for the higher rank branes. The decomposition of the 1-forms in terms of RR and F fields allows us to identify the q andq charges as This breaks SL(2, R) to the subgroup SO(1, 1), which is isomorphic to R + . We denote with w 1 the charge associated with this R + , while the charge associated to the original R + is denoted with w 2 . The actual T-duality group is a linear combination of these two R + -factors such that the weight w under the T-duality group is given by w = w 1 − w 2 . We summarise the decompositions of all the fields in Table 6. By analysing the   Table 6: The decomposition of the n-form potentials of D = 9 maximal supergravity. The U-duality is SL(2, R) × R + . We denote with (w 1 , w 2 ) the weights associated to R + × R + .
The weight under T-duality is w 1 − w 2 .
and for the 4-form A 4 is (0, 2) : ǫ αβq α q β q 2 . (4.47) The expression for the RR and Fundamental 2-forms is thus and similarly for the higher rank forms. Denoting with n(q), n(q) and n ′ (q) the number of times the chargesq α , q α and q respectively occur in the decomposition of a given charge, one has the relations n(q) − n(q) = w 1 n ′ (q) = w 2 . (4.49) Using this and the actual SL(2, R) representation to which each field belongs, the reader can identify the charges corresponding to the higher rank fields. For instance, the 9-form in the quadruplet is A 9,αβγ , and its (3,4) component is projected by (3,4) : (ǫ αβq α q β )q γ q δ q ǫ q 4 , (4.50) which is a RR field (n(q) = 1), while for the 9-form in the second doublet, A ′ 9,α , the component (−1, 2) is projected by which has n(q) = 4. Finally, the relation between the charges w 1 and w 2 of an n-form and the dilaton scaling of the tension of the corresponding (n − 1)-brane (if any) in the string frame is which is in agreement with (4.33) using (4.49) and n = n(q) + n(q) + n ′ (q) .
and we denote their fields strengths as (4.55) The Fundamental 2-form B 2 transforms as where the relative normalisation between the two terms in brackets has been chosen for convenience (one can always change it by a field redefinition of the form B 2 → B 2 + B 1 B ′ 1 ). The gauge invariant field strength is and in the next section we will be needing the Bianchi identity From Table 6 one obtains that the gauge transformations of the C fields of even rank are while the gauge transformations of the C fields of odd rank are Table 7: The decomposition of the n-form potentials of D = 8 maximal supergravity. The U-duality symmetry is SL(3, R) × SL(2, R) and the T-duality is SL(2, R) × SL(2, R), while the subscript denotes the R + -charge w.
Next, we consider the 1-branes. We have This forms a single triplet in the (3, 1) representation of the U-duality group: Finally, we consider the 2-branes. These are The two fields are SL(3, R) singlets, which form an SL(2, R) doublet A 3,α . By looking at equations (4.63), (4.65) and (4.66) one can see that there is an SL(2, R) inside SL(3, R) which leaves the D0-branes and the D2-branes invariant and transforms covariantly the D1-branes (in our choice this is the SL(2, R) that rotates the first two components of the triplet). It is also evident that the D0-branes and the D2-branes transform covariantly with respect to the other SL(2, R), while the D1-branes are invariant. The T-duality is thus SL(2, R) × SL(2, R). In Table 7 we give the decomposition of all the fields in terms of the T-duality group. We also give in the table the corresponding R + -charge as a subscript. The conventions for this charge in this case and in all the lower dimensional ones are taken from [31].
The table allows us to identify the q charges. In particular, from eq. (4.63) we see that the D0-branes are a doublet of the U-duality SL(2, R), and we thus write whereα denotes the doublet of the SL(2, R) inside SL(3, R). All other charges projecting the higher rank forms can be uniquely expressed as products of these basic charges following the general analysis at the beginning of this section. Denoting with n(q) and n(q) the number of times the chargesq M and q Ṁ α occur in the decomposition of a given field, the R + -charges of the n-form fields (n = n(q) + n(q)) are related to these numbers by w = −2n(q) + n(q) . (4.68) In case the form is associated to a brane, the dilaton scaling of the tension of a D-(n − 1)brane is given in terms w as It is instructive to identify all the q charges for the higher rank fields applying to the eight-dimensional case the general analysis at the beginning of this section. For the two forms A M 2 eq. (4.22) becomes so that the RR and Fundamental 2-forms are given by (4.71) For the 3-forms A 3,α eq. (4.28) gives One can actually determine the charge associated to all the fields in Table 7, not only the RR fields. For instance, for the 5-forms A M 5 α , one gets the charge projecting on the RR field, which is the (1, 2) 2 , which has n(q) = 1 and n(q) = 4, and the charge projecting on the (2, 2) −1 , which has n(q) = 2 and n(q) = 3. One can show that any other structure made of q's and q's with n(q) + n(q) = 5 projecting the 5-form vanishes identically.
As an example, we now wish to consider the non-trivial conjugacy classes to which the D5-branes belong. For this purpose, it is much easier to consider the truncation to the even form sector, that is all the even-form fields whose gauge transformations are generated by the gauge transformations of the 2-form. In that case the basic charges are given bỹ These basic charges define the RR and Fundamental 2-forms as (4.76) The charges of the even-form fields that survive the truncation can be expressed as products of these basic charges. In particular, for the 4-form A 4,M one gets while for the 6-form A 6,P Q one gets 12 The R + -charges are related to n(Q) and n(Q) as w = 2n(Q) − n(Q) . Similar to the D7-branes in Type IIB string theory, we may now ask which of the eight 5-branes can be reached by an SL(3, R) rotation of the two D5-branes, which correspond to the (2, 1) 3 fields. Consider the three independent three-vectors We observe that in the expression for the charge of the D-brane two of the three vectors are the same. This means that by an SL(3, R) rotation one can reach only those branes whose expressions for the charge also contains two vectors that are the same. This applies to the two (2, 1) −3 branes and to two of the three (3, 1) 0 branes. This implies that the D5-branes describe a non-linear sixplet embedded into the octoplet. The two remaining branes have charges such that the three Q's are all different. One may verify that for those branes one cannot write down a gauge-invariant WZ-term. The same conclusion can be reached considering the 0-brane charges q andq. To summarise, the q's andq's select covariantly an SL(2, R) inside SL(3, R), and rotating these charges under SL(3, R) corresponds to choosing a different embedding. All the components of a representation of SL(3, R) that can be reached this way form a conjugacy class. This applies to any dimension.  Table 8: The decomposition of the n-form potentials of D = 7 maximal supergravity. The U-duality group is SL(5, R) and the T-duality group is SL(4, R). We denote as a subscript the R + -charge (notation from [31]).
We now consider the seven-dimensional case. From eq. (4.39) one finds that the 1-form fields associated to the D0-branes are There is a manifest SL(4, R) symmetry associated to the torus, and using the corresponding epsilon symbol these fields can be written as (4.82) Using the SL(5, R) symmetry enhancement, and denoting with 6 ′ the extra index of SL(5, R), one can now use the corresponding epsilon symbol on the first vector to obtain The 2-forms that result from the reduction of (4.39) are and the last component can be rewritten as A µν6 ′ , which makes the 5 of SL(5, R). From (4.39) it follows that the first component A µν♯ corresponds to the Fundamental string, while the other components are associated to D1-branes. This is in agreement with eq. (4.22), which in this seven-dimensional case says that the 2-forms A M 2 are projected on the RR 2-forms in the 4 of SL(4, R) by and on the Fundamental 2-form by (4.87) The decomposition of all the fields under T-duality is given in Table 8. The relation between the R + -charge w and the q's is w = −3n(q) + 2n(q) . (4.88) From this one can determine all the R + -charges of the various T-duality representations given in Table 8. Table 9: The decomposition of the n-form potentials of D = 6 maximal supergravity. The U-duality is SO (5,5), while the T-duality is SO (4,4).

D=6
In six dimensions the U-duality group is SO (5,5) which is decomposed under the T-duality group SO (4,4). This is given in Table 9. We follow the group theory conventions of [31], and therefore the RR 2-forms belong to the 8 V of SO (4,4). This is not in contradiction with the general case, in which all RR fields are in the spinor representations of the T-duality group, because of triality of SO (4,4). As usual, we denote with w the R + -charge. For the cases in which the n-form can be associated to a brane, the corresponding tension scales in the string frame as Only one of the two singlets that arise in the decomposition of the 2-forms is a Fundamental string. The other singlet corresponds to a string scaling like g −2 s , which is the magnetic dual of the Fundamental string.

D=5
In five dimensions the 1-forms, which belong to the 27 of E 6 , decompose into 16 + 10 + 1 under T-duality. The decomposition of all the fields is given in Table 10, where the subscript denotes the R + -charge w. For the cases in which an (n − 1)-brane is associated to the n-form, the corresponding tension scales in the string frame as

D=4
We first consider the 1-forms, see Table 11. The RR 1-forms occur as a 32-dimensional spinor representation of the T-duality group SO(6, 6), as expected. There are 24 remaining 1-forms, half of them correspond to Fundamental 0-branes, the other half correspond to Solitonic 0-branes with dilaton coupling 1/g 2 s . To pick out the Fundamental fields one must therefore undo the symmetry enhancement by making the decomposition (4.5). Under this decomposition the 1-forms branch as where the sub-index indicates the weight w under R + . One next uses the rule that the brane tension corresponding to an n-form scales as This confirms that the 32 0 charges describe 32 D0-branes. We furthermore deduce that the 12 −1 charges describe 12 Fundamental 0-branes and that the 12 1 charges describe 12 Solitonic 0-branes. Like in D = 5 dimensions, the 1-forms are decomposed not only in RR and Fundamental fields but also in Solitonic fields D. We therefore introduce the three basic charges (q M a , q M A , q ′M A ) and decompose the 1-forms as follows: In terms of these charges the weight w of an n-form is given by We next consider the 2-forms A 2,α . Under the decomposition (4.5) the 2-forms in the RR column branch according to (32 ′ This shows that the 32 ′ −1 2-forms describe D1-branes but that the 32 ′ 1 2-forms describe exotic objects with 1/g 3 s dilaton coupling. Their charges are given by Similarly, the 2-forms in the F column decompose according to They describe a Fundamental string, a Solitonic string and an exotic object with 1/g 4 s coupling, respectively. Their charges are given by They correspond to Solitonic strings with charges given by Similarly, one can discuss the higher n-forms.

D-brane WZ Terms
In this section we will derive the main result of this paper, i.e. the expression (1.14) for the general D-brane WZ term in 3 ≤ D ≤ 10 dimensions. Our starting point is the U-duality covariant expressions for the RR C fields in terms of the covariant A fields, derived in the previous section. These RR fields have the important property that under gauge transformations they transform only into themselves, see eq. (4.14). They form the building blocks for our construction of a gauge-invariant WZ term. It is well-known that in ten dimensions it is not possible to construct a gauge-invariant WZ term using the RR fields C alone. The reason is that the C fields not only transform into a total derivative but also into a term containing the Fundamental curvature H 3  which has to be cancelled. Trying something of the form e X 2 C with dX 2 = H 3 would solve the problem. Taking X 2 = B 2 is not allowed since X 2 has to be gauge-invariant by itself. This is the reason that we need to introduce a BI vector such that B 2 can be interpreted as a term inside the gauge-invariant curvature of the BI vector. We therefore take X 2 = F 2 = dV 1 + B 2 . In D < 10 dimensions a similar reasoning works except that the C fields now not only transform to H 3 but also to H 2,A , the curvature of the Fundamental 1-forms B 1,A . This suggests that we need to introduce not only a BI vector V 1 but also 2(10 − D) worldvolume scalars V 0,A with corresponding gauge invariant field strengths given by This is gauge-invariant provided that the worldvolume scalars transforms as Furthermore, we need to adapt the definition of F 2 since B 2 also transforms under Σ 0,A , see eq. (4.12). The following expression is gauge-invariant provided that V 1 transforms under gauge transformations as It is instructive to compare the above with the expected number of bosonic worldvolume degrees of freedom. In general a Dp-brane has 8 bosonic worldvolume degrees of freedom. For instance, in ten dimensions a Dp-brane has (p − 1) d. We find that the WZ term contains twice as much extra scalars than expected, i.e. 2(10−D) instead of (10 − D). We will comment about this in the conclusions section. Given these ingredients, we can now write a compact expression for the gauge-invariant WZ term for any D-brane in any dimension. The result is given in (1.14) which we repeat here: Like in the previous section we denote with C the sum of all the RR potentials. The D = 9 case is a bit special in the sense that the combination F 1,A Γ A only contains the self-dual (anti-self-dual) part of F 1,A when projected on the even (odd) form sector. These 1-forms contain the (1, 0) and (0, 1) 1-forms in Table 6. It is also easier to explicitly write out the Γ A matrices for this case. We therefore treat this case separately below, see (5.15) for an expression for the WZ term in that case.
To proof that the WZ term (5.7) is gauge-invariant we need the Bianchi identities of the worldvolume curvatures: The proof of gauge-invariance is now remarkably simple. Given that the F 's are gauge invariant, we have The H 3 term cancels up to a total derivative precisely like in ten dimensions by using the second Bianchi identity of (5.8). We are now left with the following three terms, leaving out an overall e F 2 factor and the gauge parameter λ: 10) The first term arises from the fact that we applied the second Bianchi identity of (5.8) when cancelling the H 3 term. The second term arises from partially differentiating the exterior derivative in (5.9) when it hits the e F 1,A Γ A term. Finally, the third term is just the last term of (5.9). To show that the three terms given in (5.10) cancel amongst each other it is convenient to first expand the exponential in the first and second term and similarly expand the exponential in the second term. In the second term one next uses the first Bianchi identity of (5.8). Now all terms are linear in H 2 . After expanding the exponentials the first and second term of (5.10) are written as a sum of completely anti-symmetric Gamma matrices. The third term becomes the sum of products of an antisymmetric Gamma matrix with a single Γ B matrix. Working out this product leads to two types of terms containing a single anti-symmetric Gamma matrix. In the first type H 2 is contracted with one of the indices of the anti-symmetric Gamma matrix. Such terms cancel against the second term of (5.10). In the second type H 2 is contracted with one of the F 1 curvatures. These terms cancel against the first term of (5.10). This completes the proof of gauge invariance of the WZ term (5.7). The form of the WZ term (5.7) clearly suggests that, like in D = 10, the RR scalars C 0,ȧ can be included too in the expression although they are not needed for gauge-invariance. Like in IIB supergravity all RR scalars are axionic, i.e. the D-dimensional maximal supergravity theory is invariant under constant shifts of these scalars.
Given that in nine dimensions the T-duality group is abelian, we consider this case explicitly although it does not differ from the general analysis. We introduce the worldvolume scalars V 0 and V ′ 0 , such that are gauge invariant. From (4.56) one also defines so that the Bianchi identities are From the gauge transformations (4.59) and (4.60) one can then write a gauge invariant WZ term as the formal expression L WZ (D = 9) = e F 2 e F 1 +F ′ 1 C . (5.15) It is understood here that when we expand e F 1 +F ′ 1 we only take F 1 acting on even forms and F ′ 1 acting on odd forms. That is e F 1 +F ′ 1 C = (1 + F 1 + 1 2 F ′ 1 F 1 )C even + (1 + F ′ 1 + 1 2 F 1 F ′ 1 )C odd . (5.16) Using this notation the proof of gauge-invariance is straightforward and will not be repeated here. The alternating occurrence of F 1 and F ′ 1 is easily explained if one considers the IIA and IIB origin of the WZ term. Indeed, an even form in nine dimensions is unwrapped from the IIB perspective and wrapped from the IIA perspective, and vice-versa for an odd form. Any time there is a wrapped coordinate, this corresponds to a world volume scalar appearing in the WZ term. In particular the scalar V 0 can be seen as arising from a wrapped world-volume vector V 1 in IIB, while the scalar V ′ 0 arises as a wrapped V 1 in IIA.

Conclusions
In this paper we have constructed gauge-invariant and U-duality covariant expressions for Wess-Zumino terms corresponding to general Dp-branes (0 ≤ p ≤ D − 1) in arbitrary 3 ≤ D ≤ 10 dimensions. We did this in two steps. First, we considered the target space background fields. In particular, we constructed expressions for the RR potentials in terms of the U-duality covariant fields. For this we introduced two types of charge vectors that project the 1-forms onto the RR and Fundamental 1-forms. 13 We showed how, for D < 10, the charges of all Dp-branes with p ≥ 1 could be expressed as products of these basic charges and we derived general expressions for these higher-dimensional charges. The cases D = 3 and D = 4 required special attention due to symmetry enhancements that take place in these dimensions. Since the extra symmetries put the D-branes together with other objects into the same multiplet it is natural, for the purposes of this paper, to undo theses symmetry enhancements. We discussed the D = 4 case in quite some detail. We refrained from giving the formulae for the D = 3 case as well but we expect that it follows the same pattern we found in higher dimensions. In a second step we considered the worldvolume fields needed for the construction of the WZ term. Since, for D < 10, the WZ term for general D-branes contains both even-form and odd-form potentials it is clear that one needs also even-form and odd-form worldvolume curvatures. We therefore introduced, besides the usual worldvolume 2-form curvature for the BI vector, additional worldvolume 1-form curvatures for the extra worldvolume scalars that correspond to the compactified dimensions. These scalars are on top of the usual embedding scalars. Using the expressions for the RR potentials and the worldvolume curvatures it was then a relatively straightforward task to construct a gauge-invariant and duality-covariant expression for the WZ term in D < 10 dimensions.
The fact that for D < 10 the charges of the higher-dimensional branes can be expressed in terms of products of the D0-brane and Fundamental 0-brane charges has important consequences for the non-trivial conjugacy classes to which these D-branes may belong. As an example, we analysed in detail the case of D5-branes in D = 8 dimensions. To simplify matters we did this for the truncated case of even forms only. Making use of the observation that the charge of the standard D5-brane contains the product of two charge vectors that are the same we showed that under a general U-duality transformation they form a non-linear six-plet embedded into an octo-plet of 5-branes. This is similar to the D7-branes in IIB string theory which form a non-linear doublet inside a triplet of 7-branes. Other examples of non-trivial conjugacy classes may be discussed similarly.
A noteworthy feature of the general WZ term is that it contains twice as many extra scalars as compactified dimensions. One set of scalars has a natural IIA origin, the other set has a natural IIB origin. Together they transform as a vector of the T-duality group SO(10 − D, 10 − D). This doubling of compactified dimensions is typical for doubled geometry [32,33,34,35] but now applied to the worldvolume of the D-branes in a curved background. In the same way that in dimensions lower than ten the Wess-Zumino term of the D = 10 fundamental string, L F1 = B 2 , gets modified by F 1,A to [32] L F1 (D < 10) = B 2 + η AB F 1,A B 1,B , (6.1) which is invariant under the NS-NS gauge transformations (4.12), we have derived that in D < 10 dimensions the WZ term of the D = 10 D-branes, L Dp = e F 2 C gets modified by F 1,A to the expression (5.7). In the case of the fundamental string a correct counting of the worldvolumne scalars is re-obtained by imposing a self-duality condition on the scalars [32]. The challenge will be to see what kind of condition on the worldvolume scalars must be imposed, in the case of D-branes, to obtain a correct counting of the worldvolume degrees of freedom. Given that such a condition can be imposed for the fundamental string, we expect this to be a solvable problem [41]. There are two natural extensions of our work. One extension is to consider the coupling of D-branes to maximal gauged supergravities. This requires the introduction of the socalled embedding tensor [36,37,38]. Like in the case of the coupling of the D2-brane to massive IIA supergravity, we expect this embedding tensor to occur as the coefficient of a worldvolume Chern-Simons term [5]. More information about these extra Chern-Simons terms can be obtained by considering generalized Scherk-Schwarz reductions of the axionic RR scalars since these reductions lead to Chern-Simons terms. We plan to come back to the relation between the D-brane WZ terms and the embedding tensor in the nearby future [39].
A second extension is to construct a kappa-symmetric version of the D-brane actions. This requires of course first to construct a U-duality covariant kinetic term for all worldvolume fields. This has already been done for the D = 10 D-branes, see [9], and a kappasymmetric extension of the D = 10 D-brane actions has been constructed [40].
Finally, we observe that lower-dimensional supergravities allow for many potentials that correspond to exotic branes with unconventional dilaton couplings 1/g n s (n = 3, 4, 5, · · · ). It is not clear what the status of all these exotic branes is in string theory, and if they exist et all, but if they do exist they might help in finding a geometrical description of the individual degrees of freedom of black holes and thereby explaining the entropy of these black holes [42].